Advances in Business and Management Forecasting, Volume 4 (Advances in Business and Management Forecasting) (Advances in Business and Management Forecasting)

ADVANCES IN BUSINESS AND MANAGEMENT FORECASTING

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ADVANCES IN BUSINESS AND MANAGEMENT FORECASTING Series Editors: K.D. Lawrence, M.D. Geurts and J.G. Guerard Jr. Volume 1: Volume 2: Volume 3:

Advances in Forecasting: Advances in Forecasting:

Business and Management Forecasting Sales Business and Management Forecasting

Advances in Business and Management Forecasting

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ADVANCES IN BUSINESS AND MANAGEMENT FORECASTING VOLUME 4

ADVANCES IN BUSINESS AND MANAGEMENT FORECASTING EDITED BY

KENNETH D. LAWRENCE School of Management, New Jersey Institute of Technology, USA

MICHAEL D. GEURTS Marriott School of Management, Brigham Young University, USA

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo JAI Press is an imprint of Elsevier

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JAI Press is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2006 Copyright r 2006 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-7623-1281-8 ISBN-10: 0-7623-1281-5 ISSN: 1477-4070 (Series) For information on all JAI Press publications visit our website at books.elsevier.com Printed and bound in the Netherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1

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CONTENTS LIST OF CONTRIBUTORS

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EDITORIAL BOARD

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PART A: FORECASTING AND THE SUPPLY CHAIN FORECASTING IN SUPPLY CHAIN MANAGEMENT Kenneth D. Lawrence, Sheila M. Lawrence and Ronald K. Klimberg

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EXTRACTING FORECASTS FROM ADVANCE ORDERS Frenck Waage

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INVENTORY-SHIPMENT RATIO TIME SERIES MODELS FOR DURABLE AND NON-DURABLE PRODUCTS Supriya Mitra

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PART B: FORECASTING AND FINANCIAL APPLICATIONS AN APPLICATION OF CONFIRMATORY FACTOR ANALYSIS TO THE A PRIORI CLASSIFICATION OF FINANCIAL RATIOS Shaw K. Chen and Alan D. Olinsky

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BANK RATING CHANGE PREDICTIONS: ALTERNATIVE FORECASTING MODELS David T. Cadden and Vincent Driscoll

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CONTENTS

FORECASTING SECURITY RETURNS: THE USE OF HETEROGENEOUS EXPECTATIONS Rebecca Abraham and Charles W. Harrington

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PART C: SALES FORECASTING COMBINING MOVING AVERAGES WITH EXPONENTIAL SMOOTHING TO PRODUCE MORE STABLE SALES FORECASTS Tej S. Dhakar, Charles P. Schmidt and David M. Miller

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IMPROVED EXPONENTIAL SMOOTHING WITH APPLICATIONS TO SALES FORECASTING Tej S. Dhakar, Charles P. Schmidt and David M. Miller

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USING FLOW-THROUGH AND DIFFUSION MODELS TO FORECAST NEW PRODUCT SALES Michael D. Geurts and David B. Whitlark

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AN APPLICATION OF A REPEAT PURCHASE DIFFUSION MODEL TO THE PHARMACEUTICAL INDUSTRY Franklin J. Carter, Carol M. Motley, Alphonso O. Ogbuehi and Jacqueline A. Williams FORECASTING PRODUCT SALES WITH CONJOINT ANALYSIS DATA David B. Whitlark IMPROVING SALES FORECASTS BY TESTING UNDERLYING HYPOTHESES ABOUT CONSUMER BEHAVIOR: A PROPOSED QUALITATIVE METHOD Eric D. DeRosia, Glenn L. Christensen and David B. Whitlark

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PART D: FORECASTING METHODS AND ANALYSIS FORECASTING SALES OF COMPARABLE UNITS WITH DATA ENVELOPMENT ANALYSIS (DEA) Ronald K. Klimberg, Shelia M. Lawrence and Kenneth D. Lawrence

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DATA MINING RELIABILITY: MODEL-BUILDING WITH MARS AND NEURAL NETWORKS Rod J. Lievano and Eric S. Kyper

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SELECTING FORECASTING INTERVALS TO INCREASE USEFULNESS AND ACCURACY Michael D. Geurts

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FORECASTING SIMULTANEOUS BRAND LIFE CYCLE TRAJECTORIES Frenck Waage

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A TYPOLOGY OF PSYCHOLOGICAL BIASES IN FORECASTING ANALYSIS Paul Dishman

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A FORECAST COMBINATION METHODOLOGY FOR DEMAND FORECASTING J.Gaylord May and Joanne M. Sulek

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LIST OF CONTRIBUTORS R. Abraham

Huizenga School of Business, Nova Southeastern University, FL, USA

D.T. Cadden

Quinnipiac University, CT, USA

F.J. Carter

College of Business & Economics, Lehigh University, PA, USA

S.K. Chen

College of Business Administration, University of Rhode Island, RI, USA

G.L. Christensen

Marriott School of Management, Brigham Young University, UT, USA

T.S. Dhakar

Department of Quantitative Studies and Operations Management, School of Business, Southern New Hampshire University, NH, USA

E.D. DeRosia

Marriott School of Management, Brigham Young University, UT, USA

P. Dishman

Department of Business Management, Marriott School of Business, Brigham Young University, UT, USA

V. Driscoll

Quinnipiac University, CT, USA

M.D. Geurts

Marriott School of Management, Brigham Young University, UT, USA

C.W. Harrington

Huizenga School of Business, Nova Southeastern University, FL, USA

R.K. Klimberg

Haub School of Business, Saint Joseph’s University, PA, USA

E.S. Kyper

Department of MIS, College of Business Administration, University of Rhode Island, RI, USA ix

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LIST OF CONTRIBUTORS

K.D. Lawrence

School of Management, New Jersey Institute of Technology, NJ, USA

S.M. Lawrence

Department of Management Science and Information Systems, Rutgers University, NJ, USA

R.J. Lievano

FMIS Department, Labovitz School of Business and Economics, University of Minnesota – Duluth, MN, USA

J.G. May

Department of Mathematics, Wake Forest University, NC, USA

D.M. Miller

Department of Information Systems, Statistics, and Management Science, Culverhouse College of Commerce, University of Alabama, AL, USA

S. Mitra

Whitman School of Management, Syracuse University, NY, USA

C.M. Motley

Howard University, WA, USA

A.O. Ogbuehi

Bryant College, RI, USA

A.D. Olinsky

Bryant University, RI, USA

C.P. Schmidt

Department of Information Systems, Statistics, and Management Science, Culverhouse College of Commerce, University of Alabama, AL, USA

J.M. Sulek

School of Business and Economics, North Carolina A&T State University, NC, USA

F. Waage

Department of Management Science and Information Systems, University of Massachusetts – Boston, MA, USA

D.B. Whitlark

Marriott School of Management, Brigham Young University, UT, USA

J.A. Williams

North Carolina A&T State University, NC, USA

EDITORIAL BOARD Editors-in-Chief Kenneth D. Lawrence

Michael Geurts Brigham Young University

New Jersey Institute of Technology

Senior Editors Kenneth Cogger University of Kansas

Sheila M. Lawrence Rutgers University

Lewis Coopersmith Rider College

Essam Mahmoud American Graduate School of International Business

John Guerard Chatham, New Jersey

Daniel O’Leary University of Southern California

Douglas Jones Rutgers University

Ramesh Sharda Oklahoma State University

Ronald Klimberg Saint Joseph’s University

William Steward College of William and Mary

Stephan Kudbya New Jersey Institute of Technology

David Whitlark Brigham Young University

STATEMENT OF PURPOSE Advances in Business and Management Forecasting is a blind refereed serial publication published on an annual basis. The objective of this research annual is to present state-of-the-art studies in the application of forecasting methodologies to areas such as sales, marketing, and strategic decision xi

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making (an accurate, robust forecast is critical to effective decision making). It is the hope and direction of the research annual to become an applicationand practitioner-oriented publication. The topics will normally include sales and marketing, forecasting, new product forecasting, judgmentally based forecasting, the application of surveys to forecasting, forecasting for strategic business decisions, improvements in forecasting accuracy, and sales response models. It is both the hope and direction of the editorial board to stimulate the interest of the practitioners of forecasting to methods and techniques that are relevant. Editorial correspondence should be sent to: Professor Kenneth D. Lawrence School of Management New Jersey Institute of Technology Newark, NJ 07102, USA

PART A: FORECASTING AND THE SUPPLY CHAIN

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FORECASTING IN SUPPLY CHAIN MANAGEMENT Kenneth D. Lawrence, Sheila M. Lawrence and Ronald K. Klimberg ABSTRACT This paper gives an overview of the forecasting process in supply chain management. It discusses forecasting in demand chains and its relationship to the customers and their needs. Finally, forecasting at the stockkeeping unit (SKU) level is examined.

1. AN INTRODUCTION TO SUPPLY CHAIN MANAGEMENT Supply chain management is the combination of science and art that companies use to improve the way they find raw materials that are needed to make a product or service, manufacture that product or service and deliver it to customers. A supply chain is a complex network of stages including customers, retailers, wholesalers/distributors, manufacturers, and raw material suppliers. It exists for two main reasons: to satisfy customers and to generate profits for itself. There are five core stages in supply chain management: 1. Plan. This is the strategy that is used for managing the supply chain network so that all stages work toward the common goal of meeting Advances in Business and Management Forecasting, Volume 4, 3–12 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04001-8

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customer demand for an organization’s product or service. In this stage, it is crucial to have in place a set of metrics to monitor the performance of the supply chain so that the network is able to keep costs down and deliver value to customers. Source. This has to do with choosing the suppliers that will deliver the raw materials your organization needs to produce products or services. After pricing, delivery and payment processes with material suppliers are created, it is important to put together processes for managing the inventory received. Verifying shipments, transferring shipments to manufacturing facilities, and authorizing material supplier payments are also included in this component. Make. This is the manufacturing component; it encompasses the activities necessary for production, testing, packaging, and preparation for delivery. Metrics are extremely important in this phase of the supply chain as it is necessary to measure quality levels, production output, and worker productivity. Deliver. Also known as logistics, this encompasses the coordination of the receipt of orders from customers, the set-up of a network of warehouses, distributors to get products to customers, and invoice systems so that payments may be received. Return. This includes supporting customers who have problems with delivered products as well as creating a network for receiving defective and excess products back from customers.

Traditionally, supply chains started with the manufacturing of a product or service and ended with the sale to consumers. In this model, retailers sell the products or services that manufacturers make and wholesalers supply. Supply chain managers focus on the products and services that flow through the supply chain. Unfortunately, this traditional supply chain model has a major shortcoming: Manufacturers’ strengths, resources, and best instincts are driving the products that are created instead of the products being derived from customer preferences and needs. Because of this, it is a common occurrence that products are not purchased by consumers at high enough rates to be profitable for supply chain participants. Given this phenomenon, many supply chain managers are beginning to shift the way they think about supply chain management. The latest trend in supply chain management is that supply chains are being turned into demand chains. Supply chain management deals with the buy-side of an organization, while demand-based management tackles the sell-side of the company. Addressing supply and demand together is the only

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way to get a macro view of a supply chain, and have all of the information needed for successful supply chain decisions to be made. If an organization addresses supply and demand independently of one another, less than adequate supply chain performance will result.

2. DEMAND CHAINS Demand chains have a greater focus on the customer and their needs and wants than traditional supply chains do. Demand chains include all of the same parts and functions (suppliers, manufacturers, warehouses, etc.) as supply chains; however, rather than building and operating the supply chain from manufacturer to market, demand chains focus on creating alliances with channel partners best able to fulfill consumers’ needs and wants, as well as changing supply chain planning and execution to be driven by demand in real time. One of the goals of demand chains is to increase the efficiency of the supply chain so that products can be as close to being built to order as possible (Browne, 1994; Van Landeghem, 2002). Companies that are investing in demand chain efforts are realizing strong improvements in inventory, customer service, operational costs, and response times. When trying to create a successful demand chain, it is obviously important to understand demand. In order to understand demand, it is imperative to know exactly who are your customers, what products and services are important to them, how they purchase these products and services, how frequently purchases are made, and what constraints they place on the purchasing process. This knowledge will enable an organization to create a demand chain to replenish products that have the proper inputs from suppliers, and reliably and efficiently delivers the demanded products and services to customers. Ideally, this will ultimately lead to the goal of increased customer satisfaction. This focus on the consumer has shifted the spotlight from supply to demand, causing supply chain members to focus less on their strengths and traditional roles, and focus more on the needs of the entire supply/demand chain. Because of their closeness to consumers, retailers traditionally were responsible for supervising consumer research and consumer preferences to guide collaborative product and supply chain design with the manufacturer. The Internet and direct sales have changed this. Consumers now have interaction with different supply chain partners at various points along the supply chain. Because of this, it is important that customer research and information be shared with every supply chain participant so that everyone is

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able to recognize and communicate opportunities for product or packaging improvements, brand extensions, new products, or marketing opportunities.

3. FORECASTING IN SUPPLY CHAIN All organizations operate in an atmosphere of uncertainty, and despite this fact, decisions must be made that affect the future of the organization. Everyone knows that educated guesses are more valuable than uneducated guesses, and it is for this reason that companies use forecasting as a tool to aid in making supply chain business decisions (Stock, Greis, & Kasarda, 2000; Tanwari & Betts, 1999; Tyagi, 2002). Forecasting is a tool that is used to predict the uncertain future in an effort to help supply chain managers make better decisions. Demand forecasting is the most common form of forecasting used in supply chain management. Demand forecasting is the process that supply chain managers and participants use to determine what products and services are needed, where, when, and in what quantities. It plays a decisive role in an organization’s planning, budgeting and performance monitoring process and relates to virtually all functions in an organization. Accurate forecast data can drive financial planning and production, distribution and logistics management, and parts and service management (Gilbert & Ballou, 1999; Heikkila, 2002). In order to determine whether or not what the forecast predicted actually took place, one must know what actual demand is, compare the actual demand data to the demand forecast data, and determine if and what the differences are between the two. If there are large differences between actual and forecast data, it is important to analyze the assumptions that went into the forecast and determine why they were not correct. After this analysis is completed, it is also important to apply the resulting new assumptions to the current statistical forecast so that the process is improved. The newly incorporated assumptions can be more optimistic or pessimistic and even result in new or different actions being incorporated into the forecast. A common mistake that leads to inaccurate demand forecasting is that many organizations rely on one forecasting approach for all stock-keeping units (SKUs). Because demand patterns can vary drastically based on factors such as product life cycle, relying on the same forecasting model for all SKUs is risky. While standard techniques work well for steady selling SKUs, alternative forecasting approaches must be incorporated for products such as fad-sensitive products. It is also important to point out that before data are used to forecast that outliers are removed.

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While forecasting can require a lot of time and resources, it is recommended that the forecasting process be performed at least monthly. This enables organizations to better measure actual performance so that if there are performance issues, corrective action may be taken in a reasonable amount of time. However, many larger companies have hundreds or even thousands of SKUs that are subject to independent demand.

4. FORECASTING AT THE SKU LEVEL Individual SKUs and customers each have different characteristics and behaviors, and their responses to marketing tools varies. For example, similar stores in different locations can have very different consumers. Because of this, it is recommended that models be created at the level where consumer decisions are made. In other words, forecasts should be modeled off of the demand for a SKU at an individual site. For a retailer, this means modeling the demand at the store level; for a manufacturer it means modeling at the factory or distribution center level. (Fliedner & Lawrence, 1995; Zotteri & Verganti, 2001). Forecasting every single individual SKU would require tremendous resources and effort. Because of this, it is necessary for companies to incorporate a process for selecting which SKUs should be forecasted as well as ensuring that only meaningful data are forecasted. One way for an organization to determine which SKUs should be forecasted is to understand which SKUs have the greatest effect on results, and then forecasting only those SKUs. For example, if 8% of an organization’s SKUs account for slightly more than 30% of company sales and another 12% account for 25% of sales, then forecasting these 20% of all SKUs will result in a meaningful forecast of more than 50% of total company sales. Another option that reduces the number of SKUs that need to be forecasted is by forecasting only for top accounts. Similar to the dynamic we see when forecasting the group of best selling SKUs, each company also has a group of largest accounts which represent the majority of total sales volume for the organization. Focusing on only these largest accounts is another way for companies to increase forecasting efficiency. Unilever is a good example of an organization that is currently incorporating this strategy into its forecasting and demand planning system (Fliedner & Lawrence, 1995; Zotteri & Verganti, 2001). Organizations can ensure that they forecast meaningful data by approaching forecasting in a disciplined manner. It is important to point out that if a

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forecast takes too long to produce, it becomes useless as a predictive and tracking tool. By creating a description of the forecasting process that is very detailed and describes data extraction, timing, note taking, analysis, and how to apply changes, it is easier for organizations to implement the forecasting process. It is also necessary to develop specific roles and responsibilities and assign responsibility for each step in the forecasting process. Many companies incorporate these roles and responsibilities into position descriptions of the responsible functions or parties. It is important to point out that without well-developed, proper forecasting support processes in place, it will be very difficult to maintain the discipline of preparing reliable monthly forecasts, with reproducible analysis and assumption options. Organizations must identify the functions that are responsible for specific drivers. It is important that all functions partake in driver identification, as forecasting is not just a sales or marketing function. Without input from all functions, a forecast stands little chance of being accurate. Accurate forecasting can be achieved only when different business functions work collaboratively and integrate their different forecasting systems. With the development of more sophisticated forecasting techniques, along with the advent of computers and sophisticated software, forecasting is receiving more attention. Every manager now has the ability to utilize very sophisticated data analysis techniques for forecasting purposes, and an understanding of these techniques is essential for supply chain managers. For this same reason, forecast users must be aware of the improper use of forecasting techniques because inaccurate forecasts can lead to poor decisions. It is rare that the predictions of future outcomes are precisely on the mark. Therefore, it is the challenge of supply chain managers to make the inevitable errors as small as possible. Many other factors contribute to inaccurate forecasts. Internal politics, personal agendas, financial performance requirements, and lack of collaboration are just a few. Forecasting should be a collaborative effort which uses many resources including sales people, financial analysts, production and inventory planners, marketers, strategic planners, executives, and in most cases full-time forecasters. IT resources are also necessary to capture historical data, and IT managers must maintain the systems. Each member of the supply chain needs demand projections in the unit of measure that applies most to their needs. A ‘‘unit’’ changes the further you travel up the demand chain. A unit for a customer can be a bag of potato chips, a case may be the unit used for retail stores, and a truckload may be the unit used for warehouses. Inventories become more expensive to maintain as a product travels down the chain toward the consumer. Therefore,

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participants must be careful in interpreting forecasted data and make sure they understand to what type of ‘‘unit’’ is being forecasted.

5. INVENTORY DRIVERS Inventory is a useful resource to the firm. However, firms do not want to hold on to more inventory that is necessary. Inventory costs the firm in terms of both space and capital. Additional inventory posses a risk of obsolescence, particularly in supply chains with short-product life cycles. The following gives a picture of various inventory drivers and their impact on the firm. Inventory Driver The level of uncertainty in the firm’s supply and demand The mismatch between downstream demand levels and the upstream production capacity in the supply chain The mismatch between the partner’s demand the most efficient production of shipment volumes for upstream partners in the supply chain

Impact Safety stock; hedge inventory Smoothing inventory

Anticipation of inventories and transportation of inventories

The firm faces a large degree of uncertainty in managing inventory through its supply chain. In the upstream end of its supply chain, it faces supply uncertainty in terms of the interruption in the flow of components that they need for their operations. In the uncertainty in the supply of the firm, the planners need to concern themselves with the following items: A. B. C. D.

Consistency of quality items purchased. Reliability of the suppliers’ delivery estimates. Are the items subject to expected price increases? Are the items subject to shortage?

These problem areas can increase supply uncertainty, thus causing the firm to hold onto safety stock or hedging their inventories. On the downstream side, the firm faces uncertainty in terms of the risk of unpredictable fluctuations in the demand for the firm’s product. In such

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conditions, suppliers are forced to hold extra safety stock to meet unexpected fluctuations in demand or changes in the order size. In the process of dealing with the firm’s uncertainty in supply or demand there is a need to assess what form of uncertainty can be minimized. If the level of quality is a major component of the supply uncertainty, it can be minimized through a program of quality improvement. Changing forecasting methodology may help to minimize demand uncertainty; it can never eliminate it (Gardner, Anderson-Fletcher, & Wicks, 2001; Helmes, Ettkin, & Chapman, 2000; Lee & Adam, 1986; Newbury & Bhame, 1981; Ritzman & King, 1993). Another important inventory driver is the mismatch between demand and the most efficient production of logistics volumes. In order to effectively management this problem, the firm can alter its business process to reduce production or shipment mismatch. Additionally, if there is no match between overall demand level and production capacity, then the firm will be forced to holding smoothing inventories. The firm can minimize the level of smoothing inventories by varying its capacity to better match demand or by smoothing demand to better match capacity. The final inventory driver is a mismatch between the timing of the customer’s demand and the supply chain lead time. If the customer’s waiting time is less than the supply chain’s lead time, then the firm needs to have transportation and anticipation inventories to ensure product availability for a customer in a timely fashion.

6. THE BULLWHIP EFFECT The bullwhip effect is defined as an extreme change in the supply position upstream in a supply chain generated by a small change in demand downstream on the supply chain. What is the cause of such a problem? Basically, if a distributor reaches a reorder point, it places a large order. Otherwise, it does nothing. Therefore, a single unit change in demand may determine whether or not a distributor places an order. Thus, even though the distributor is following a sound inventory policy, the impact in the supply chain is to increase demand variability at a factory. This demand variability will drive up cost at the factory. Furthermore, the demand variability will drive up costs, which will then be forced to pass at least some of these costs on the distribution. These extreme changes in demand make the forecast process much more complex. Standard forecasting methods do not work under such radical changes in demand (Lee & Adam, 1986).

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In order to minimize the bullwhip effect, many supply chain partners work together to reduce order quantities removing volume discounts and reducing ordering costs. A particularly important question is to decide where to hold the inventory in the supply chain. Basically, the cost and the value of inventory increase as material moves down the supply chain. Additionally, the flexibility of inventory decreases as materials move down the supply chain. As materials work their way through the supply chain, they are transformed and moved close to their customers. All these activities add both cost and value. The value added goes well beyond transformation and packaging; it includes location. A product that is in stock and available immediately is always worth more to the customer than the same produce’s being available late. Firms are kept from pushing inventory as far down the supply chain as possible by cost considerations. By delaying the transformation and movement of materials, firms postpone their costs. Additionally, inventory levels are held back to maximize flexibility. Once material has been transformed and packaged, the supply chain becomes more complex. In similar fashion, for transportation, moving items from one location to another becomes very expensive when compared to delaying the movement of items until the certainty level of demand becomes higher. This loss of flexibility is a major reason why materials are often held back in the supply chain. In short, supply chain managers are always trying to strike a balance between costs and flexibility.

7. CONCLUSION This paper gives a broad overview of the process of supply chain management forecasting. The process of integrating customer information into planning and control is the basis of the forecasting process. Moreover, the forecasting process deals with estimating customer demand and then converting this demand into specific orders, with delivery, as well as helping balance demand with supply. Thus, the forecasting process matches the customers’ needs with the firm’s capabilities, including the physical distribution from the firm to the customer.

REFERENCES Browne, J. (1994). Analyzing the dynamics of supply and demand for goods and services. Industrial Engineering, 26(6), 18.

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Fliedner, E. B., & Lawrence, B. (1995). Forecasting system parent group formulation: An empirical application of cluster analysis. Journal of Operations Management, 12(2), 119–130. Gardner, E. S., Anderson-Fletcher, E. A., & Wicks, A. M. (2001). Further results on focus forecasting and exponential smoothing. International Journal of Forecasting, 17(2), 287–293. Gilbert, S. M., & Ballou, R. H. (1999). Supply chain benefits from advanced customer commitments. Journal of Operations Management, 18(1), 61–73. Heikkila, J. (2002). From supply to demand chain management: Efficiency and customer satisfaction. Journal of Operations Management, 20(6), 747–767. Helmes, M., Ettkin, L., & Chapman, S. (2000). Supply chain forecasting – Collaborative forecasting supports supply chain management. Business Process Management Journal, 6(5), 392. Lee, H., Padmanabhin, L. V., & Whang, S. (1997). The bullwhip effect in supply chain. Sloan Management Review, 38(3), 70–77. Lee, T., & Adam, E., Jr. (1986). Forecasting error evaluation in material requirements planning production-inventory systems. Management Science, 32(9), 1186–1205. Newbury, T. L., & Bhame, C. D. (1981). How management should use and interact with sales forecasts. Inventories and Production Magazine, July–August, 207–218. Ritzman, L. P., & King, B. E. (1993). The relative significance of forecast errors in multi-stage manufacturing. Journal of Operations Management, 11(1), 51–65. Stock, G. N., Greis, J. P., & Kasarda, J. D. (2000). Enterprise logistics and supply chain structure: The role of fit. Journal of Operations Management, 18(5), 531–547. Tanwari, A. U., & Betts, J. (1999). Impact of forecasting on demand planning. Production and Inventory Management Journal (third quarter), 31–35. Tyagi, R. (2002). How to evaluate a demand planning and forecasting package. Supply Chain Management Review, 48, 48–55. Van Landeghem, H., & Vanmaele, H. (2002). Robust planning: A new paradigm for demand chain planning. Journal of Operations Management, 20(6), 769. Zotteri, G., & Verganti, R. (2001). Multi-level approaches to demand management in complex environments: An analytical model. International Journal of Production Economics, 71(1–3), 22.

EXTRACTING FORECASTS FROM ADVANCE ORDERS Frenck Waage ABSTRACT This paper presents a stochastic dynamic model that is capable of guiding a forecast along that path which the true future trajectory will actually follow, when realized. The guidance capability of the model derives from the use of two independent sources of information about the future: (1) one is a conventional econometric sales forecast and (2) partial independent information about planned future events known presently. The paper develops the model, and presents a real application. The benefits of this model are (1) significant reductions in forecasting errors for both the near and the far future and (2) a solution to the problem of predicting the turning point of a product’s life cycle curve months before the turning point will occur.

1. ONE VIEW OF THE FUTURE We shall be operating with two independent forecasts in this paper. The first to be discussed is the corporate rolling monthly sales forecast. Quantities that are sold in any two adjacent months, t and t
1, are measured by the

Advances in Business and Management Forecasting, Volume 4, 13–26 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04002-X

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variables yt and yt
1. The forecasts yt are generated by the econometric model (1) in which the variable ut is a driving force, f and g constant coefficients and qt the forecast error. yt ¼ f  yt
1 þ g  ut þ qt

(1)

To forecast sales for months t ¼ 1, 2, y , H, we use simultaneous equations (2). (2) is obtained by expanding (1) by t ¼ 1, 2, 3, y , H. 10 1 1 0 1 0 0 yt yt
1 1 0 0 0 0 0 1 0 0 0 0 0 CB C C B C B B B ytþ1 C B 0 1 0 0 0 0 CB yt
0 C B 0 1 0 0 0 0 C CB C C B C B B CB C C B C B B B ytþ2 C B 0 0 1 0 0 0 CB utþ2 C B 0 0 1 0 0 0 C CB C C B C B B CB C C B C B B B ytþ3 C ¼ B 0 0 0 1 0 0 CB utþ3 C þ B 0 0 0 1 0 0 C CB C C B C B B CB C C B C B B B .. C B .. .. .. .. .. .. CB .. C B .. .. .. .. .. .. C B . C B . . . . . . CB . C B . . . . . . C A@ A A @ A @ @ ytþH

0 0 0 0 0 1 utþH 1 0 1 0 qt ut C B C B B utþ1 C B qtþ1 C C B C B C B C B B utþ2 C B qtþ2 C C B C B C B C B B utþ3 C þ B qtþ3 C C B C B C B C B B .. C B .. C B . C B . C A @ A @ utþH qtþH

0

0

0 0

0

1

ð2Þ

(3) is the vector matrix representation of (2) where the column vector Yt ¼ (y1, y2, y , yt, y , yH), the column vector Ut ¼ (u1, u2, y , ut, y , uH ) and the column vector qt ¼ (q1, q2, y qt, y qH ). F and G are both H  H constant coefficient matrices. Y t ¼ F  Y t
1 þ G  U t þ qt

(3)

The probability density function that governs Yt is a random vector with a mean E(Yt): EðY t Þ ¼ F  Y t
1 þ G  U t The forecast error is qt: qt ¼ Y t
EðY t Þ

Extracting Forecasts from Advance Orders

15

qt is assumed normally distributed with expected value E(qt) ¼ 0 and a variance co-variance matrix Q(t), measured from history and given by


1=2   Nðqt j0; QðtÞÞ ¼ ð2pÞ
H=2
QðtÞ
exp
12f½qðtÞT QðtÞ
1 ½qðtÞg (4) We have derived the density of qt. We need the density of Yt, however. To get it, we transform the density (4) of qt into the density (5) of Yt using the fact that variance(qt) ¼ variance(Yt) and the Jacobi transform qqt/qYt. These transformations are discussed in advanced texts on Econometrics and Probability Theory (Maddala, 1977; Maybeck, 1979; Wilks, 1950). NðY t jF  Y t
1 þ G  U t ; QðtÞÞ

(5)

The corporate rolling monthly sales forecast Yt is one source of information about future events. Other sources about future events, independent of the corporate forecast, sometimes exist. When such sources of information about the future exist, we shall take advantage of them in the manner described in the following section. Doing so will create an alternative and independent sales forecast, a second view of the future as it were.

2. A SECOND AND ALTERNATIVE VIEW OF THE FUTURE Sometimes, early signals about planned future events are available in the present. This information is about commitments made in the present to execute activities in the future. The commitments can be breached and altered, and must therefore be considered random variables. Examples are advance orders (orders received now and accepted now, committing a supplier to ship certain quantities in the future) and building permits (permits to build houses sometime in the future). To be specific, we shall focus on how to use advance orders information. A supplier who accepts advance orders, will have in his ‘‘order fill files’’, at the beginning of month t, orders committing him to ship a quantity zt+1 in month t+1, to ship a quantity zt+2 in month t+2, y , and to ship a quantity zt+H in month t+H. The quantity zt+j represents, on average, a percentage pt+j of the total quantity yt+j that the supplier has promised to ship in month t+j. The closer to a ship date one comes, the closer zt+j will come to yt+j, and the closer pt+j will be to 1.00. When empirical data is examined, we find that p(k) as a function of ‘‘k months till shipment’’ moves as shown in the following figure as

16

FRENCK WAAGE Percentage of sales in the future month H covered by advance orders t months before the ship date H

1.000 0.900 0.800

Percentage

0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 1

2

3

4

5

6

7

Months before Shipment at H

The variables zt+j and yt+j, are related via the percentage pt+j in equation (6). zt+j is a random variable. The estimate errors are vt+j. pt+j and vt+j were measured by regression analysis using historical observations on zt+j and yt+j. ztþj ¼ ptþj  ytþj þ vtþj

for j ¼ 1; 2; 3; . . .

(6)

The forecast implied by the advance orders is obtained by dividing pt+j through (6). The result is p
1 tþj ztþj ¼ ytþj þ rtþj

for j ¼ 1; 2; 3; . . .

(7)

Expand (7) by j ¼ 1, 2, y , H obtaining (8). (8) measures the ‘‘forecasts from the advance orders’’ for the H future months. 1 0 1 0 ytþ0 1 0 rtþ0 ðtÞ 1 0 ptþ0 ðtÞ
1 ztþ0 ðtÞ 1 0 0 0 C B C B B ptþ1 ðtÞ
1 ztþ1 ðtÞ C B 0 1 0 0 C B ytþ1 C C B rtþ1 ðtÞ C C B C B B C C B B C C B B
1 ytþ2 C B rtþ2 ðtÞ C B ptþ2 ðtÞ ztþ2 ðtÞ C ¼ B 0 0 1 0 C  B þ C C (8) B B C B C B B B . C .. C B .. C C B B . C B . . A @ . A @ . C C @ B . A . A @
1 0 0 0 1 ytþH rtþH ðtÞ ptþH ðtÞ ztþH ðtÞ In vector matrix notation, (8) becomes (9). P
1 t  Z t ¼ M  Y t þ rt

(9) P
1 t

and M both where Zt, Yt and rt is each an H-element column vector and H  H coefficient matrices. The quantity ðP
1  Z Þ is the alternative t t forecast, independent of the corporate rolling forecast.The forecast P
1 t 

Extracting Forecasts from Advance Orders

17

Z t in (9) is a random vector with a mean EðP
1 t  Z t Þ given by EðP
1 t  Z t Þ ¼ M  Y t þ rt

(9.1)

The forecast error is rt
1 rt ¼ P
1 t  Z t
EðPt  Z t Þ

(9.2)

rt is assumed normally distributed with expected value E(qt) ¼ 0 and a variance co-variance matrix R(t), measured from history and symbolically given by


1 Nðrt j0; RðtÞÞ ¼ ð2pÞ
H=2
RðtÞ
2 expð
12f½rðtÞT RðtÞ
1 ½rðtÞgÞ We know that variance(rt) ¼ varianceðP
1 t  ZðtÞÞ: The Jacobi transform @rt =@ðP
1  Z Þ ¼ 1:00 on (9.2) shows that the density governing P
1 t t t  Z t is the normal density function. NðP
1 ðtÞ  ZðtÞjMðtÞ  Y t ; RðtÞÞ

(10)

3. A NEEDED CONDITIONAL VIEW OF THE FUTURE The Normal density (10) measures the likelihood that sales, forecasted from the advance orders, will be P
1 ðtÞ  ZðtÞ conditionally on Yt. However, we are interested in the obverse; namely, the probability that sales will be Yt given that the advance orders have been observed to be Z(t). We shall calculate this conditional density function in Section 6 using the two critical density functions (5) NðY t jF  Y t
1 þ G  U t ; QðtÞÞ and (10) NðP
1 ðtÞ  ZðtÞjMðtÞ  Yt; RðtÞÞ (Raiffa & Schlaifer, 1961; Gelb, 1974; Harrison & Stevens, 1971; Aoki. 1967).

4. MORE PRECISE PRIOR POSTERIOR NOTATION The distinction between prior and posterior will become important. Therefore, double notation is introduced here. Y(t|t
1) replaces Y(t) in (3) and (9) and is the prior forecast of sales when the previous month’s advance orders Z(t
1) alone are known. S(t|t
1) measures the variance covariance matrix about Y(t|t
1) knowing only Z(t
1). Y(t|t) measures the posterior sales forecast conditionally on knowing the current advance orders Z(t). S(t|t) measures the revised variance co-variance matrix about Y(t|t) after Z(t) is known. Eqs. (3) and (9) are re-stated in the new notation

18

FRENCK WAAGE

immediately below. Y ðtÞ ¼ F ðtÞ  Y ðt
1Þ þ GðtÞ  UðtÞ þ qðt
1Þ

in original notation

Y ðtjt
1Þ ¼ F ðtÞ  Y ðt
1jt
1Þ þ GðtÞ  UðtÞ þ qðt
1Þ in new double notation P
1 ðtÞ  ZðtÞ ¼ MðtÞ  Y ðtÞ þ rðtÞ P
1 ðtÞ  ZðtÞ ¼ MðtÞ  Y ðtjt
1Þ þ rðtÞ

in original notation in new double notation

(3) (30 ) (9) (90 )

A mathematical model will be developed which uses the market signals P
1 ðtÞ  ZðtÞ to guide the official forecast Y(t|t) closer to the correct, but unknown, trajectory.

5. GUIDING THE SALES TRAJECTORY YðtjtÞ WITH THE SIGNALS P
1 ðtÞ  ZðtÞ FROM THE ADVANCE ORDERS We shall in this section develop the model, which will guide the forecast Y(t|t) given knowledge of Z(t) and P(t). Our approach relies on stochastic control theory. A very accessible source on this subject is (Astrom, 1970; Jazwinski, 1974; Lewis, 1986; Sorensen, 1970). From the dynamic state space equations (3) and (9), we create the stochastic feedback-guidance control equation (11) that guides the posterior estimate Y(t|t). Y ðtjtÞ ¼ F ðtÞ  Y ðt
1jt
1Þ þ UðtÞ þ KðtÞ  ½P
1 ðtÞ  ZðtÞ
MðtÞ  Y ðtjt
1Þ

ð11Þ

Eq. (11) has two components. The first is the prior forecast Y(t|t
1) ¼ F(t) * Y(t
1|t
1)+U(t) from (3). The second is the feedback-guidance control P
1(t) * Z(t)
M(t) * Y(t|t
1) of (14) multiplied by the weight K(t). The posterior forecast Y(t|t) of (11) equals the prior forecast Y(t|t
1) plus the correction factor K(t) * [P
1(t) * Z(t)
M(t) * Y(t|t
1)]. If the future signal P
1(t) * Z(t) is signaling lower sales than do the prior official forecast Y(t|t
1), the difference P
1(t) * Z(t)
M(t) * Y(t|t
1) will be negative. The posterior forecast Y(t|t) will be guided downward to below the prior Y(t|t
1) by the difference times K(t). The matrix K(t) is calculated from history in a manner that minimizes the squared difference between actual sales Y(t) and the posterior forecasted

Extracting Forecasts from Advance Orders

19

P sales Y(t|t); by the least squares criterion (Y(t)–Y(t|t))2. Proofs of correctness are found in the original papers (Kalman, 1960, Astrom, 1970; Kalman & Buchy, 1961; Bozic, 1994; Brown & Hwang, 1992; Chui & Chen, 1987; Proakis & Manolakis, 1992; Cipra, 1993). (11) is a recursive equation, and to calculate Y(t|t) of (11), an efficient recursive algorithm is needed. Eqs. (12) through (17) define such an algorithm. The algorithm was first developed by Kalman (Kalman, 1960; Antoniou, 1993; Mehra, 1979; Singer & Behnke, 1971). The sales trajectory Y(t|t) and the variance co-variance S(t|t) will now be calculated sequentially from equations (12) through (16). The calculations start with initializing the algorithm with start-up values for the matrices Q(0), R(0) and P(0|0), and the vectors Y(0|0) and U(1). For ‘‘round 1’’ set t ¼ 1. At the beginning of month t calculate K(t
1): Kðt
1Þ ¼ F ðt
1ÞSðt
1jt
1ÞMðt
1ÞT ½Mðt
1ÞSðt
1jt
1ÞMðt
1ÞT þ Rðt
1Þ
1

ð12Þ

At the beginning of month t calculate the prior forecast Y(t|t
1): Y ðtjt
1Þ ¼ F ðt
1Þ  Y ðt
1jt
1Þ þ BðtÞ  UðtÞ

(13)

After the current advance orders Z(t) have been counted, calculate the feedback control signals e(t): eðtÞ ¼ P
1 ðtÞ  ZðtÞ
MðtÞ  Y ðtjt
1Þ

(14)

Thereafter, substitute (13) and (14) into (11) to obtain the posterior mean Y(t|t) given by (15). Eq. (15) provides the trajectory guidance mechanism. Y ðtjtÞ ¼ Y ðtjt
1Þ þ Kðt
1Þ  eðtÞ

(15)

Calculate the posterior variance co-variance matrix S(t|t) about the mean Y(t|t): SðtjtÞ ¼ ½F ðt
1Þ
Kðt
1ÞMðt
1ÞSðt
1jt
1Þ  F ðt
1ÞT þ QðtÞ (16) Eqs. (15) and (16) jointly define the normal distribution (17).


1 NðY ðtÞjY ðtjtÞ; SðtjtÞÞ ¼ ð2pÞ
H=2
SðtjtÞ
2 expð
12f½Y ðtjt
1Þ þ Kðt
1Þ  eðtÞT SðtjtÞ
1 ½Y ðtjt
1Þ þ Kðt
1Þ  eðtÞgÞ

ð17Þ

At the end of month t, the first element of each of the vectors Y(t), Z(t) will have become history. Use these known observations to update the estimates

20

FRENCK WAAGE

of P(t), Q(t) and R(t). Advance the time index to t+1. Go back to (12). Repeat the calculations from equations (12) through (17) for t+1. Having completed the (t+1) round, do the (t+2) round and so on (end of recursive algorithm). As t becomes t+1, t+2, t+3, y , the posterior forecast Y(t|t) describes a trajectory of future sales. This forecasted trajectory will be close to the true trajectory that the actual sales will follow in the future when the future occurs. The following application elucidates and illustrates.

6. AN APPLICATION The application dynamically forecasted the sales of a large expensive special purpose computer as market conditions were changing with time. There were several competing computers in the market. The computer whose sales we forecasted was the market leader and a growth product when the application started. A conventional econometric forecast was run at the beginning of each month. All orders received by the supplier were ‘‘advance orders’’ received up to six months in advance of delivery. At the beginning of any month t the model was thus able to look six months into the future, and to guide the sales forecast over these six future months closer to where actual sales will end up. The model was initialized with the following diagonal matrix values. The off-diagonal matrix values are all zero in this application in order to have arithmetic simplicity. In the real life models, we worked with full matrices. All vectors were six-element vectors and the matrices were six rows by six columns. Diagonal initializing values Q11 ð0Þ ¼ 1 Q22 ð0Þ ¼ 16 Q33 ð0Þ ¼ 36 Q44 ð0Þ ¼ 81

R11 ð0Þ ¼ 0:25 S 11 ð0j0Þ ¼ 1 R22 ð0Þ ¼ 9 S 22 ð0j0Þ ¼ 2 R33 ð0Þ ¼ 25 S 33 ð0j0Þ ¼ 4 R44 ð0Þ ¼ 49 S 44 ð0j0Þ ¼ 16

P
1 11 ð0Þ ¼ 1=0:9542 P
1 22 ð0Þ ¼ 1=0:7505 P
1 33 ð0Þ ¼ 1=0:1997 P
1 44 ð0Þ ¼ 1=0:0915

Y tþ0 ð0j0Þ ¼ 185 Y tþ1 ð0j0Þ ¼ 286 Y tþ2 ð0j0Þ ¼ 202 Y tþ3 ð0j0Þ ¼ 151

Q55 ð0Þ ¼ 100 Q66 ð0Þ ¼ 144

R55 ð0Þ ¼ 81 R66 ð0Þ ¼ 100

P
1 55 ð0Þ ¼ 1=0:0502 P
1 66 ð0Þ ¼ 1=0:0205

Y tþ4 ð0j0Þ ¼ 279 Y tþ5 ð0j0Þ ¼ 82

S 55 ð0j0Þ ¼ 32 S 66 ð0j0Þ ¼ 64

The initializing forecast at the beginning of December: round t ¼ 0. This initializing forecast Y(0|0) is graphed in Fig. 1 and tabulated in Table 1. It forecasts the future sales of a product for the year, and is shown as the dotted line in Figs. 1 and 2.

Extracting Forecasts from Advance Orders

21

400

Y(0|0) 350

Units

300 250 200

Y(1|1)

150 100 50 0

1 2 3 4 5 6 7 1 = January, 2 = February etc

Fig. 1.

109 185 286 202 151 279

10 11

Y(0|0) and Y(1|1).

3 U(1)

4 Y(1|0) Prior

5 Z(1)

6 P
1(1)nZ(1)

8 K(1)ne(1)

7 Y(1|1) Posterior

76 100 86 70 70 82

185 285 200 132 221 197

177 215 40 15 8 4

185 286 200 164 159 195

0.39 0.27 0.04 7.86
17.45
0.73

185 285 200 140 204 196

400

Y(0|0)

350 300

Units

December January February March April May June

2 Y(0|0)

9

The January Forecast View.

Table 1. 1

8

250 200

Y(1|1)

150 100

Y(2|2)

50 0

1

2

Fig. 2.

3 4 5 6 7 8 1 = January, 2 = February etc

Y(0|0), Y(1|1), Y(2|2).

9

10

11

22

FRENCK WAAGE

The forecast at the beginning of January: round t ¼ 1. Table 1 shows the given initializing forecast Y(0|0), given here without showing the calculations, and the values of the driving force U(1) and the prior forecast Y(1|0) developed from (13) at the beginning of January. Table 1 further shows the advance data Z(1) that became available sometime during January, the forecast P
1(1) * Z(1) that the advance data implied given P
1(1), the guidance signal K(1) * e(1) calculated from (12) and (14) and the posterior sales forecast Y(1|1) calculated from (15). Fig. 1 graphs the prior Y(0|0) and the January update of the forecast Y(1|1). This update has been steered downward by the advance order information in Z(1). Fig. 2 graphs the same prior Y(0|0), the same January update Y(1|1) and adds the February revision of the forecast Y(2|2). This update has been steered downward further by the advance order information in Z(2). With a full five-month lead-time we were forewarned by Y(1|1) that the sales in the future months of May and June will drop significantly! Y(1|1) and Y(2|2) suggest that the product life cycle will peak in March or April and thereafter decline. This is how life cycle turning points are forecasted by this model. The forecast at the beginning of February: round t ¼ 2. The time index is increased from t ¼ 1 to 2. Table 2 shows the same variables that Table 1 did after t has become t ¼ 2. Table 2 repeats Y(1|1) and calculates Y(2|1) and Y(2|2) from (13) and (15). Fig. 2 graphs Y(0|0), Y(1|1) and Y(2|2) and these have already been discussed. The customers continued to signal us, through the advance orders Z(2) that they would purchase less in May and June than had been forecasted in the original forecast. Also, the life cycle peak will occur near March. The forecast at the beginning of March: round t ¼ 3. The time index t is increased from t ¼ 2 to 3. Table 3 shows the forecasting results. The Table 2. 1

January February March April May June July

2 Y(1|1) 185 285 200 140 204 196

The February Forecast View.

3 U(2)

4 Y(2|1) Prior

5 Z(2)

6 P
1(2)Z(2)

8 K(2)ne(2)

7 Y(2|2) Posterior

100
86
70 90
28
90

285 199 130 230 176 106

270 152 30 17 9 2

283 203 150 186 179 98


2 +2 +12
69 +2
5

283 201 142 201 178 101

Extracting Forecasts from Advance Orders

23

Table 3. The March Forecast View. 1

2 Y(2|2)

February March April May June July August

283 201 142 201 178 101

3 U(3)

4 Y(3|2)

5 Z(3)

6 P
1(3)Z(3)

8 K(3)ne(3)

7 Y(3|3)


75
80 85
40 38 21

208 121 227 161 216 122

190 84 39 13 3 2

199 112 195 142 60 98


7
6
21
13
101
17

201 115 206 148 115 105

300

Y(2|2)

Units

250 200 150

Actual 100

Y(3|3)

50 0

1

Fig. 3.

2

3 4 5 6 7 8 9 1= January, 2 = February etc

10 11

Y(2|2), Y(3|3) and Actual Sales.

advance orders Z(3) continued to guide the forecast downward. The life cycle turning point by now stood confirmed. Fig. 3 graphs the revised forecasts Y(2|2) and Y(3|3). Fig. 4 graphs the revised forecasts Y(4|4), but we do not show the calculations behind Fig. 4.

7. CONCLUSIONS, MANAGERIAL RAMIFICATIONS AND FUTURE RESEARCH 7.1. Guided Forecasts Converge on the True Sales Trajectory In real life, actual customers often possess special information and advanced insights on products and competitive developments not generally available

24

FRENCK WAAGE 300

Units

250 200

Actual Sales

150

Y(4|4)

100 50 0

1

2

Fig. 4.

3 4 5 6 7 8 9 1 = January, 2 = February etc

10 11

Y(4|4) and Actual Sales.

to forecasters. They use this information dynamically to revise their current and future order and purchase decisions. Revisions of their purchase decisions lead to changes in their advance orders for future deliveries of all competing products. Products that are increasingly becoming ‘‘preferred’’ will be experiencing increases in advance orders now and in future sales. Products that are falling out of favor will experience decreasing advance orders and decreasing future sales. The dynamic changes in the advance orders predict and track the changing locus of the true future sales trajectory. The advance orders guide the trajectory of the sales forecast, therefore, to the path that sales will actually be following in each future month before the future is known. The sales forecast, when guided by the advance orders, will converge on the actual sales trajectory even before the actual trajectory is known.

7.2. Early Warnings of Future Changes The advance orders allow the forecasters to see more clearly as far forward into the future as the advance orders reach (six months in the above application). The advance orders change dynamically in response to competitive information. The advance orders record such changes and signal these future changes many months before they take place. The present model captures the signals. The signals are early warnings on how sales will change in the future. The model delivers these early warnings. Column 8 of all the tables offers examples.

Extracting Forecasts from Advance Orders

25

7.3. Predicting the Turning Point of a Brand’s Life Cycle Trajectory Assume that a given product is a growth product, and that therefore it will be on the increasing-slope of its life cycle curve. Assume that a competing product is introduced into the market to compete with the former product. Assume further that the customers begin to gradually switch their purchases and advance orders to the new product and away from the old growth product. Under these circumstances, the sales of the old product will reach a peak after the new product’s introduction and decline thereafter. The model can predict this peak from the advance order signals. When the new product enters the market and becomes the preferred product, the customers will decrease their advance orders for the old product and increase their advance orders for the entering product. The advance orders for the old product will guide the forecast Y(t|t) along a declining trajectory. The moment of the life cycle peak is forecasted by this model. Such a life cycle peak was predicted in the above application.

7.4. Research Opportunities We have assumed the error structures, measured by Q(t), R(t) and S(t|t), to be those of normal density functions. This assumption permitted the use of very tractable state space difference equations. It will be useful to continue developing filters for error structures that are not normal and not linear.

REFERENCES Antoniou, A. (1993). Digital filters. New York: McGraw-Hill. Aoki, M. (1967). Optimization of stochastic systems–topics in discrete-time systems. New York: Academic Press. Astrom, K. J. (1970). Introduction to stochastic control theory. New York: Academic Press. Bozic, S. M. (1994). Digital and Kalman filtering (2nd ed.). London: Edward Arnold of Hodder Headline Group. Brown, R. G., & Hwang, P. Y. C. (1992). Introduction to random signals and applied Kalman filtering (2nd ed.). New York: Wiley. Chui, C. K., & Chen, G. (1987). Kalman filtering with real-time applications. Heidelberg: Springer. Cipra, B. (1993). Engineers look to Kalman filtering for guidance. SIAM News, 26(5). Gelb, A. (1974). Applied optimal estimation. Cambridge, MA: MIT Press. Harrison, P. J., & Stevens, C. F. (1971). Bayesian forecasting. Journal of the Royal Statistical Society, Series B, 3B, 205–247.

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Jazwinski, A. H. (1974). Stochastic processes and filtering theory. New York: Academic Press. Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Transactions on ASME, Journal of Basic Engineering, 82, 34–45. Kalman, R. E., & Buchy, R. S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83, 95–108. Lewis, R. (1986). Optimal estimation with an introduction to stochastic control theory. New York: Wiley. Maddala, G. S. (1977). Econometrics. New York: McGraw-Hill Chapter 17. Maybeck, P. S. (1979). Stochastic models, estimation and control. New York: Academic Press. Mehra, R. K. (1979). Kalman filters and their applications to forecasting. TIMS Studies in the Management Sciences, 12, 75–94. Proakis, J. G., & Manolakis, D. G. (1992). Digital signal processing (2nd ed.). New York: Macmillan Publishing. Raiffa, H., & Schlaifer, R. (1961). Applied statistical decision theory. Boston, MA: Harvard Business School Press. Singer, R., & Behnke, K. (1971). Real time tracking filter evaluation and selection for tactical evaluations. IEEE Transactions on Aerospace Electron Systems, AES-7(January), 100–110. Sorensen, H. W. (1970). Least squares estimation: From Gauss to Kalman. IEEE Spectrum, 7, 63–68. Wilks, S. S. (1950). Mathematical statistics. New Jersey: Princeton University Press.

INVENTORY-SHIPMENT RATIO TIME SERIES MODELS FOR DURABLE AND NON-DURABLE PRODUCTS Supriya Mitra ABSTRACT This paper proposes the inventory-shipment (I-S) ratio time series model to capture the dynamics of inventory and sales simultaneously over time. We discuss the business implications of trends in this ratio and also illustrate the effectiveness of this ratio in capturing just-in-time trends. The main contribution of this paper lies in analyzing and comparing the I-S ratio time series of durable, non-durable and total products over the time period from 1992 to 2003.

1. INTRODUCTION An appropriate time series model is much needed for effective management of inventory in supply chains. Most studies till date have focused on using sales forecasts for appropriating inventory requirements. We discuss the inadequacy of such models and propose an inventory-shipment (I-S) ratio model to capture the dynamics of inventory and sales simultaneously over Advances in Business and Management Forecasting, Volume 4, 27–53 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04003-1

27

28

SUPRIYA MITRA

time. We discuss the business implications of trends in this ratio and also illustrate the effectiveness of this ratio in capturing just-in-time (JIT) trends. The main contribution of this paper lies in analyzing and comparing the ratio time series behavior of durable and non-durable products over the time period from 1992 to 2003. We show that a comparison of the models for durable and non-durable products provides interesting insights on the supply chain behavior of these products. The comparison shows that both the durable and total products (durable plus non-durable) series have an ARIMA(3,1,0) model structure, whereas the non-durable products have an ARIMA(1,1,0) model structure. Durable products having higher I-S ratios are prone to be slow-moving goods when compared to non-durables. Thus, the durable series tends to depend on values even three periods back whereas the non-durable series tends to depend on only the last period values. For total products, the model is comparable to that of durable products. One striking similarity across all the three product groups is the modeling of the squared residuals by an ARCH(1) model. Finally, we also compare the forecast of a simple average (and a modified simple average) of the durable and non-durable products with that of the total products.

2. LITERATURE REVIEW Brownstein (1984) observed the major domestic automobile manufacturers carried between $700 and $1000 worth of inventory per car with plans to reduce by another 20–30%. However, this target was still below the Japanese average of $200 inventory per car. According to Chen, Frank, and Wu (2003), the proponents of JIT were concerned that American manufacturing firms needed to reduce their inventories. Using balance sheet data from 6,077 manufacturing firms, the authors found strong empirical evidence that inventories have actually declined (as recommended by the proponents of JIT). However, the authors note that inventories may also be influenced by macroeconomic conditions. For example, when there is high gross domestic product (GDP) growth, low inventory levels occur since ‘‘firms may have trouble keeping up with demand.’’ Similarly, in a recession, inventories tend to build up. The authors note that interest rate effects are rather ambiguous. When interest rates increase, inventories become expensive and hence a retailer may prefer lower inventory levels. However, higher interest rates might motivate suppliers to increase sales by providing lower prices to the retailer. Thus, the retailer faces a dilemma with regard to increasing or

Inventory-Shipment Ratio Time Series Models

29

decreasing inventory levels in such a situation. The authors use the inventory-to-asset ratio (inventory divided by total assets) and a modified inventory day’s (length of time goods are held) measure for testing their hypothesis. Rajagopalan and Malhotra (2001) conducted the first empirical study to comprehensively investigate inventory trends (from 1961 to 1994) in all manufacturing sectors and at all stages of manufacturing (raw materials, work-in-process and finished goods). The authors investigated inventory ratio (inventory value/cost of goods sold) trends at an industry level rather than at a firm level to eliminate any firm-specific biases. The use of ratios instead of actual inventory values was to control for differences in firmspecific/industry-specific inventory valuation methods. The authors hypothesized that for the total manufacturing sector: 1. Inventory ratios would show a decreasing trend from 1961 to 1994. 2. Inventory ratios would show a higher rate of improvement in the 1980– 1994 period vis-a`-vis the pre-1980 period. This second hypothesis was based on the premise that JIT adoption started in the 1980s. The authors found empirical support for the first hypothesis in all stages of manufacturing. However, the second hypothesis was not supported at any stage of manufacturing. The results purvey a mixed picture of inventory reduction in U.S. manufacturing. The authors attributed this anomaly to the increase in product and process variety after the 1980s. The need for mass customization resulted in more variety in sub-assembly and assembly processes. The industry also experienced a rapid growth in the number of competing firms and this added to the overall inventory of the sector. The author also discussed how increased imports by U.S. firms in certain sectors contributed to higher levels of inventory needed as a buffer for the long lead times associated with international shipments. Gaur, Fisher, and Raman (2005) assert that there is considerable interest in the operations management community in evaluating time trends in the inventory turnover ratio (the ratio of the firm’s cost of goods sold to its average inventory level). However, the authors affirm that there are few empirical studies on this topic. The authors study this ratio in retail services by using firm-level panel data. They use a log–linear model to test the relationship between inventory turnover as the dependent variable and gross margin, capital intensity and sales surprise as the independent variables. They hypothesized that inventory turnover is negatively correlated with gross margin (as items with higher margins are given lower inventory turnover target than items with lower margins). They also hypothesized that higher capital intensity increased inventory turnover. This was based on the

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SUPRIYA MITRA

fact that an addition of a new warehouse should result in a decrease in total inventory (safety stock being lower) at the retailer and thus increase inventory turnover. Their third hypothesis stated that inventory turnover was positively correlated to sales surprise (or unexpectedly high sales). This is based on the fact that if sales in a given period were higher than forecasted, then the average inventory level for the period would be lower (since periodending inventory is low). Thus, the inventory turnover ratio is high due to the higher cost of goods sold and lower average inventory. The authors found strong empirical support for all the three hypotheses. However, the authors actually found that the overall inventory turnover ratio had a downward slope during 1987–2000, even though 43% of the firms (with a greater increase in capital intensity) showed a large increase in the ratio. Hanssens (1998) used time series modeling to compare the behavior of factory orders and consumer sales. The author asserts that the inefficient matching of factory orders and consumer sales could either lead to excess inventory or shortages. This was exemplified by reference to the secondquarter losses (April, 1996) of Apple Computers due to inventory write-offs in excess of $388 million. Similarly, industry-wide shortages of DRAM memory chips illustrated inventory shortages due to forecasting inefficiencies. The author conjectures that though orders and sales evolve over time, their movements are not necessarily independent. The author estimated a linear regression equation with orders in time period t as the dependent variable and sales in time period t as the independent variable. The author also estimated a model in which orders were a function of previous orders, consumer sales and the manufacturers’ marketing mix. The manufacturers’ marketing mix included variables such as product profit margin (for retailer), supply allocations and advertising. Higher product profit margins encourage retailers to carry the company’s product over the competition. Supply allocations or rationing by the manufacturer lead to larger retailer orders since the retailers believe that they would only be allocated a fraction of the original order quantity. Manufacturer-sponsored advertising signals the manufacturer’s confidence in the product and hence stimulates larger orders from the retailer. Fildes and Beard (1992) noted that the non-durables market is more sensitive than the durables market to changes in marketing mix parameters like advertising campaigns, price, promotions and in-store displays. Snyder, Koehler, and Ord (2002) explore the non-constant variance in inventory control forecasts. Their research was motivated by the statement of Brown (1959) on the proportionality of the standard deviation of demand to the total annual usage. Steffens (2001) explored sales models for durable

Inventory-Shipment Ratio Time Series Models

31

products while incorporating a component of sales due to replacement. The author challenged traditional models that implicitly assume that the older unit being replaced cannot be resold in the second-hand market (the resold units could cannibalize net sales of new units). The author considers two types of replacements: ‘‘forced’’ replacements which occur due to a failed product and ‘‘unforced’’ replacements which are discretionary and depend on time-varying factors such as product reliability/durability, price, repair costs, scrapping values and economic values which change the mean replacement age of products. The author modeled this time-varying replacement behavior of durable products and empirically confirmed a substantial increase in the average aggregate replacement age for motor vehicles. Fildes and Beard (1992) assert that the most common use of quantitative forecasting techniques is in the areas of production and inventory systems. However, several authors have recognized the problematic nature of forecasting production and inventory levels (Albertson & Aylen, 2003). Fildes and Beard (1992) assert that researchers have mostly concentrated on forecasting sales activity rather than inventory and production. The authors suggested that forecasting systems should be designed to be equally useful to both production and marketing activities especially with the advent of enterprise resource planning (ERP) systems. Thus, the importance of accurate sales forecasts for efficient inventory and production management cannot be undermined (Frees & Miller, 2004; Ching-Wu & Zhang, 2003).

3. MOTIVATION AND BACKGROUND An appropriate time series variable is much needed for the appropriate management of inventory in supply chains. As discussed, most studies till date focus on sales series trends for appropriating the inventory and production requirements. However, a cursory look at the I-S ratio of diverse product types reveals the inefficiency of using the sales series for forecasting inventory requirements. Presently, this ratio is typically higher than 1 and sometimes higher than 2 for certain product types. The conflicting dynamics between inventory and sales (shipments) leads one to believe that an appropriate index that preserves the relationship between these two variables and can yet be modeled by a single time series needs to be established. In our view, the I-S ratio is an appropriate ratio for this purpose. The I-S ratio is calculated by dividing the total inventory by the total shipment for the same month. It is reasonable to assume that shipments made by the manufacturer approximate actual sales. This

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assumption is valid if (1) the retailer is not allowed to forward-buy for future periods and (2) the retailer does not have a return option wherein unsold inventory may be returned to the manufacturer. The I-S ratio, if monitored properly can be used to gauge some useful signals. An increasing trend in the I-S ratio from one period to the next indicates the following: (1) investment in inventory is growing more rapidly than sales, (2) declining sales, (3) oncoming cash-flow problem, (4) (certain) product lines are taking longer to sell, (5) a (certain) inventory should be written off. Businesses respond to this unintended buildup of inventories by postponing orders and cutting down production rates. The result is a slowdown in economic activity. Short-term interest rates reach their cyclical peak. Similarly, a decreasing trend in the I-S ratio from one month to the next indicate: (1) investment in inventory is shrinking in relation to sales, (2) effective management of business’s inventory levels and its cash flow, (3) business conditions are improving and interest rates are close to reaching their cyclical low, (4) more effective production control. Businesses respond to meet the increase in sales by speeding up their orders and their production rates. Retailers often make adjustments in their merchandise assortment based on the I-S ratio. Departments with a high inventory/sales mix ratio often use more discounting to sell off their products, and therefore receive a lower return. Departments with low ratios often receive a higher return. In many businesses, this ratio remains relatively unchanged from year to year, so even a small change requires investigation. Thus, the I-S ratio is not only an appropriate measure of the dynamics between inventory and sales, but also merits modeling as a separate entity. As we have shown in the present literature, only two papers, i.e., (Rajagopalan & Malhotra (2001) and Gaur, Fisher, & Raman (2005)), have attempted to empirically model and analyze this ratio. Gaur and co-workers actually modeled the inverse of the ratio, i.e., the inventory turnover ratio instead of the I-S ratio, whereas Rajagopalan and Malhotra modeled the I-S ratio. Both the authors used regression models to explore the effect of different independent variables on the dependent ratio variable. However, neither of the two authors analyzed the time series model for this ratio. Our objective is to provide businesses of durable and non-durable goods an appropriate time-series model for understanding the dynamics of the I-S ratio over time. We look at the I-S ratio time-series data for durable and non-durable goods from January 1992 to February 2003 available from the Census Bureau data (http://www.economagic.com). This provided us with

Inventory-Shipment Ratio Time Series Models

33

134 monthly data points for time-series analysis. Even though the Census Bureau has ratio data available from January 1958, the choice of time period for analysis was motivated by the recency of the data points. The durable products included industries such as computer, machinery, appliance and transportation among others. The non-durable products included industries such as food, beverage, coal, paper and petroleum among others. Since the data aggregates across various industries in the two product types, this helps to eliminate ‘‘curious features of the data’’ (Fildes & Beard, 1992). Thus, data aggregation helps to nullify over-predictions and underpredictions in product-specific forecasts. Such aggregation also mitigates seasonal effects idiosyncratic to certain products.

4. DESCRIPTIVE STATISTICS We used the Scientific Computing Associates (SCA) software for modeling the time series. The mean of the durable time series is 1.6516 with variance 0.0142. The minimum is 1.47 and the maximum is 2.03 with a range of 0.56. The mean of the non-durable time series is 1.1525 with variance 0.0010. The minimum is 1.09 and the maximum is 1.24 with a range of 0.15. The mean of the non-durable series is lower since non-durables have higher shipment values and lower average inventory values (can be verified by data available at the Census Bureau). This stems from the essential characteristic of nondurables like food, beverages or petroleum that have less shelf life and hence cannot be kept in inventory for long. Hence, retailers tend to keep less of non-durables in their inventory compared to durables. Also, non-durables are often daily necessities and hence sales of such products tend to be much higher than those of the durable products. We also look at the total of the durables and non-durables (referred to as ‘‘total’’). This series is obtained by adding the total durable plus non-durable inventories and dividing by the total durable plus non-durable shipments during each period. The mean of the total time series is 1.4225 with variance 0.0050. The minimum is 1.31 and the maximum is 1.64 with a range of 0.33. As is seen in the plot (Fig. 1), the ratios have decreased over the period from January 1992 to February 2003. As discussed earlier in Rajagopalan and Malhotra (2001), this is a good sign for the economy and has been achieved by implementation of JIT methods and lean manufacturing systems which attempt to reduce the raw materials, work-in-progress (WIP) and finished products’ inventory while satisfying customer demands. The JIT goal of zero inventory can be achieved by simultaneous production and

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Ratio

2

Durable Total Non-Durable

1.5

1

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Year

Fig. 1. Monthly Ratio Data (from January 1992 to February 2003).

sales as demand arises thus minimizing the need for inventory. Dell has been using JIT quite effectively in the durable industries sector. An example of such a practice in the non-durables industry is the fast-food restaurants. However, such restaurants do maintain some non-durable raw material inventory in the form of food ingredients. Though all the three ratios have been steadily decreasing over time, the ratio has increased noticeably during the years 2000–2002 for the durables and total product series. In a recent monthly survey of manufacturing (The Daily, 2003), the article attributed the increase in I-S ratio during this period to economic slowdown which resulted in a sharp decrease in shipments. However, as observed, none of the time series is stationary. We also check the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the original time series for all the three product types. The ACF is used for identifying the order of a moving average (MA) model. The kth order autocorrelation rk is defined as the correlation between the observation at time period t, i.e., Zt and the observation at time period t
1, i.e., Zt
1. Thus rk ¼ corrðZ t ; Z t
1 Þ: For an MA(q) model (i.e., an MA model of order q), one observes that the ACF cuts off after lag q. Such cut-off patterns enable us to identify the order of an MA-type model. The PACF is used for identifying the order of an autoregressive (AR) model. The sample ^ ; k ¼ 1, 2, y, where f ^ is the kth coefficient for an PACF denoted by f kk kk AR(k) fit. Thus, if the true AR order is p, one will observe that the sample PACF f^ pp a0; but f^ kk ¼ 0 for all k4p. Hence the PACF cuts off after the

Inventory-Shipment Ratio Time Series Models

35

correct order p. By observing the cut-off pattern one can identify the order of an AR-type model. On checking the ACF pattern (see Fig. 2), we find that the ACF has a r1 (lag 1) of 0.93. This value decreases slowly for subsequent rk (0.87, 0.83, y.). Similarly for the PACF (see Fig. 3), we observe that f11 is 0.93 and the otherfs (0.1, 0.06, y) are insignificant compared to f11. These are signs of non-stationarity, hence we use differencing. We show the ACF and PACF patterns for the durable products only below. We do not show the ACF/PACF patterns for non-durable and total products since AUTOCORRELATIONS

1 2 3 4 5 6 7 8 9 10 11 12

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 0.93 + IXXX+XXXXXXXXXXXXXXXXXXX 0.87 + IXXXXXX+XXXXXXXXXXXXXXX 0.83 + IXXXXXXXX+XXXXXXXXXXXX 0.78 + IXXXXXXXXX+XXXXXXXXX 0.74 + IXXXXXXXXXX+XXXXXXX 0.70 + IXXXXXXXXXXX+XXXXX 0.66 + IXXXXXXXXXXXX+XXXX 0.62 + IXXXXXXXXXXXX+XXX 0.57 + IXXXXXXXXXXXXXX 0.52 + IXXXXXXXXXXXXX+ 0.48 + IXXXXXXXXXXXX + 0.43 + IXXXXXXXXXXX +

Fig. 2.

ACF Pattern.

PARTIAL AUTOCORRELATIONS

1 2 3 4 5 6 7 8 9 10 11 12

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 0.93 + IXXX+XXXXXXXXXXXXXXXXXXX 0.10 + IXXX+ 0.06 + IXX + -0.09 + XXI + 0.09 + IXX + -0.05 + XI + 0.06 + IXX + -0.09 + XXI + -0.11 +XXXI + 0.00 + I + -0.02 + XI + -0.05 + XI +

Fig. 3.

PACF Pattern.

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Fig. 4.

Durable Goods – Differenced Time Series.

the results are quite similar to durable products. The differenced series (Fig. 4) for durables has a negative mean of
0.0036 and variance of 0.0009. The minimum is
0.0700 and the maximum is 0.0800. The negative mean makes sense in view of the decreasing trend seen in the original series. The differenced series (Fig. 5) for non-durables has a negative mean of
0.0009 and variance of 0.0002. The minimum is
0.0500 and the maximum is 0.0400. As can be observed, the absolute mean value of the differenced durable series is higher than the absolute mean of the differenced nondurable series. This implies that the average decrease (slope) in I-S ratio is higher in the durable series than in the non-durable series. It appears that JIT and lean manufacturing have been more effective on the durable sector than on the non-durable sector. However, this difference again stems from the nature of durable goods vis-a`-vis non-durable goods. It may not be feasible to decrease the I-S ratio of non-durables at the same rate as durable products. The differenced series for totals (Fig. 6) has a negative mean of
0.0023 and variance of 0.0003. The minimum is
0.0500 and the maximum is 0.0500. Overall, all the three differenced series appear to have a constant mean. However, they appear to have some variance changes. The reason for such variance changes stems from the reaction curves of manufactures or retailers to this ratio over time. Usually, when the ratio starts increasing, manufacturers

Inventory-Shipment Ratio Time Series Models

Fig. 5.

37

Non-Durable Goods – Differenced Time Series.

Fig. 6.

Total Goods – Differenced Time Series.

start to cut back on production in order to reduce inventory. However, this being a very dynamic scenario on a very narrow timescale (often measured in days and shifts), one soon lands up with a scenario with insufficient inventory to satisfy demand, which in turn triggers increased production.

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Thus, manufacturers must constantly perform this ‘‘balancing act’’ thus resulting in variance changes over time. In view of this, it becomes necessary to model the variance change over time.

5. MODELING We proceed through the initial rough-cut steps needed to commend suitable models for comparison in each product type.

5.1. Durable Products We evaluate the two best models – ARIMA (1,1,3) and ARIMA (3,1,0) for durable products. ARIMA(1,1,3). The t values of the constant term and the first and second orders of the AR were insignificant and hence dropped. Only the third order of the MA and first order of the AR are significant (t values
3.07 and
2.70, respectively). The coefficient values are
0.2565 and
0.2295, respectively. The R2 value is 0.940 and the residual standard error is 0.01717. The results are quite good. We also check for the normality of the residuals and find a symmetric bell-shaped curve in the histogram. The ACF, PACF and the extended autocorrelation function (EACF) of the residuals do not indicate any reasonable correlations. Thus, the ARIMA(1,1,3) model can be written as

1
f1 B ð1
BÞ lnðZt Þ ¼ 1
y3 B3 at (1) or
ð1 þ 0:2295BÞð1
BÞlnðZ t Þ ¼ 1 þ 0:2565B3 at

(2)

Note that ‘B’ denotes the back-shift operator typically used in time series models. We now check the fitted plot with the logged series and find a reasonable enough fit. Next, we attempt to model the variance changes by modeling the square of the residuals. Even though the residuals per se do not indicate any reasonable correlation, the square of the residuals that is indicative of the variance needs to be modeled in view of our earlier observations on the differenced series. The PACF and EACF suggest an AR(1) model for the squared residuals. We model the squared residuals as an AR(1) model and obtain significant t values for both the constant term and the AR(1) terms (t values 5.68 and 2.20, respectively). The respective

Inventory-Shipment Ratio Time Series Models

39

coefficients are 0.0002 and 0.1888. The R2 is low at 0.030 and the residual standard error is 0.379710E
03. Thus, we have an ARCH(1) model for the squared residuals. The residuals of this ARCH(1) model are insignificant as indicated by the EACF of the residuals. Thus the ARCH(1) model is a2t ¼ 0:0002 þ 0:1888a2t
1 þ wt

(3)

ARIMA(3,1,0). We model this ARIMA(3,1,0) model and obtain significant AR(1) and AR(3) parameters (t values are
3.27 and 3.42, respectively). The respective coefficients are
0.2640 and 0.2782. The R2 is 0.946 which is better than that obtained using the ARIMA(1,1,3) model. However, the residual standard error (0.0277119) is higher. The histogram gives a symmetric bell-shaped curve and the probability plot gives a fairly straight line indicating normality. There is no significant correlation between the residuals as indicated by the EACF of the residuals. The model is thus: ð1 þ 0:2640B
0:2782B3 Þð1
BÞZ t ¼ at

(4)

We now check the fitted plot (Fig. 7) with the original series and find a reasonable enough fit. The EACF and PACF of the squared residuals indicate an AR(1) model for the squared residuals. We model the squared residuals as an AR(1) model and obtain significant t values for both the constant term and the AR(1) terms (t values 5.53 and 2.34, respectively). The respective coefficients are 0.0006 and 0.2018. The R2 is low at 0.034 and the residual standard

Fig. 7.

Fitted Plot for Durables using ARIMA (3,1,0).

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SUPRIYA MITRA

error is 0.100865E
02. Thus, one has an ARCH(1) model for the squared residuals. The residuals of this ARCH(1) model are insignificant as indicated by the EACF of the residuals. Thus the ARCH(1) model is a2t ¼ 0:0006 þ 0:2018a2t
1 þ wt

(5)

As shown in Table 1, both the ARIMA(1,1,3) and the ARIMA(3,1,0) models perform well on most of the requirements. Both the models exhibit equal parsimony since there are only two significant parameters in both the models. The squared residuals of both the models have an ARCH(1) model. The ARIMA(3,1,0) model has a higher R2 but lower residual standard error than the ARIMA(1,1,3). Also, the ARIMA(3,1,0) model does not require taking logarithm of the original series. It depends on the previous Zt values (which are observable) and not the previous at values as in the ARIMA(1,1,3). In view of this, practitioners may prefer the ARIMA(3,1,0) model owing to its relative simplicity and is hence the preferred choice for comparison with the non-durables and total models.

5.2. Non-Durable Products We evaluate the two best models – ARIMA(0,1,1) and ARIMA(1,1,0) for non-durables ARIMA(0,1,1). On modeling the ARIMA(0,1,1), we find that the t value of the constant term is insignificant (t value
1.35) and so we drop the same. The re-estimated model has a significant MA(1) term with coefficient 0.4100 and t value of 5.33. The R2 of 0.810 is slightly low when compared with the durable products model that had an R2 of about 0.946. The residual standard error is 0.0136427. The histogram gives a symmetric bell-shaped curve and the probability plot gives a fairly straight line indicating normality. There is no significant correlation between the residuals as indicated by the EACF of the residuals. The model is thus: ð1
BÞZ t ¼ ð1
0:41BÞat

(6)

We also check the fitted plot with the original series and find a reasonable enough fit. ARIMA(1,1,0). The t value of the AR(1) parameter is significant at
4.78 and the coefficient is
0.3862. The R2 is 0.806 and the residual standard error is 0.013755. The histogram is reasonably bell shaped, even though it appears slightly uneven. The probability plot is a straight line indicating normality. The EACF of the residuals shows no correlation. ð1 þ 0:3862BÞð1
BÞZ t ¼ at

(7)

Model R2 Residual standard error Requires logarithm Parsimony Diagnostic checking Fitted plot Squared residuals

Durable Goods ARIMA (1,1,3) vis-a`-vis ARIMA (3,1,0). ARIMA (1,1,3)

ARIMA (3,1,0)

(1+0.2640B
0.2782B3)(1
B)Zt ¼ at 0.9400 0.0172 Yes 1 AR and 1 MA(3) parameter Satisfactory Satisfactory a2t ¼ 0:0002 þ 0:1888a2t
1 þ wt

(1+0.2295B)(1
B)ln(Zt) ¼ (1+0.2565B3)at 0.9460 0.0277 No AR(1) and AR(3) Satisfactory Satisfactory a2t ¼ 0:0006 þ 0:2018a2t
1 þ wt

Inventory-Shipment Ratio Time Series Models

Table 1.

41

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On checking the fitted plot with the original series (Fig. 8), one finds a reasonable enough fit. In Table 2, both the ARIMA(0,1,1) and the ARIMA(1,1,0) models perform well on most of the requirements. Also both the models exhibit equal parsimony. The ARIMA(1,1,0) model has a slightly lower R2 and residual standard error than the ARIMA(0,1,1) model. However, the ARIMA(1,1,0) model depends on the previous Zt values (which are more observable) and not the previous at values as in the ARIMA(0,1,1) model. In view of this, practitioners may prefer the ARIMA(1,1,0) model due to its relative simplicity and is hence the preferred choice.

Fig. 8.

Fitted Plot Using ARIMA (1,1,0) Model.

Table 2. Non-Durables ARIMA(0,1,1) vis-a`-vis ARIMA(1,1,0).

Model R2 Residual standard error Requires logarithm Parsimony Diagnostic checking Fitted plot

ARIMA (0,1,1)

ARIMA (1,1,0)

(1
B)Zt ¼ (1
0.41B)at 0.810 0.013643 No 1 parameter Satisfactory Satisfactory

(1+0.3862B)(1
B)Zt ¼ at 0.806 0.013755 No 1 parameter Satisfactory Satisfactory

Inventory-Shipment Ratio Time Series Models

43

Next we check for squared residuals in the chosen model ARIMA(1,1,0). The EACF of the squared residuals indicates that an AR(2) model is appropriate. However, on modeling, we find that the t value of the AR(2) parameter is insignificant (t value ¼ 0.97) and hence we drop the same. Thus, we have an AR(1) model or ARCH(1) model. The t values of the constant term and the AR(1) terms are both significant (t values 5.20 and 4.24, respectively). The respective coefficients are 0.0001 and 0.2830. The R2 is 0.412 and the residual standard error is 0.223954E
03. The ACF/PACF of the residuals for this ARCH(1) model indicates no significant correlations. We also try an ARMA(1,1) model for the squared residuals as indicated by the EACF of the squared residuals. However, we find that the MA(1) parameter is insignificant (t value
0.75). Dropping this parameter would make the model behave like an ARCH(1) model. Thus the ARCH(1) model is: a2t ¼ 0:0001 þ 0:2830 a2t
1 þ wt

(8)

5.3. Total Products ARIMA(3,1,0). The t values for the ARIMA(3,1,0) are
3.00 for the AR(1) parameter and 2.75 for the AR(3) parameter. The respective coefficients are
0.2475 and 0.2218. The R2 is 0.944 and the residual standard error is 0.0166273. The histogram gives a symmetric bell-shaped curve and the probability plot gives a fairly straight line indicating normality. There is no significant correlation between the residuals as indicated by the EACF of the residuals. The model is thus: ð1 þ 0:2475B
0:2218B3 Þð1
BÞZ t ¼ at

(9)

We now check the fitted plot (Fig. 9) with the original series and find a reasonable enough fit. Next we check for squared residuals in our chosen model ARIMA(3,1,0). We find from the PACF of the squared residuals that an AR(1) model is appropriate. The EACF also suggests an ARMA(1,1). However, on modeling the ARMA(1,1) model, we find that the t values of both the MA(1) and AR(1) parameters are insignificant and negative (t ¼
1.54,
0.51) and hence we do not use the ARMA(1,1) model for modeling the squared residuals. Thus, we have an AR(1) model or ARCH(1) model. The t values of the constant term and the AR(1) terms are both significant (t values 4.90 and 2.77, respectively). The respective coefficients are 0.0002 and 0.2370. The R2 is 0.049 and the residual standard error is 0.407451E
03. The ACF/PACF/

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SUPRIYA MITRA

Fig. 9.

Fitted Plot Using ARIMA (3,1,0) Model.

EACF of the residuals for this ARCH(1) model indicates no significant correlations. Thus the ARCH(1) model is: a2t ¼ 0:0002 þ 0:2370a2t
1 þ wt

(10)

As shown in Table 3, both the durable and total product series have a very similar model structure ARIMA(3,1,0), whereas the non-durable products have an ARIMA(1,1,0) model structure. The results are intuitive and realistic. Durable products are prone to be slow-moving goods and have a higher I-S ratio when compared to non-durables. Thus, the series tends to depend on the value three periods back. In the case of non-durables, the series tends to depend on only the last period values. There is a subtle link between product inventory turns and the models found. Durable goods appear to be turning every three periods, whereas non-durables turn almost every month. This could also be attributed to the lower shelf life of nondurables and the higher shelf life of durable products. For total products, the model is similar to that of durable products. However, as shown by the coefficients of the total product model, the effect of previous periods is less for total products (compared to durable) owing to the averaging effect of durables’ and non-durables’ behavior. One striking similarity across all three product types is the modeling of the squared residuals by an ARCH(1) model. Thus, the effect of changes in variance appears to be similar in all three cases.

Model R2 Residual standard error Requires logarithm Parsimony Diagnostic checking Fitted plot Squared residuals

Best Model Comparison across Product Groups.

Durable ARIMA (3,1,0)

Non-Durable ARIMA (1,1,0)

Total Products ARIMA (3,1,0)

(1+0.264B
0.2782B3)(1
B)Zt ¼ at 0.946 0.027712 No AR(1) and AR(3) Satisfactory Satisfactory a2t ¼ 0:0006 þ 0:2018a2t
1 þ wt

(1+0.3862B)(1
B)Zt ¼ at 0.806 0.013755 No 1 parameter Satisfactory Satisfactory a2t ¼ 0:0001 þ 0:2830a2t
1 þ wt

(1+0.2475B
0.2218B3)(1
B)Zt ¼ at 0.944 0.016627 No 2 parameters (AR(1) and AR(3)) Satisfactory Satisfactory a2t ¼ 0:0002 þ 0:2370a2t
1 þ wt

Inventory-Shipment Ratio Time Series Models

Table 3.

45

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6. FORECAST COMPARISONS Durable products. Fig. 10 represents the forecasts for durables. The forecast oscillates in the initial forecasted periods (135–147) and finally reaches a steady state of 1.5533 after the 147th period. The accuracy of the forecast is measured by plotting the upper and lower confidence levels (95%). As shown in Table A1 (appendix), the forecast is quite accurate within the 72 standard error bounds (upper and lower confidence levels at 95%). Non-durable products. Fig. 11 represents the forecasts for non-durables. The forecast merely oscillates from time period 135 to 140 before reaching a steady state of 1.1044. Once again as shown in Table A2 (appendix), the forecast is quite accurate within the 72 standard error bounds (upper and lower confidence levels at 95%).

Durables forecast oscillations before steady state

Forecast

1.5570 1.5520 1.5470

7 14

5 14

3 14

1 14

13 9

13 7

13 5

1.5420 Time

Fig. 10.

Durables Forecast.

Non-durable forecast oscillations before steady state

Forecast

1.1050 1.1040 1.1030 1.1020 135

Fig. 11.

136

137 138 Time

139

Non-Durables Forecast.

140

Inventory-Shipment Ratio Time Series Models

47

Total products. Fig. 12 represents the forecasts for non-durables. The forecast merely oscillates from time period 135 to 143 before reaching a steady state of 1.3373. Once again as shown in Table A3 (appendix), the forecast is quite accurate within the 72 standard error bounds (upper and lower confidence levels at 95%). A question worth considering from a practitioner’s viewpoint is whether it is necessary to forecast the total products’ ratio using the total products’ time series or would a simple average of the durable and non-durable product forecast suffice for this purpose. This question is further motivated by the computer-intensive nature of ARIMA models (Fildes & Beard, 1992). Thus, we averaged the durable and non-durable forecasts and compared this simple average with the forecast of the total product series. As can be observed in Fig. 13, both these forecasts behave in a similar fashion. They oscillate between periods 135 and 143. However, the simple average reaches a steady state of 1.3288, whereas the total products reach a steady state of 1.3373. This difference stems from the fact that the total products’ ratio is not a simple average but takes the total durable plus non-durable inventory and divides by the total durable plus non-durable shipments. We also do a paired two sample for means t-test to determine whether the simple average time series forecast and the total products’ time series forecast come from distributions with equal population means. This t-test form does not assume that the variances of both populations are equal. However, it does assume normality. The diagnostic checking of the three forecasted models for durable, non-durable and total products has validated the normality assumptions for the three models. Hence, a simple average of the durable and non-durable product forecasts would not violate this normality assumption. Based on the t-test, both the p values (one tailed and two tailed) are less than Total products forecast oscillations before steady state

Forecast

1.3390 1.3370 1.3350 1.3330 135 136 137 138 139 140 141 142 143 144 145 Time

Fig. 12. Total Products Forecast.

48

SUPRIYA MITRA Forecast Comparison

Forecast

1.3390

1.3340

1.3290

14 5

14 3

14 1

13 9

13 7

13 5

1.3240 Time Simple Average Total Forecast

Fig. 13.

Forecast Comparison.

0.05 (for the 95% confidence level) suggesting that we reject the Null Hypothesis that the means are equal. Thus, a significant difference in means does exist between the two forecasts. The gap between the two forecasts is fairly constant with an average gap value of 0.008529. This average value tends to the steady-state difference in forecast between the two methods. We added this constant 0.008529 to the simple average forecast and ran a paired two sample for means t-test for the following two forecasts: (1) simple average forecast +0.008529, (2) total products’ forecast. Both the p values (one-tailed: 0.425857 and two-tailed: 0.851714) are more than 0.05 (for the 95% confidence level) suggesting that we cannot reject the Null Hypothesis that the means are equal. Thus, if the steady-state difference between the simple average forecast and total forecast is known, one can simply use a simple average (of durables and nondurables) forecast and add this steady-state difference to obtain a reasonable total products’ forecast.

7. DISCUSSION This study has important implications for the supply chain. By modeling an apparently trivial indicator such as I-S ratio, one has been able to explain how the results intuitively relate to supply chain dynamics. As we have shown in the present literature, only two papers, i.e., Rajagopalan and

Inventory-Shipment Ratio Time Series Models

49

Malhotra (2001) and Gaur et al. (2005), have attempted to empirically model and analyze this ratio. However, neither of the two authors analyzed the time series models for this ratio. In view of this, we could not compare the results with any other similar models existing in literature. The study shows that both the durable and total product series have a similar model structure – ARIMA(3,1,0), whereas the non-durable products have an ARIMA(1,1,0) model structure. We have also shown why this makes sense intuitively based on the nature of the products. One also finds that all three models have an ARCH(1) model which explains the variance changes over time. As discussed earlier, this variance stems from production adjustments made for the ‘‘balancing act.’’ It is intuitive that only adjustments made in the recent period would affect this variance. We have next tried to compare the forecast of a simple average of the durable and non-durable products with that of the total products. We find that even though both these forecasts behave in a similar fashion with respect to direction and magnitude of oscillations about the mean, they differ with respect to their absolute means. A paired t-test of sample means confirms that the means of the two forecasts are significantly different. However, a modified average that essentially adds a constant to the simple average makes the two mean differences statistically insignificant. Thus, a forecasting manager who has the need to forecast durable, non-durable and the total inventory may find it sufficient to use the time-series models for durables and non-durables alone for forecasting all the three product groups. We have used generalized data aggregated at the durable/non-durable product level and not a standard industry classification (SIC) or firm-specific level. However, one can extend and compare the study to the specific industry/firm level. Such an extension to a manufacturer would enable him to use the forecasts and trends in the ratio to decide on whether to speedup or cut down production. To a retailer, it would help in decisions regarding which item prices to discount or escalate. One can also envisage an extension to a multi-echelon inventory system in which a distributor stocks different classes of products at different levels of the echelon. An extension to such an inventory system could enable the distributor to decide on how much and what to stock at each level of the echelon.

ACKNOWLEDGMENT The author would like to thank Dr. Kasing Man, Associate Professor (Syracuse University) for his helpful comments.

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REFERENCES Albertson, K., & Aylen, J. (2003). Forecasting the behavior of manufacturing inventory. International Journal of Forecasting, 19(2), 299–312. Brown, R. G. (1959). Statistical forecasting for inventory control. New York: McGraw-Hill. Brownstein, V. (1984). The war on inventories is real this time. Fortune, 109(June 11), 20–25. Chen H., Frank M. Z., & Wu O. Q. (2003). The JIT Inventory Revolution. What actually happened to the inventories of American Companies between 1981 and 2000? Working Paper. Ching-Wu, C., & Zhang, G. P. (2003). A comparative study of linear and nonlinear models for aggregate retail sales forecasting. International Journal of Production Economics, 86, 217–231. Fildes, R., & Beard, C. (1992). Forecasting systems for production and inventory control. International Journal of Operations & Production Management, 12(5), 4–27. Frees, W. W., & Miller, T. W. (2004). Sales forecasting using longitudinal data models. International Journal of Forecasting, 20, 99–114. Gaur, V., Fisher, M. L., & Raman, A. (2005). An econometric analysis of inventory turnover performance in retail services. Management Science, 51(2), 181–194. Hanssens, D. M. (1998). Order forecasts, retail sales, and the marketing mix for consumer durables. Journal of Forecasting, 17, 327–346. Rajagopalan, S., & Malhotra, A. (2001). Have US manufacturing inventories really decreased? An empirical study. Manufacturing and Service Operations Management, 2(1), 14–24. Snyder, R., Koehler, A. B., & Ord, J. K. (2002). Forecasting for inventory control with exponential smoothing. International Journal of Forecasting, 18, 5–18. Steffens, P. R. (2001). An aggregate sales model for consumer durables incorporating a timevarying mean replacement age. Journal of Forecasting, 20, 63–77. The Daily. (2003). Monthly survey of manufacturing. Wednesday, January 22. http://www. statcan.ca/Daily/English/030122/d030122b.htm.

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APPENDIX Forecast comparisons are shown in Tables A1–A3. Table A1. Forecast Table for Durable Products. Time (Month) 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158

Durable Forecast

Std Error

Forecast+2  Std Error

Forecast–2  Std Error

1.5560 1.5433 1.5550 1.5536 1.5504 1.5545 1.5530 1.5525 1.5538 1.5530 1.5531 1.5534 1.5531 1.5532 1.5533 1.5532 1.5533 1.5533 1.5532 1.5533 1.5533 1.5532 1.5533 1.5533

0.0277 0.0344 0.0410 0.0505 0.0566 0.0628 0.0691 0.0743 0.0794 0.0843 0.0888 0.0931 0.0973 0.1012 0.1051 0.1088 0.1123 0.1158 0.1192 0.1224 0.1256 0.1287 0.1318 0.1347

1.6114 1.6121 1.6370 1.6546 1.6636 1.6801 1.6912 1.7011 1.7126 1.7216 1.7307 1.7396 1.7477 1.7556 1.7635 1.7708 1.7779 1.7849 1.7916 1.7981 1.8045 1.8106 1.8169 1.8227

1.5006 1.4745 1.4730 1.4526 1.4372 1.4289 1.4148 1.4039 1.3950 1.3844 1.3755 1.3672 1.3585 1.3508 1.3431 1.3356 1.3287 1.3217 1.3148 1.3085 1.3021 1.2958 1.2897 1.2839

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Table A2. Time (Month) 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158

Forecast Table for Non-Durable Products.

Non-Durables Forecast

Std Error

Forecast+2  Std Error

Forecast-2  Std Error

1.1023 1.1053 1.1041 1.1046 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044 1.1044

0.0138 0.0161 0.0193 0.0216 0.0238 0.0257 0.0276 0.0293 0.0310 0.0325 0.0340 0.0354 0.0368 0.0381 0.0394 0.0406 0.0418 0.0429 0.0441 0.0452 0.0463 0.0473 0.0483 0.0493

1.1299 1.1375 1.1427 1.1478 1.1520 1.1558 1.1596 1.1630 1.1664 1.1694 1.1724 1.1752 1.1780 1.1806 1.1832 1.1856 1.1880 1.1902 1.1926 1.1948 1.1970 1.1990 1.2010 1.2030

1.0747 1.0731 1.0655 1.0614 1.0568 1.0530 1.0492 1.0458 1.0424 1.0394 1.0364 1.0336 1.0308 1.0282 1.0256 1.0232 1.0208 1.0186 1.0162 1.0140 1.0118 1.0098 1.0078 1.0058

Inventory-Shipment Ratio Time Series Models

Table A3. Time (Month) 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158

53

Forecast Table for Total Products.

Total Forecast

Std Error

Forecast+2  Std Error

Forecast-2  Std Error

1.3373 1.3335 1.3389 1.3369 1.3366 1.3379 1.3371 1.3372 1.3375 1.3373 1.3373 1.3374 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373 1.3373

0.0166 0.0208 0.0248 0.0301 0.0337 0.0372 0.0407 0.0437 0.0466 0.0494 0.0520 0.0544 0.0568 0.0591 0.0612 0.0634 0.0654 0.0674 0.0693 0.0712 0.0730 0.0748 0.0765 0.0782

1.3705 1.3751 1.3885 1.3971 1.4040 1.4123 1.4185 1.4246 1.4307 1.4361 1.4413 1.4462 1.4509 1.4555 1.4597 1.4641 1.4681 1.4721 1.4759 1.4797 1.4833 1.4869 1.4903 1.4937

1.3041 1.2919 1.2893 1.2767 1.2692 1.2635 1.2557 1.2498 1.2443 1.2385 1.2333 1.2286 1.2237 1.2191 1.2149 1.2105 1.2065 1.2025 1.1987 1.1949 1.1913 1.1877 1.1843 1.1809

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AN APPLICATION OF CONFIRMATORY FACTOR ANALYSIS TO THE A PRIORI CLASSIFICATION OF FINANCIAL RATIOS Shaw K. Chen and Alan D. Olinsky ABSTRACT Financial ratios play an important role in financial forecasting. There has been much controversy over the appropriate classification of financial ratios. Exploratory factor analysis has often been used. Unfortunately, many of the resulting factors are difficult to interpret from a theoretical point of view, as well as difficult to evaluate from an empirical perspective. This paper makes use of confirmatory factor analysis (CFA) to measure the goodness of fit of the theoretical classification system of Koh and Killough. An earlier study in Finland reports poor results when using CFA with Lev’s traditional classification system. This paper, using the classification scheme by Koh and Killough, yields excellent results when examining various measures of fit for several U.S. industries. The superiority of Koh and Killough’s scheme is further substantiated by a comparison with the results obtained by using Lev’s classical scheme. In addition, applying this technique across several different industries

Advances in Business and Management Forecasting, Volume 4, 57–75 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04004-3

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permits the determination of the industry specificity of this classification model.

INTRODUCTION Analyzing financial ratios is usually the first step in the financial analysis of a company’s financial structure. An investor examines financial ratios in order to predict the future ability of the company in question to pay a good return on his/her investment. Banks and creditors concerned with the ability of a company to pay off its debts use financial ratios to predict bankruptcy. Managers use financial analysis both to predict the future and also as a beginning point for planning approaches to improve the future for their company. There are many ratios that are available for analyzing the financial condition of a company and financial ratios are typically grouped into various categories. Lev (1974) presented a classification pattern that has become traditional in the field. He divides financial ratios into four categories: profitability ratios, liquidity ratios, financial leverage (long-term solvency) ratios, and activity (efficiency) ratios. The profitability ratios allow for the evaluation of a firm’s operational performance. Liquidity ratios determine a firm’s ability to repay its short-term financial obligations. Financial leverage ratios indicate the ability of a firm to meet its principal and interest payments on their long-term obligations. The activity (efficiency) ratios determine the efficiency of a firm’s operations. However, it should be noted that this is only one such classification scheme. Others include those of Horrigan (1967), Courtis (1978), and Tamari (1978). These use different categories (constructs) and there is much debate as to which ratios best measure the constructs in such classification schemes. In recent years, several studies have attempted to empirically group these ratios by the use of exploratory factor analysis. Unfortunately, problems often arose. Assigning ratios to factors proved difficult as every financial ratio was allowed to load on each theoretical factor. Many times the factors that were statistically determined were not interpretable in financial analysis terms in spite of the authors’ best attempts. It was also difficult to assess how well the factor patterns represented the actual data. In addition, the assumption of no correlation between the resultant factors has often been assumed in the determination of such factors by the use of orthogonal rotations, when, in fact, it is known that there are correlations among the factors.

Confirmatory Factor Analysis of Financial Ratios

59

It would certainly seem more appropriate to select a theoretical grouping that makes intuitive sense and then apply a more rigorous procedure, confirmatory factor analysis (CFA), to this model. In this way, loadings of ratios on factors will be pre-determined by appropriate theory. If the specification of a loading does not make sense, it will not be allowed. In this way, a measure of the goodness of fit of an hypothesized model can be determined and compared to the goodness of fit of other hypothesized models. In other words, rather than loading every measured variable on every factor, we would just allow the model to specify loadings for measured variables (financial ratios) on factors (constructs) which make theoretical sense. We would also allow these constructs (factors) to be correlated as previous research indicates. Previous researchers used exploratory factor analysis to group the ratios. They then utilized regression models with the resulting factors to predict the financial condition of the firm. However, if our CFA approach yields models with a significantly better fit, it would be appropriate for future studies to simultaneously use these factors to predict the firm’s financial condition through the use of a structural model. In this way, the factor loadings and regression parameters for the model can be simultaneously estimated. This use of a predetermined theoretical rationale would seem to have advantages over the ‘‘fishing expedition’’ approach. The structure of this paper is organized as follows: In the next section, we summarize previous research. A data and modeling considerations section follows in which we discuss issues related to two popular theoretical classification systems of financial ratios. Past research by Kanto and Martikainen (1992) has utilized the scheme of Lev (1974) and applied a CFA to a sample of 26 non-Financial Finnish firms. This study will make use of the scheme of Koh and Killough (1990), but will, in addition, compare our results with those obtained using the scheme of Lev with large samples of U.S. companies. Since financial ratios tend to be different across industries, Gupta and Huefner (1972), CFAs will also be applied to the ratios in several industries. It will be interesting to note similarities and differences among industries in relation to classification schema and ratio loadings on the theoretical constructs. Finally, the results of the study and concluding remarks are explored in the last two sections.

LITERATURE REVIEW Horrigan (1967) and Courtis (1978) suggest that it is inconceivable that accounting data can be analyzed without transferring it into ratios in one

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way or another. This assertion is even more true today with the number of financial statements that are available and the number of managers and investors interested in examining them. Financial statement data are converted into ratio form for several reasons. Ratios control for size differences across firms as well as time. They facilitate the drawing of statistical inferences. Especially relevant to this paper, researchers examine observed regularities between financial ratios and they make use of this information in the prediction of variables of interest, such as the financial condition of a firm. In order to make effective use of financial ratios, it is necessary for the researcher to classify the large number of possible ratios into meaningful groups. Unfortunately, there are many such possible classification systems. Also, with the large number of possible ratios, one ratio must be selected that is a good surrogate of a particular dimension of the firm. In addition to the traditional scheme of Lev (1974), several other classification patterns have been developed by theoretical considerations (e.g., Horrigan, 1967; Courtis, 1978; Tamari, 1978; and Koh & Killough, 1990). Another method of classifying financial ratios has become popular as of late. This involves the selection of many financial ratios and empirically classifying them. This is usually accomplished through the use of exploratory factor analysis. A seminal study was done by Pinches, Mingo, and Carruthers (1973) in which 48 ratios of 221 firms were analyzed. Using exploratory factor analysis, in which every ratio was allowed to load on every factor, they developed seven empirical categories, namely, return on investment, capital intensiveness, inventory intensiveness, financial leverage, receivables intensiveness, short-term liquidity, and cash position. Johnson (1978) used principal components factor analysis and an orthogonal rotation to empirically classify ratios. Such orthogonal rotation, which is common, does not allow for the correlation of resulting factors when, in fact, it makes intuitive sense that many of these resulting factors would be correlated. In Johnson’s study, 9 empirical ratios were determined from 41 financial ratios. Similar methods were used by Laurent (1979), resulting in 10 factors, and by Gombola and Ketz (1983), resulting in 8 factors. It should be noted that, Yli-Olli and Virtanen (1985) made use of a non-orthogonal quartimin rotation and reported some correlation among factors, although the factor pattern resulting from an orthogonal varimax rotation did not differ significantly from that of the non-orthogonal rotation. Even so, it would seem reasonable intuitively and theoretically to expect an improved model by allowing correlations among factors Table 1 (Kanto & Martikainen, 1992).

Confirmatory Factor Analysis of Financial Ratios

Table 1.

61

Koh and Killough’s Classification Scheme for Financial Ratios.

Liquidity ratios determine a firm’s ability to repay its short-term financial obligations: (1) Current ratio ¼ current assets/current liabilities. This ratio indicates the extent to which the claims of short-term creditors are covered by assets expected to be converted to cash in the near future. (2) Net working capital to total assets ¼ (current assets—current liabilities)/Total assets. This ratio indicates the extent to which total assets of the firm are being utilized to provide working capital, which is the amount by which current assets exceed current liabilities. (3) Quick ratio ¼ current assets minus inventory, divided by current liabilities (also called the acid test ratio). This ratio is similar to the current ratio, but provides a better measure of overall liquidity when a firm’s inventory cannot easily be converted into cash. (4) Cash ratio ¼ cash/current liabilities. This ratio measures the extent to which a firm’s cash can satisfy the claims of short-term creditors. Profitability ratios allow for the evaluation of a firm’s operational performance. (5) Net profit margin ¼ net profit after taxes/sales. This ratio measures the percentage of each sales dollar remaining after all expenses, including taxes, have been removed (6) Return on investment ¼ net profit after taxes/total assets. This ratio measures the overall effectiveness of management in generating profits with its available assets. (7) Return on equity ¼ —net profit after taxes/stockholders’ equity. This ratio measures the return earned on the owners’ (both preferred and common stockholders’) investment. (8) Basic earning power ¼ earnings before interest and taxes/total assets. This ratio measures how much operating income a company is getting from its assets. (9) Retained earnings to total assets ¼ retained earnings/total assets. This ratio measures how much of a firm’s earnings that have been saved, rather than paid out as dividends, has been generated from its assets. Financial leverage ratios indicate the ability of a firm to meet its principal and interest payments on their long-term obligations: (10) Debt to total assets ¼ total debt/total assets. This ratio measures the percentage of total funds provided by creditors (11) Time interest earned ¼ earnings before interest and taxes/interest charges. This ratio measures the extent to which earnings can decline without resultant financial embarrassment to the firm because of an inability to meet annual interest costs. (12) Debt equity ratio ¼ total debt/stockholders’ equity. This ratio indicates the relationship between the total amount of funds provided by creditors and those provided by the firm’s owners. (13) Long-term debt to equity ¼ long-term debt/stockholders’ equity. This ratio indicates the relationship between the long-term funds provided by creditors and those provided by the firm’s owners. Activity (efficiency) ratios determine the efficiency of a firm’s operations: (14) Inventory turnover ¼ cost of goods sold/inventory. This ratio measures how quickly inventory is sold or, alternatively, how long inventory is held prior to sale.

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Table 1. (Continued ) (15) Total asset turnover ¼ sales/total assets. This ratio measures the efficiency with which the firm has been using its assets to generate sales. (16) Accounts receivable turnover ¼ sales on account/average accounts receivable. This ratio measures how many times a year the accounts receivable balance turns over, i.e., how many times old receivables are collected and replaced by new receivables. (17) Sales to net working capital ¼ sales/(current assets—current liabilities). This ratio measures the effectiveness of the firm in generating net sales from its net working capital, or its excess of current assets over current liabilities. Returns and market ratios relate a firm’s stock price to its earnings and book value per share: (18) Earnings per share ¼ —earnings available for common stockholders/no. of shares of common stock outstanding. This ratio represents the number of dollars earned on behalf of each outstanding share of common stock. (19) Dividends per share ¼ dividends paid/no. of shares of common stock outstanding. This ratio, which is of great interest to many investors, indicates the annual dollar amount of cash dividends paid per share of common stock. (20) Book value per share ¼ common stockholders’ equity/no. of shares outstanding. This ratio indicates the amount each share of stock would receive if the company were liquidated at the amounts reported in its balance sheet, i.e., it represents the amount of net assets owned by a single share. (21) Market value to book value of equity ¼ market price per share/book value per share. This ratio gives an indication of how investors regard the company. Companies with relatively high rates of return on equity generally sell at higher multiples of book value.

In 1990, Koh and Killough presented a classification system that was developed through a careful study of the literature (articles and textbooks) and seems to make theoretical sense. The latent constructs and their surrogate measured variables (in parentheses) using the classification system of Koh and Killough (1990) are as follows: liquidity ratios (current ratio, net working capital to total assets, quick ratio, and cash ratio), profitability ratios (net profit margin, return on investment, return on equity, basic earning power, and retained earnings to total assets), leverage ratios (debt to total assets, time interest earned, debt equity ratio, and long-term debt to equity), activity ratios (inventory turnover, total asset turnover, accounts receivable turnover, and sales to net working capital), and returns and market ratios (earnings per share, dividends per share, book value per share, and market value to book value of equity ratio). Liquidity ratios determine a firm’s ability to repay its short-term financial obligations. Profitability ratios allow for the evaluation of a firm’s operational performance. Financial leverage ratios indicate the ability of a firm to meet its principal and interest payments on their long-term obligations. Activity (efficiency) ratios

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63

determine the efficiency of a firm’s operations. Returns and market ratios relate a firm’s stock price to its earnings and book value per share. Details of Koh and Killough’s scheme and definitions of the ratios are presented in Table 1.

DATA AND MODELING CONSIDERATIONS The main advantage of CFA is that it allows us to empirically test specific hypothesized factor structures. In this study the null hypothesis is specified as follows: The underlying a priori classification scheme of Koh and Killough provides a good factor structure with the financial ratios adequately defining their predetermined constructs with little error. Using a CFA, the structure of three parameter matrices including factor loadings, factor intercorrelations, and unique variances is specified on an a priori basis. The method of maximum likelihood (ML) is widely preferred and was utilized for all models in this paper for estimation. In addition to the w2 significance test based on the ML fitting function, Root Mean Square Residual (RMSR) and Comparative Fit Index (CFI) are used in this paper. RMSR is the square root of the mean of the squared residuals between observed and estimated input matrices, and CFI assesses how well a model fits compared to a null model. As a basis for comparison with Koh and Killough’s scheme, we also used Lev’s (1974) classification system, as in Kanto and Martikainen (1992). The latent constructs and their surrogate measured variables (in parentheses) are the following: profitability ratios (return on assets, return on investments, and earnings to sales), liquidity ratios (quick ratio, current ratio, and defensive interval), financial leverage ratios (debt to equity, debt to sales, and equity to capital), and activity ratios (inventory turnover, accounts receivable turnover, and accounts payable turnover). Details of Lev’s scheme and definitions of the ratios are presented in Table 2. All data were generated from Compact Disclosure (1994) using a PC with the program and database on a CD-ROM. Compact Disclosure is a CDROM database that offers detailed financial and management data on over 12,000 public companies. Company data is extracted from company reports filed with the Securities and Exchange Commission (SEC). The information available includes corporate profiles, balance sheets, income statements, cash flow statements, and financial ratios. Quarterly data are given for the last 6–8 quarters, and most annual data goes back 5–7 years. Also available are earnings estimates and detailed ownership information, including data

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Table 2.

Lev’s Classification Scheme for Financial Ratios.

Profitability ratios allow for the evaluation of a firm’s operational performance. (1) Return on assets ¼ net profit after taxes/total assets. This ratio measures the overall effectiveness of management in generating profits with its available assets. (2) Return on investments ¼ - net profit after taxes/stockholders’ equity. This ratio measures the return earned on the owners’ (both preferred and common stockholders’) investment. (3) Earnings to sales ¼ earnings before interest and taxes/sales. This ratio measures the profitability of the firm’s sales. Liquidity ratios determine a firm’s ability to repay its short-term financial obligations: (4) Quick ratio ¼ current assets minus inventory, divided by current liabilities (also called the acid test ratio). This ratio is similar to the current ratio, but provides a better measure of overall liquidity when a firm’s inventory cannot easily be converted into cash. (5) Current ratio ¼ current assets/current liabilities. This ratio indicates the extent to which the claims of short-term creditors are covered by assets expected to be converted to cash in the near future. (6) Defensive interval ¼ - current assets—inventories/average daily expenditures to operations. While the quick ratio is a balance sheet/balance sheet ratio, the defensive interval measure is a balance sheet/income statement ratio. This measure indicates how well the liquid assets cover the expenditures needed to keep the operations running. Financial leverage ratios indicate the ability of a firm to meet its principal and interest payments on their long-term obligations: (7) Debt equity ratio ¼ total debt/stockholders’ equity. This ratio indicates the relationship between the total amount of funds provided by creditors and those provided by the firm’s owners. (8) Debt to sales ¼ total debt/sales. This ratio measures the ability of a firm’s sales to cover its debt obligations (9) Equity to capital ¼ —total equity/invested capital. This ratio indicates the portion of invested capital attributable to stockholders’ equity. Activity (efficiency) ratios determine the efficiency of a firm’s operations: (10) Inventory turnover ¼ cost of goods sold/inventory. This ratio measures how quickly inventory is sold or, alternatively, how long inventory is held prior to sale. (11) Accounts receivable turnover ¼ sales on account/average accounts receivable. This ratio measures how many times a year the accounts receivable balance turns over, i.e., how many times old receivables are collected and replaced by new receivables. (12) Accounts payable turnover ¼ purchases/average accounts payable. This ratio measures how many times a year the accounts payable balance turns over, i.e., how many times accounts payable are paid and replaced by new payables.

on institutional and insider ownership. The industries that were utilized in this study are as follows: (number of companies in parentheses): chemicals (493), communications (236), computers (592), drugs (260), durables (273), electronics (250), machinery (609), medical (287), metals (132), nondurables

Confirmatory Factor Analysis of Financial Ratios

65

(181), office equipment (250), and petroleum (313). These industries were selected because of their large sample sizes, which are required in structural modeling. It was expected that missing values might pose somewhat of a problem and would have to be addressed prior to conducting analyses. However, as it turned out, the data extracted were very complete with only a few cases that had to be deleted due to missing values. The CFA applied to all industries combined, as well as to each industry individually, involved 5 latent variables in accordance with Koh and Killough’s categories, namely, liquidity, profitability, financial leverage, activity (efficiency), and returns (market). The manifest variables are the 21 ratios listed above (see Table 1). The CFA was constructed so that the ratios (manifest variables) under each of the above headings were only allowed to load on the latent variables as indicated by this schema (see Fig. 1). In other words, the manifest variables current ratio, net working capital to total assets, quick ratio, and cash ratio were only allowed to load on the latent variable of liquidity, and so on. Next, the CFA applied to all industries D1

NPM

D2 D3

ROI

D4 D5 D6 D7 D8 D9

0.1428 0.2332 0.1507 0.4232 0.6733

ROE BEP RE/TA D/TA

0.3537 0.0671 0.0454 0.0196

TIE D/E LTD/E

D10

CURR

D11 D12 D13

NWC/A

D14 D15 D16 D17 D18 D19 D20 D21

Profit KSI1

QUICK

1.0016 0.7235 0.8527 0.2287

Leverage KSI2

Liquidity KSI3

CASHR INVT TATURN ARTURN

0.2362 0.0410 0.0927 0.0078

Activity KSI4

S/NWC EPS

0.4252 0.3134 0.6776 0.0213

DPS BV/S

Market KSI5

MV/BV

Fig. 1.

CFA model (Koh & Killough, 1990).

66

SHAW K. CHEN AND ALAN D. OLINSKY

combined, as well as to each industry individually, involving the 4 latent variables in accordance with Lev’s categories, namely, profitability, liquidity, financial leverage, and activity, was examined. The manifest variables are the same 12 ratios used for Lev’s scheme in the Finnish study (Kanto & Martikainen, 1992). The CFA was constructed so that the ratios (manifest variables) under each of the above headings were only allowed to load on the latent variables as indicated by Lev’s schema. In other words, the manifest variables return on assets, return on investments, and earnings to sales were only allowed to load on the latent variable of profitability, and so on. In this manner, a good fitting model would allow a meaningful interpretation and use of the pre-determined latent variables or factors. Unlike with orthogonal rotations, correlations among the factors were allowed and, to some degree, expected. The accompanying Figs. 1 and 2 show the set-up for these models (except for the residuals of the manifest variables and the correlations of the latent constructs). In running a successful model, there are certain requirements (Hatcher, 1994). Exogenous variables (latent variables or factors), but not endogenous D1

ROA

0.1957

D2

ROI

0.0272

D3

E/S

D4

D/E

D5

D/S

D6

E/C

D7

CURR

D8

DI

D9

Quick

D10

INVT

D11

APTURN

D12

ARTURN

1.1057

Profit KSI1

0.026 0.0730 0.0005

Leverage KSI2

0.8062 0.3055 1.1419

Liquidity KSI3

0.0201 0.0012 5.5996

Fig. 2.

Activity KSI4

CFA model (Lev, 1974).

Confirmatory Factor Analysis of Financial Ratios

67

(manifest) variables, are allowed to have covariances. However, a residual term must be identified for each endogenous (manifest) variable, while exogenous (latent) variables have no residual terms. For a recursive model, such as the model used in this study, covariances are not estimated for residual terms. In addition, variances should be estimated for every exogenous variable in the model (including residual terms). However, this will not apply to the factors (latent variables) in a CFA since there is a basic indeterminancy involving the variance of the factors (latent variables) and the factor loadings for the manifest variables that measure those factors. Since the factor is a hypothetical construct, it has no established metric or scale. If we do not resolve this problem of scale indeterminancy, it will be impossible to distinguish between the case in which the factor has a large variance and the factor loadings are small and the case in which the factor has a small variance and the factor loadings are large. One way of resolving this problem is to give all factors unit variances. Another is to assign a factor loading of 1 to the loading of one ratio on each latent variable (factor). In this situation, the option of assigning unit variances to all factors was selected. In regard to the selection of software, we used SAS’s PROC CALIS for the analyses presented in this paper. It should be noted that the LINEQS option of PROC CALIS, which is essentially equivalent to the statement format used in EQS, was used. After analyzing the models using the rather robust ML estimation, we examined parameters of interest, including factor loadings, variances of factors, and covariances among factors. In the CFA models, we were particularly interested in examining the indicator loadings for statistical significance and assessing the construct’s reliability and variance extracted. Overall goodness of fit measures were also carefully examined. Specifically, we looked at the w2 value as a measure of the overall fit of the model to the data. However, since this measure is dependent on sample size, it is useful to examine other measures, such as the CFI and the Root Mean Square Standardized Residuals.

RESULTS Using the method of ML, a CFA was first conducted using all data from all industries as a single sample. Other CFAs were then conducted for each of the component industries: chemical, communications, computers, drugs, durables, electronics, machinery, medical, metals, nondurables, office equipment, and petroleum. This procedure was first examined using Koh and

68

SHAW K. CHEN AND ALAN D. OLINSKY

Killough’s financial ratio classification system and then using Lev’s system for comparison. In assessing the goodness of fit of the CFA to the data, we first examine the results of the w2 test. For all industries selected as well as in each industry the w2 statistic was significantly large enough to reject the null hypothesis of a good fit. In fact, the p value for each test was less than 0.0001. However, with large samples and real-world data, the w2 statistic tends to be significant even when the model provides a good fit (Bentler & Bonnett, 1980; James, Mulaik, & Brett, 1982). Therefore, it is only appropriate to look at other measures of goodness of fit. Various measures of fit used in this study are presented in Table 3. They are the w2 and its degrees of freedom (df), Bentler’s (1990) Comparative Fit Index (CFI), and RMSRs. The CFI is a non-statistical overall goodness-of-fit measure that ranges in value from 0 for a poor fit to 1 for a perfect fit. It represents a comparison between the estimated model and a null or independence model. It is been suggested that values over 0.9 on the CFI would indicate significant acceptable fit (Bentler, 1993). We would like to remind the readers that these indices do not lend themselves to tests of significance as no theoretical distributions can be derived. Nevertheless, we can use the recommended guidelines in assessing goodnessof-fit. Reviewing Table 3, these measures of goodness of fit seem very promising. In examining the CFI, it is clear that the CFA for all companies in all industries selected indicates a good fit (X0.9). In addition, looking at specific industries, the chemical, drug, medical, and metals industries indicate a good fit (X0.9). It also appears that the CFAs for the communications industry, the computer industry, and possibly even the nondurables industry seem to give a reasonably good fit (X0.8) (Tables 3 and 4). In reviewing the significance tests for factor loadings, most, if not all, of the loadings are significant at the 0.05 level of significance. (Fig. 1 includes standardized factor loadings when all industries are included in analysis.) It is difficult to be more precise as there are some differences depending on the industry selected. However, it would seem inappropriate to modify the model based on the significance of loadings that are industry specific. Since a factor loading is equivalent to the coefficient on a path from the factor to the manifest (endogenous) variable, a significant loading gives some evidence that the manifest variable is doing a good job of measuring the latent construct and should be retained. The covariance matrix when all industries are combined is presented in Table 5. The ratios were subjected to descriptive data analysis to examine if underlying distribution assumptions were met. Of particular concern, a lack of multivariate normality can substantially increase the value of the w2 fitness

Confirmatory Factor Analysis of Financial Ratios

69

Table 3. Goodness-of-Fit results for Confirmatory Factor Analysis Based on Categorization Scheme of Koh and Killough. Industry All Industries Chemical Communications Computers Drugs Durables Electronics Machinery Medical Metals Nondurables Office Equipment Petroleum

Table 4.

No. of Companies

w2

df

Bentler’s CFI

RMSR

3876 493 236 592 260 273 250 609 287 132 181 250 313

2918.22 337.59 439.42 1003.60 326.47 1756.69 1237.77 2876.32 339.61 262.95 576.66 1700.32 1461.27

179 179 179 179 179 179 179 179 179 179 179 179 179

0.93 0.97 0.89 0.86 0.95 0.53 0.61 0.73 0.91 0.90 0.80 0.72 0.68

123.71 208.01 44.47 194.25 352.14 12.53 14.22 43.60 78.64 2.97 49.73 513.14 54.46

Goodness of Fit Results for Confirmatory Factor Analysis Based on Categorization Scheme of Lev.

Industry All Industries Chemical Communications Computers Drugs Durables Electronics Machinery Medical Metals Nondurables Office Equipment Petroleum

No. of Companies

w2

Df

Bentler’s CFI

RMSR

4162 493 236 592 260 273 250 609 287 132 181 250 313

62279.65 15613.14 4790.26 9889.60 9881.04 2926.39 3440.20 9603.96 3062.53 1486.71 3632.48 2679.82 8503.16

48 48 48 48 48 48 48 48 48 48 48 48 48

0.39 0.27 0.37 0.50 0.24 0.53 0.46 0.30 0.51 0.47 0.44 0.53 0.36

378.78 659.07 47.83 284.96 908.29 78.54 1.87 306.11 265.42 2.02 808.88h 8.40 480.44

measure. It can also result in an upward bias in the critical values for determining the significance of coefficients (Hair, Anderson, Tatham, & Black, 1992). If necessary, log transformations can usually be applied to data to improve normality, remove outliers, and reduce heteroskedasticity. However, since many of the ratios result in negative values, it was not possible to take logarithms in spite of the fact that there was strong evidence

CURR 2.10 0.19 1.26 0.51 0.03 0.01 1.11 0.08
0.13 1043.41
1.26
0.28
8.08
0.03
4.63 8.44
0.20
0.20
1.21
0.05

CURR NWCTOTA QUICK CASHRAT NETPFTMG ROI ROE RETOTA TDTOTA

RETOTA 0.080 0.021 0.028 -0.016 0.092 0.030 2.205 0.581
0.078

Overall Covariance Matrix (Koh and Killough’s Scheme).

NWCTOTA 0.188 0.035 0.105 0.030 0.009 0.003 0.279 0.021
0.021 379.720
0.204
0.045
1.604 0.031
0.705 1.875
0.030
0.036
0.164
0.022 TDTOTA
0.135
0.021
0.092
0.029 0.006
0.006
0.354
0.078 0.056

QUICK 1.26 0.10 0.99 0.36 0.01 0.00 0.99 0.03
0.09 1021.15
1.26
0.26
1.47
0.12
6.01 1.93
0.07
0.11
0.66
0.05 TIMESINT 1043.408 379.7201 1021.147 256.2628 647.9511 230.8121 2324.691 880.3497
351.57

CASHRAT 0.506 0.030 0.361 0.294
0.024
0.003 0.226
0.016
0.029 256.263
0.290
0.044
0.089
0.068
0.166
0.317
0.061
0.045
0.608 0.010 DEBTEQ
1.26
0.20
1.26
0.29
0.16
0.59
573.88
1.42 0.44

NETPFTMG 0.033 0.009 0.007
0.024 1.762 0.036 0.843 0.092 0.006 647.951
0.157
0.052
0.638 0.057 0.390
1.446 0.248 0.040 0.815
0.130 LTDEBTEQ
0.28
0.05
0.26
0.04
0.05
0.17
159.25
0.33 0.12

ROI 0.006 0.003 0.003
0.003 0.036 0.016 0.837 0.030
0.006 230.812
0.594
0.174
0.127 0.003
0.398
0.509 0.130 0.013 0.178
0.206 INVTURN
8.1
1.6
1.5
0.1
0.6
0.1 13.9
0.5 0.7

ROE 1.11 0.28 0.99 0.23 0.84 0.84 760.25 2.20
0.35 2324.69
573.88
159.25 13.93
0.66
193.61
125.64 3.35 0.53 12.70
310.56 TOTATURN
0.026 0.031
0.120
0.068 0.057 0.003
0.658
0.027 0.026

SHAW K. CHEN AND ALAN D. OLINSKY

CURR NWCTOTA QUICK CASHRAT NETPFTMG ROI ROE RETOTA TDTOTA TIMESINT DEBTEQ LTDEBTEQ INVTURN TOTATURN ARTURN SALTONWC EPS DIVPERSH BVPERSH MVTOBVEQ

70

Table 5.

CURR NWCTOTA QUICK CASHRAT NETPFTMG ROI ROE RETOTA TDTOTA TIMESINT DEBTEQ LTDEBTEQ INVTURN TOTATURN ARTURN SALTONWC EPS

ARTURN
4.6
0.7
6.0
0.2 0.4
0.4
193.6
0.7 0.7
12745.8 316.4 89.9 97.1 6.1 1474.5 312.2
3.9

SALTONWC 8 2 2
0
1
1
126
5
1
45543 218 76
132 24 312 456188
10

DIVPERSH BVPERSH MVTOBVEQ

CURR
2.3
36.6 157.8

NWCTOTA
11 56 134


351.570 0.436 0.119 0.747 0.026 0.662
0.603
0.049
0.000
0.539 0.016

Note: Determinant ¼ 3.64E24 (ln ¼ 56.554).

1.0097E9
3267.16
1216.08
15549.7 434.7576
12745.8
45543 1975.576
56.2964 2601.439 750.4353 EPS
0.20
0.03
0.07
0.06 0.25 0.13 3.35 0.31
0.05 1975.58
4.71
1.58
2.86
0.11
3.86
10.41 6.22 QUICK 0.58 11.75
1.75


3267.16 1008.37 278.19
3.30 1.72 316.41 218.39
4.71
0.35
8.37 515.97


1216.08 278.19 84.43 4.87 0.36 89.86 76.29
1.58
0.11
1.75 141.50

DIVPERSH
0.2003
0.0360
0.1064
0.0453 0.0395 0.0130 0.5290 0.0948
0.0003
56.2964
0.3525
0.1074
0.9675
0.1184
2.2725
11.3541 0.5806

BVPERSH
1.21
0.16
0.66
0.61 0.81 0.18 12.70 2.23
0.54 2601.44
8.37
1.75
16.23
1.38
36.63 56.15 11.75

CASHRAT 0.4438 4.0774
0.1097

NEFPFTMG 4.08 130.34
7.95


15549.7
3.3 4.9 1174.2 1.6 97.1
131.9
2.9
1.0
16.2
3.3

434.758 1.717 0.357 1.618 0.672 6.119 24.334
0.109
0.118
1.383 0.708

MVTOBVEQ
0.047
0.022
0.053 0.010
0.130
0.206
310.565
0.038 0.016 750.435 515.973 141.500
3.295 0.708 157.814 134.337
1.747 ROI
0.110
7.948 317.987

ROE

71

880.350
1.419
0.329
0.527
0.027
0.735
5.199 0.308 0.095 2.230
0.038

Confirmatory Factor Analysis of Financial Ratios

TIMESINT DEBTEQ LTDEBTEQ INVTURN TOTATURN ARTURN SALTONWC EPS DIVPERSH BVPERSH MVTOBVEQ

72

SHAW K. CHEN AND ALAN D. OLINSKY

of kurtosis in working with financial ratios. However, several robustness studies (e.g., Boomsma, 1983; Harlow, 1985) report that ML estimation appears robust with large samples (e.g., n X 400) when skewness and kurtosis are present. The use of alternative transformations may be the subject of further analysis. As another indication of fit, we examined the normalized residuals with the expectation of them being centered on zero, symmetrical, and containing no or few large residuals. Hatcher (1994) suggests that normalized residuals, thus showing a lack of significance at the 0.05 level (i.e., z ¼ 1.96), should generally not exceed 2. A SAS plot of the distribution of normalized residuals for the chemical industry (using Koh and Killough’s classification system) was examined. The normalized residuals satisfy the requirements of being centered at zero and are symmetrical. Although there are some normalized residuals that exceed an absolute value of 2, the majority of the residuals are in the acceptable range. In assessing the goodness of fit of the CFA to the data when using Lev’s system, we also examined various measures of fit for all industries selected as well as in each industry. (Fig. 2 includes standardized factor loadings when all industries are included in analysis.) Again, all w2 tests were significant at the 0.0001 level. However, as in Koh and Killough’s system, James et al (1982) allows us to pursue other measures of goodness of fit. Table 4 contains measures of goodness of fit for this model. Unfortunately, as in the Finnish study (Kanto & Martikainen, 1992), none of the CFAs could be considered a good fit to the industry data. When compared to those indices in Table 3 for Koh and Killough’s scheme, we observed a pattern of consistently lower levels of fit when using Lev’s scheme. (The covariance matrix when all industries are combined is presented in Table 6.) It would, therefore, not be appropriate to pursue additional interpretations with this model. To summarize the results, we have built upon the same technique as Kanto & Martikainen (1992) in their study of companies in Finland. We also had very little success in using the CFA with Lev’s classification scheme. However, we had great success in applying confirmatory factor analyses with the more recently developed classification system of Koh and Killough. This would seem to be a major breakthrough in testing a hypothesized classification model through confirmatory factor analytic techniques and achieving an excellent fit with such a system of financial ratios. We were also able to successfully compare two classification systems, namely that of Lev and Koh and Killough, and found that the system of Koh and Killough appears to be superior to that of Lev in its ability to model underlying constructs that adequately define their underlying manifest variables (financial ratios).

Confirmatory Factor Analysis of Financial Ratios

Overall Covariance Matrix (Lev’s Scheme).

Table 6. CURR CURR QUICK DEFINT ROILEV ROA EARNSAL DEBTEQ INVTURN ARTURN SALTONWC DEBTSAL EQUICAP

3.5767494 2.4995315 0.5937124 0.9941858 0.0149436 0.0282009
2.2947121
6.6624421
5.6139330 8.1560979
0.2556192 0.0714480

DEBTEQ CURR
2.2947 QUICK
1.8628 DEFINT
0.3869 ROILEV
401.7160 ROA
0.5409 EARNSAL
0.0660 DEBTEQ 1465.5101 INVTURN
5.0719 ARTURN 273.6895 SALTONWC 181.0084 DEBTSAL 0.8844 EQUICAP
0.9853

73

QUICK

DEFINT

ROILEV

2.4995315 2.0617770 0.6136297 0.8995017 0.0106810 0.0167257
1.8627875
0.1071352
6.6055826 2.2912426
0.0934962 0.1179314

0.5937124 0.6136297 1.3911363 0.2304987 0.0014368 0.0073950
0.3868611
1.9334245
2.9942086
0.9554152 0.0986881
0.1238880

0.99419 0.89950 0.23050 594.89921 0.74470 0.78953
401.71596 10.45862
155.27167
96.78484
0.07734 0.12964

INVTURN
6.6624
0.1071
1.9334 10.4586
0.1079
0.4943
5.0719 1085.3904 120.9181
120.5160 1.8939 6.7203

ARTURN
5.6139
6.6056
2.9942
155.2717
0.2714 0.0879 273.6895 120.9181 1792.8056 280.5770
2.3593 5.3665

ROA

EARNSAL

0.01494363 0.01068095 0.00143676 0.74469625 0.02259936 0.03849279
0.54088246
0.10787312
0.27141042
0.38980084
0.02918854 0.09735379

0.0282009 0.0167257 0.0073950 0.7895256 0.0384928 1.3820041
0.0660440
0.4943361 0.0879182
1.9544221
1.2767622 0.2470621

SALTONWC DEBTSAL 8.16
0.255619 2.29
0.093496
0.96 0.098688
96.78
0.077338
0.39
0.029189
1.95
1.276762 181.01 0.884409
120.52 1.893865 280.58
2.359303 403775.69
14.536622
14.54 1.588174
3.72
0.110352

EQUICAP 0.07145 0.11793
0.12389 0.12964 0.09735 0.24706
0.98529 6.72028 5.36653
3.71919
0.11035 355.03639

Note: Determinant ¼ 2.27E18 (ln ¼ 42.266).

In addition, we were able to note differences among industries with Koh and Killough’s classification system. Although there were differences among the various industries, we did find an excellent fit for the chemical, drug, medical, and metals industries and reasonably good fits for the communications and computer industries, and possibly even for the nondurables industry. This would tend to suggest that such classification systems might very well be industry specific. It may, therefore, become necessary to use different classification systems when modeling different industries.

CONCLUDING REMARKS This study has successfully demonstrated the value of using CFA in the classification of financial ratios. Koh and Killough’s scheme of classification

74

SHAW K. CHEN AND ALAN D. OLINSKY

provides far superior levels of fit when compared to Lev’s scheme of classification. Most of the CFIs for Koh and Killough’s scheme ranged from 0.7 to values close to 1, whereas most of the CFIs for Lev’s scheme ranged from 0.3 to 0.5. Although we found that the results were very promising when all industries were examined, we found different patterns when individual industries were studied. This finding provides support to the hypothesis that classification schemes for financial ratios may well be industry specific. Obviously, our finding lays the foundation for other promising future research. Some of the ways in which this study can be extended in the future include data transformations, an experimental design, and the use of a structural model for the prediction of the financial condition of a firm. Future research can concentrate on ways of transforming ratios, without the use of logarithms, to satisfy the assumption of multivariate normality. Also, different classification systems could be tested and multi-sample techniques could be used to better compare results across industries. In addition, further research could involve expanding the models to include various points in time (e.g., 5–10 years apart). This would allow us to examine the ability of each financial construct to predict its equivalent in a successive time period and thus would allow the testing of the stability of our financial ratios over time. It might also prove fruitful to proceed to a full structural model for predicting the financial condition of a firm. This would include our CFA of the measurement model for the constructs as well as path analysis (a regression model) which uses our factors (financial constructs) to predict the financial condition of firms. This would have many advantages over the use of exploratory factor analysis and the subsequent regression on factors that may or may not make sense. In this case, we would use theoretical constructs that make sense and proceed to simultaneously estimate all parameters in this complete structural model. The dependent variable would be a surrogate for the financial condition of the firm: Standard & Poors Bond Ratings. If possible, other measures of the financial condition of companies should be obtained to allow for more than one measured variable loading on a surrogate factor for financial condition. This provides more reliable and valid assessment of the financial condition. In summary, our application of CFA and, more generally, structural equation modeling to the a priori classification of financial ratios demonstrates the possibilities for this relatively new multivariate technique in the more quantitative side of business.

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REFERENCES Bentler, P. B. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246. Bentler, P. B. (1993). EQS structural equations program manual. Los Angeles, CA: BMDP Statistical Software, Inc. Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness-of-fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606. Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation) against small sample size and nonnormality. Unpublished doctoral dissertation, University of Eroningen, The Netherlands. Compact Disclosure. (1994). CD-ROM database which offers access to extracts of reports filed by publically owned companies with the Securities and Exchange Commission. Disclosure Incorporated, Bethesda, MD. Courtis, J. (1978). Modeling a financial ratio category framework. Journal of Business Finance and Accounting, 5(4), 371–386. Gombola, M. J., & Ketz, E. J. (1983). A note on cash flow and classification patterns of financial ratios. Accounting Review, 63(1), 105–114. Gupta, M., & Huefner, R. (1972). A cluster analysis study of financial ratios and industry characteristics. Journal of Accounting Research, 10(1), 77–95. Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1992). Multivariate data analysis with readings (3rd ed.). New York: Macmillan Publishing Co. Harlow, L. L. (1985). Behavior of some elliptical theory estimators with nonnormal data in a covariance structural framework: A Monte Carlo study. Unpublished doctoral dissertation. University of California, Los Angeles. Hatcher, L. (1994). A step-by-step approach to using the SAS system for factor analysis and structural equation modeling. Cary, NC: SAS Institute Inc. Horrigan, J. O. (1967). An evaluation of financial ratio analysis. Unpublished doctoral dissertation, University of Chicago. James, L. R., Mulaik, S. A., & Brett, J. M. (1982). Causal analysis. Beverly Hills, CA: Sage Publications. Johnson, W. B. (1978). The cross-sectional stability of financial patterns. Journal of Business Finance and Accounting, 5(2), 207–214. Kanto, A. J., & Martikainen, T. (1992). A test on a priori financial characteristics of the firm. European Journal of Operational Research, 57(1), 13–23. Koh, H. C., & Killough, L. N. (1990). The use of multiple discriminant analysis in the assessment of the going-concern status of an audit client. Journal of Business Finance and Accounting, 17(2), 179–192. Laurent, C. R. (1979). Improving the efficiency and effectiveness of financial ratio analysis. Journal of Business Finance and Accounting, 6(3), 401–413. Lev, B. (1974). Financial statement analysis. Englewood Cliffs, NJ: Prentice-Hall. Pinches, G. E., Mingo, K. A., & Caruthers, J. K. (1973). The stability of financial patterns in industrial organizations. Journal of Finance, 28(3), 389–396. Tamari, M. (1978). Financial ratio analysis and prediction. London: Paul Elek. Yli-Olli, P., & Virtanen, I. (1985). Modeling a financial ratio system on the economy-wide level. Acta Wasaensia, 21.

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BANK RATING CHANGE PREDICTIONS: ALTERNATIVE FORECASTING MODELS David T. Cadden and Vincent Driscoll INTRODUCTION The phenomena of corporate failure is not, unfortunately, infrequent nor of minor economic consequence. Every year thousands of firms fail with a concomitant total liability measuring in billions of dollars. It is obvious that such a crucial issue in corporate finance warrants careful investigation. This has lead to a large literature devoted to predicting corporate and bank failure. During the last three decades new statistical approaches have been applied as part of an overall strategy to better predict failure. Critical as this search is, there is a subset of prediction that has received significantly less attention – changes in the overall financial position of an organization. Bank failure studies, generally, follow a similar research design; however, a bank’s health can be categorized in more than two groups. Federal Depositors Insurance Corporation’s examiners can place a bank in one of the five groups. This is done on the basis of hard accounting data and evaluator judgment. Sheshunoff’s Bank Quarterly: Ratings and Analysis has a rating system that has 10 categories; it is based on statistical data of financial ratios, but little or no research has been done with respect to classification of financial institutions beyond the basic binary classification of failed and

Advances in Business and Management Forecasting, Volume 4, 77–91 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04005-5

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non-failed. This paper presents some preliminary research on utilizing two alternative statistical tools – discriminant analysis and neural networks – to predict the change in the Sheshunoff’s bank rating. We present the results for competing models to predict changes in the ratings of 47 Connecticut savings banks and discuss the implications for continued research.

BACKGROUND Contemporary bankruptcy research examines accounting data for matchedpairs, failed and non-failed, of firms. It employs statistical tests to detect the accounting data that best discriminates between failed and non-failed firms. One of the most commonly used statistical tests is discriminant analysis. Recently, neural networks – specifically, back-propagation neural networks – have been added to the statistical techniques in bankruptcy studies. Neural networks represent an approach in artificial intelligence research and are often referred to as a connectionist approach. The term connectionist comes from its use of the brain, and its myriad interconnections, as the basis for information processing. The fundamental unit in the brain is the neuron, which in connectionist terms finds its analog in the processing element or node. Just as neurons are interconnected by dendrites and axons so nodes in neural networks are interconnected. The phenomenal number of interconnections of neurons through the dendrites and axons in the human brain provides for a number of characteristics that are desirable in a computer system. These include fault tolerance, learning by example, adaptability, and pattern recognition. Neural networks can be of different architectural forms; however, one of the most common forms possesses an input layer and an output layer and sometimes one or more hidden layers. In addition to the architecture, their respective learning algorithm can distinguish neural networks. Further, each learning algorithm may be characterized by a wide number of parameters. Once a particular learning algorithm and parameter values are initially selected, the neural network is repeatedly exposed to a training data set. This may involve thousands of iterations until the system converges to a solution. This research examines two issues (1) how effective – vis-a`-vis standard statistical methods – could back-propagation and other neural network paradigms are in discriminating amongst more than two-group classifications; and (2) what is the accuracy of these classifications for future periods.

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LITERATURE REVIEW The premiere multivariate study of bankruptcy was Altman’s (1968) paper. It has become the benchmark against which most other bankruptcy studies are measured. Altman utilized the statistical technique of multiple discriminant analysis and found that bankruptcy could be explained quite completely by using a combination of five (selected from an original list of 22) financial ratios. Linear Discriminant Analysis (LDA) is a statistical technique, developed by Fisher (1936), which is used to classify an observation into one of several a priori groupings dependent on the observation’s individual characteristics. It is used primarily to classify and make predictions in problems where the categories are dependent on the observation’s individual characteristics. It is used primarily to classify and/or make predictions in problems where the dependent variable appears in qualitative form, e.g. male or female, bankrupt or non-bankrupt. After group classifications have been established, LDA attempts to derive a linear combination of these characteristics that ‘‘best’’ discriminates between the groups. LDA requires certain assumptions about the data: (1) each group follows a multivariate normal distribution; (2) the variance–covariance matrices of the groups are equal; and (3) the prior probabilities are known. Altman (1977) applied quadratic discriminant analysis to predicting performance in the Savings and Loan industry. This study differed from the classic corporate failure studies in two important points (1) it utilized three classification groups – banks with serious problems, temporary problems, and no problems; and (2) the use of trends of accounting ratios. Pettway and Sinkey (1980) proposed using both accounting data and market information as a means of an early warning system for banks with problems. There have been several attempts to apply expert systems methodologies to the bankruptcy problem and the allied problem of creditor evaluation. Elmer and Borowski (1988) developed an expert system to evaluate the financial health of Savings and Loan (S&L) institutions and predict their failure. Their expert system took publicly available information and produced a single index to measure an institution’s health. The rules were derived from the Federal Home Loan Bank Board (FHLBB) Examination Objectives and Procedures Manual and individuals’ expertise. This system worked with five ratios drawn from CAMEL framework – CAMEL being an acronym for (C)apital, (A)ssets, (M)anagement, (E)arnings, and (L)iquidity. It, however, excluded the (M)anagement component since that was a subjective, quality measure. The S&L industry is seen as not being

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homogeneous; this system had the ability to identify thrifts with unusual characteristics and thus improve its own reliability. The system’s single index is a weighted average of scores for the four characteristics – (C)apital, (A)ssets, (E)arnings, and (L)iquidity measures. The relative importance (the weights) for the four was derived from a poll of S&L presidents. Ten ratios are used to generate the scores for the four characteristics. The production rules treat these ratios either in the context of peer group comparison or with respect to fixed cutoff values. These rules provide sufficient flexibility to allow for changes in the industry. The authors tested this expert system’s predictive capability against a logit analysis based on an Altman (1977) study of S&Ls and another study. Their test used 60 matched pairs of failed and non-failed S&Ls from 1986. On this data set, the two statistical approaches outperformed the expert system by a very slight margin. A second test was conducted. Here the models were used to predict failure 1–6, 7–12, and 13–18 months prior to failure. In the earliest time period prior to failure the expert system was as good a classifier as the Altman model and better than the second statistical model. As one moved further away from the failure date, the correct classification rates for the three models declined; however, the expert system’s declined at a more modest rate. For the period 13–18 months prior to failure, the expert system correctly classified nearly 62% of the sample while Altman’s model’s value was approximately 48% and the second statistical model’s value was 33%. The authors conclude that the expert system approach appears to be robust and that correlation studies have difficulty adapting to new circumstances and are subject to error due to samples from which they are derived. A study that compared the performance of a neural network model with a logit regression was Salchenberger, Cinar, and Lash’s (1993) study of S&L Thrift failures. As with the case of corporate failure, there have been numerous studies of thrift institutions. These studies have used LDA, quadratic discriminant analysis, logit, and probit. Salchenberger et al. drew upon these prior studies to select an initial list of 29 financial variables. These were reduced down to five by means of stepwise regression. The training data consisted of 100 failed S&Ls, and 100 non-failed S&Ls. These were matched by both asset size and geographical region. They used, in effect, four-holdout sample, which consisted of matched pairs of thrifts. The first three samples consisted of failed and non-failed institutions 6, 12, and 18 months prior to failure. The total sample sizes were 116, 94, and 48, respectively. The fourth sample consisted of 75 failures matched with 329 non-failures. This fourth holdout sample was designed to more accurately represent the proportions of failed to non-failed institutions. They used a back propagation neural

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network with one hidden layer that had three nodes. In addition, a logit model was run on the initial training set. For both the logit and neural network models, two cutoff points (0.5 and 0.2) were used. As previously mentioned, the lower cutoff point reduces the chance of a Type I error. For the training set and the 18-month holdout sample, the neural network statistically outperformed the logit model in forecasting failures. For the training set, the neural network model was also more robust when it came to lowering the cutoff point to 0.2, and misclassifying fewer numbers of nonfailures. For the fourth holdout sample, the neural network, again, was statistically superior in classifying failed institutions and non-failed institutions when the cutoff point is equal to 0.2. The authors conclude that the neural network model yield more useful results than the logit model, particularly when the data is reflective of the total population of thrift institutions. Tam and Kiang (1992) have published two studies in which they applied neural networks to the study of commercial bank failure. The latter is perhaps the most comprehensive study in comparing neural network methodology to alternative approaches. In it they compare a neural network model’s performance with a linear discriminant model, a logistic model, the ID3 algorithm, and the k Nearest Neighbor (kNN) approach. This last approach is a non-parametric classification technique. It does not have any requirement for functional form nor does it assume normality in the distributions. The data were collected for the period 1985–1987 and consisted of 59 failed and 59 non-failed banks. They were matched not only on the basis of assets but also on charter type and number of branches. Nineteen ratios, drawn from prior studies, were selected for use in this research. Although 15 of the 19 ratios were not normal in their distribution, they were used ‘‘as is’’ since attempts at transformation did not produce normal distributions. Tam and Kiang (1992) used two back propagation architectures – one with no hidden layer and another with one hidden layer that contained 10 nodes. They modified the learning function to consider both the differing probabilities for failure and non-failure and the differing costs of misclassification. The study considered two probabilities for failure and eight misclassification costs. The models were tested on data one and two years prior to failure. For the training set, one year prior to failure the neural network with the hidden layer outperformed all other approaches; however, two years prior to failure discriminant analysis had the lowest total misclassification rate followed by the hidden layer neural network. Both neural networks had lower substitution risks (expected cost of misclassification) than the discriminant analysis across all combinations. The neural network with the hidden layer tended to outperform the two-layer neural network.

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The models were tested on a holdout sample of 44-paired failed and nonfailed banks. The neural net with the hidden layer the best overall classifier one year prior to failure, while the logit model scored best two years prior to failure with the hidden layer neural network came in second. Since the results for the models for the training and holdout sets were inconsistent with regard to relative accuracy for the two time periods, Tam and Kiang (1992) used a jackknife method of estimation. Utilizing this method, the hidden layer neural network produced smaller total misclassification rates, for both time periods, than the other models. The neural network with no hidden layers tended to perform at a rate comparable to the discriminant function model. These last studies clearly indicate the potential benefit to be derived from using neural networks in the study of bank failure.

DATA, RESEARCH DESIGN, AND RESULTS The object of the first phase of the research centered on predicting changes in a bank’s Sheshunoff’s rating. This rating can assume a value of 1–10 and is based upon the bank’s financial ratios. Our current research has limited itself to the banks of Connecticut, starting with the savings banks. Data were obtained from several sources – state published databases; Sheshunoff’s The Bank Quarterly: Ratings and Analysis. Financial data were collected for 47 savings banks for the period 1990–1995. For each of the banks a set of financial ratios was computed. Based upon prior studies, the researchers used 26 financial ratios – categorized as CAMEL ratios – a listing of these ratios is given in Appendix A. In addition, we computed the quarterly and annual rates of change for the 26 ratios. These variables were examined to determine the degree of correlation amongst them. In addition, T-tests and factor analysis were conducted to examine which variables were most significant in terms of differentiating among the Sheshunoff’s ratings. This was done to reduce the data set (See Appendix B). The authors employed several neural network back-propagation architectures designed to classify the banks according to the annual changes in the Sheshunoff’s ratings. By changes, we mean the change either up or down in the Sheshunoff’s rating – refer to Table 1. For the time horizon under investigation, no bank had a change in its ranking greater than one (either up or down) for any one year. The architecture of the first model (NN1) consisted of six nodes in one hidden layer; the second model (NN2) had 11 nodes in one hidden layer; the third model (NN3) had six nodes in the first hidden layer and five nodes in a

Bank Rating Change Predictions

Table 1.

Annual Changes in the Sheshunoff’s Rating.

One Rank Improvement in Rating

No Change in Rating

One Rank Decline in Rating

3 1 1 1

38 45 44 42

6 1 2 4

1992 1993 1994 1995

Table 2.

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Overall Number of Banks Correctly and Incorrectly Classified by Models.

Model NN1 Model NN2 Model NN3 Model NN4 Linear discriminant model

Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified

1992

1993

1994

1995

45 2 45 2 45 2 41 6 43 4

43 4 45 2 43 4 41 6 41 6

41 6 41 6 41 6 39 8 39 8

41 6 41 6 37 10 35 12 37 10

second hidden layer; and the fourth model (NN4) had 10 nodes in the first hidden layer and 10 nodes in a second hidden layer. These results were compared to a discriminant analysis model using the same set of 11 ratios. All models were built on 1991 data. In Table 2, we present the overall number of banks correctly and incorrectly classified by the four neural network models and the linear discriminant model for the years 1992, 1993, 1994, and 1995. In Table 3, we present the overall percentage of banks correctly and incorrectly classified by the four neural network models and the linear discriminant model for the years 1992, 1993, 1994, and 1995. If we examine the results in Table 2, we find that NN1, NN2, and NN3 were better predictors with respect to overall correct classification than the linear discriminant model for all four years. Model NN4 was inferior, in terms of overall prediction, to the linear discriminant model for two of the four years. If we examine Table 3, the overall percentage correctly classified, it appears the most accurate of the five approaches is model NN2 with model NN1 being a close competitor. Both models have overall classification accuracy in excess of 85% four years

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Table 3.

Overall Percentage of Banks Correctly and Incorrectly Classified by Models.

Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified Correctly classified Incorrectly classified

Model NN1 Model NN2 Model NN3 Model NN4 Linear discriminant model

1992(%)

1993(%)

1994(%)

1995(%)

95.7 4.3 95.7 4.3 95.7 4.3 87.2 12.8 91.5 8.5

91.5 8.5 95.7 4.3 91.5 8.5 87.2 12.8 87.2 12.8

87.2 12.8 87.2 12.8 87.2 12.8 83.0 17.0 83.0 17.0

87.2 12.8 87.2 12.8 78.7 21.3 74.5 25.5 78.7 21.3

Misclassification Errors for Model NN1.

Table 4.

Moved up One Rating

Stayed in Same Rating

Moved down One Rating

1992 Actual Predicted Error

3 2 1

38 39 1

6 6 0

45 47 2

1 0 1

44 42 2

2 5 3

42 45 3

4 2 2

1993 Actual Predicted Error

1 0 1 1994

Actual Predicted Error

1 0 1 1995

Actual Predicted Error

1 0 1

out. Now it must be pointed out that these results are skewed to give the majority of the cases that did not change their status from year to year. While overall accuracy in identifying the correct classification is important it is equally important to identify the misclassification errors by type. Next we will review how each model misclassified the banks into each of the three categories. The results are provided in Tables 4–8. Each table summarizes

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Misclassification Errors for Model NN2.

Table 5.

Moved up One Rating

Stayed in Same Rating

Moved down One Rating

1992 3 2 1

Actual Predicted Error

38 39 1

6 6 0

45 46 1

1 1 0

44 42 2

2 5 3

42 45 3

4 2 2

1993 1 0 1

Actual Predicted Error

1994 1 0 1

Actual Predicted Error

1995 1 0 1

Actual Predicted Error

Misclassification Errors for Model NN3.

Table 6.

Moved up One Rating

Stayed in Same Rating

Moved down One Rating

1992 Actual Predicted Error

3 2 1

38 39 1

6 6 0

45 47 2

1 0 1

44 47 3

2 0 2

42 47 5

4 0 4

1993 Actual Predicted Error

1 0 1 1994

Actual Predicted Error

1 0 1 1995

Actual Predicted Error

1 0 1

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DAVID T. CADDEN AND VINCENT DRISCOLL

Misclassification Errors for Model NN4.

Table 7.

Moved up One Rating

Stayed in Same Rating

Moved down One Rating

1992 Actual Predicted Error

3 1 2

38 41 3

6 5 1

45 43 2

1 4 3

44 41 3

2 6 4

42 37 5

4 10 6

1993 Actual Predicted Error

1 0 1 1994

Actual Predicted Error

1 0 1 1995

Actual Predicted Error

Table 8.

1 0 1

Misclassification Errors for Linear Discriminant Model. Moved up One Rating

Stayed in Same Rating

Moved down One Rating

1992 Actual Predicted Error

3 1 2

38 40 2

6 6 0

45 43 2

1 4 3

44 41 3

2 6 4

42 38 4

4 9 5

1993 Actual Predicted Error

1 0 1 1994

Actual Predicted Error

1 0 1 1995

Actual Predicted Error

1 0 1

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the misclassification errors by type for each model by each year. One pattern appears to be consistent across all five models is the underestimation of the banks that were moved up one rating. It should also be noted that after 1993 most of the models NN1, NN2, NN4, and the linear discriminant model overestimated the number of banks that would be dropped one rating. In actual practice misclassification errors should not be considered of equal importance. In the future one might consider the use of different penalty values for each type of misclassification.

FUTURE RESEARCH This study should be seen as a preliminary work. The authors will pursue several additional areas of investigation. These are discussed below. Ratio Selection – In addition to the neural network models and linear discriminant model, the data will be tested using upon quadratic discriminant analysis (QDF). This classification scheme was used because of evidence of its superiority to LDA in bankruptcy studies (Mahmood & Lawrence, 1987). However, the authors plan to test the data for deviations from multivariate normality to which QDF analysis is sensitive. We also plan to Probit and Logit analyses to evaluate the data. Alternative Learning Algorithms – Most neural network studies of bankruptcy have relied upon back propagation. Although this has been linked to discriminant analysis and found to produce gains over discriminant analysis when the number of hidden odes is greater than two, there is a need to investigate other neural network models. Possible alternatives include: brainstate-in-a-box; general regression neural network; fuzzy neural network, learning vector quantization network; and probabilistic neural networks. Normalization – Neural networks such as back propagation require that the data that they work will be of a specific form. This may mean that the data be recalibrated so that it lies between 0 and 1 or between 0.5 and 0.5. Software packages such as NeuralWare provides for easy data transformation. However, in the case of classification prediction there may be additional consideration with respect to data transformation. Raghupathi, Schkade, and Raju (1993) normalized their data by subtracting the minimum value for a particular ratio from each observation and dividing by the range (maximum value – minimum value) for that ratio. It should be noted that there could be wide variation in the values of financial ratios not only between banks. This variability has been substantiated in several empirical studies. Chudson (1945) discussed how different industrial groupings have

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different values for selected ratios. Horigan (1967) gave further evidence that this variability existed for many ratios. Scott (1977) in his examination of corporate financial structure found how the equity as a percentage of total capitalization ratio was statistically significantly different for 12 industries in each of the 10 years examined for the period 1959–1968. Gupta and Huefner (1972) utilized the technique of cluster analysis to show that industries could be grouped on the basis of their financial ratios. Further, they were able to provide a series of conceptual linkages between the groupings and the ratios. Future studies might benefit from a normalization method, which combines Rughupathi et al. (1993) approach with an industry-relative ratio proposed by Platt and Platt (1991). Their industry-relative ratio is a firm’s ratio divided by the average for the industry. Hidden Layers and Number of Nodes – Perhaps one of the most serious problems with applying neural networks is its black-box nature. There is little in the way of solid theory to guide the user in specifying the architecture of the network or the initial value for the network’s parameters. There has been work done on attempting to find the best format, and how to improve classification accuracy. There are also some general heuristics for determining the maximum number of nodes. As we have seen researchers often dealt with this problem by employing different architectures. The use of response surfaces can be helpful in deterring the impact of changing parameters.

REFERENCES Altman, E. (1968). Financial rations, discriminant analysis, the prediction of corporate bankruptcy. Journal of Finance, 23(September), 589–609. Altman, E. (1977). Predicting performance in the savings and loan association industry. Journal of Monetary Economics, 3, 443–466. Chudson, W. A. (1945). The pattern of corporate financial structure: A cross-section of manufacturing, mining, trade, and construction. New York: National Bureau of Economic Research. Elmer, P., Borowski, D. (1988). An expert system approach to financial analysis: The case of S&L bankruptcy. Financial Management, 17(Autumn), 66–75. Fisher, R. (1936). The use of multiple measures in taxonomy problems. Annals of Eugenics, 7, 179–188. Gupta, M. C., & Huefner, R. J. (1972). A cluster analysis of financial ratios and industry characteristics. Journal of Accounting Research, 10(Spring), 77–95. Horrigan, J. O. (1967). An evaluation of financial ratio analysis. Ph.D. dissertation, University of Chicago. Mahmood, Mo. A., & Lawrence, E. C. (1987). A performance analysis of parametric and nonparametric discriminant approaches to business decision making. Decision Sciences Atlanta, 18(2), 308.

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Pettway, R. H., & Sinkey, J. F., Jr. (1980). Establishing on-site bank examination priorities: An early-warning system using accounting and market information. The Journal of Finance Cambridge, 35(1), 137. Platt, H., & Platt, M. (1991). A note on the use of industry relative ratios in bankruptcy prediction. Journal of Banking and Finance, 15, 1183–1194. Raghupathi, W., Schkade, L., & Raju, B. (1993). A neural network approach to bankruptcy prediction. In: R. Trippi & E. Turban (Eds), Neural networks in finance and investing (pp. 141–158). Chicago: Probus Publishing Company Reprint. Salchenberger, L., Cinar, E., & Lash, N. (1993). Neural networks: A new tool for predicting thrift failures. In: R. Trippi & E. Turban (Eds), Neural networks in finance and investing (pp. 229–254). Chicago: Probus Publishing Company Reprint. Scott Jr., J. H. (1977). Bankruptcy, secured debt and optimal capital structure. The Journal of Finance, 32(March), 1–19. Tam, K., & Kiang, M. (1992). Managerial applications of neural networks: The case of bank failure predictions. Management Science, 38(7), 926–947.

APPENDIX A

Entire Ratio Set Capital adequacy Core capital – unreal. loss/assets Core capital/assets (leverage ratio) Core capital+loan loss reserves/total loans Risk adjusted capital ratio Tangible equity ratio Core capital/risk weighted assets

% Change in total assets % Change in total equity Equity/total assets Equity/total capital Dividend payout Debt & ltd-life pref. stock/ total capital

% Change in core capital Asset quality Adjusted non-performing Loans+OREO/ loans+OREO Adjusted non-performing asset (included OREO)/assets Non-performing loans/gross loans Non-performing loans & securities/core capital+reserve Loans 90 days past due/gross loans

% Change in total assets % Change in total equity Equity/total assets Equity/total capital Dividend payout

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APPENDEX A. (Continued ) Entire Ratio Set Non-accrual loans/gross loans

Debt & ltd-life pref. stock/ tot. cap

Restructured loans/gross loans Earnings profitability Operating profit/average assets Return on average assets (R.O.A.) Return on average equity (R.O.E.) Effective tax rate Yield/average earning assets Yield/average earn assets (tax adj.) Rate/average earning assets Spread/average earning assets Cost of funds Cost of borrowings Adj. bank inc. per $ of salary exp. Securities gains (losses)/IBEI Net income/salary & benefits exp.

Cost of borrowings Cost of deposits Yield on loans Yield on loans (tax adj.) Yield on securities Yield on securities (tax adj.) Non-interest income/ average assets Overhead expense/average assets Net overhead expense/ average assets Non-interest income/total overhead % Change in inc. before extra. Items % Change in non-interest income % Change in overhead expense

Service charge income/salary exp. Liquidity Liquid assets/liabilities Liquid assets – lg. liab./assets $100M plus time deposits/assets Borrowings & for. dep./assets

Total IPC deposits/total deposits $100M+time dep./total deposits Public funds/total deposits Brokered deposits/total deposits

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APPENDEX A. (Continued ) Entire Ratio Set Net fed funds purch./assets Brokered deposits/assets Total loans/deposits – pub. funds

APPENDIX B

Reduced Ratio Set Capital adequacy Core capital/assets (leverage ratio) Risk adjusted capital ratio Tangible equity ratio Equity/total assets % Change in core capital Asset quality Loans 90 days past due/gross loans Earnings profitability Operating profit/average assets Return on average assets (R.O.A.) Yield on loans Liquidity Liquid assets – lg. liab./assets Borrowings & for. dep./assets

% Change in total IPC deposits Pledged securities/total securities Fair value to amortized cost-total

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FORECASTING SECURITY RETURNS: THE USE OF HETEROGENEOUS EXPECTATIONS Rebecca Abraham and Charles W. Harrington ABSTRACT This study proposes a novel method of forecasting security returns. Prior to earnings announcements, both analysts and individuals make forecasts of corporate earnings which in turn, drive the price of the stock. The greater the consensus between analysts forecasts and individual forecasts (termed earnings whispers) the less likely that returns will be excessively optimistic, or the more likely the forecasts will be accurate estimates of earnings. Empirically, low-differential stocks, or stocks with lower differentials between earnings whisper forecasts and analysts consensus forecasts had significantly higher future security returns following the earnings announcement than stocks with higher differentials indicating that they were more rationally priced. When placed within the framework of the Capital Asset Pricing Model (CAPM), earnings forecast differentials explained a significant amount of the variance in security returns beyond the market risk premium. Low-differential stocks are shown to resemble value stocks, while high-differential stocks display the characteristics of glamour stocks.

Advances in Business and Management Forecasting, Volume 4, 93–115 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04006-7

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INTRODUCTION Forecasts of security returns have been either direct or indirect with early studies (see Stollis, 2005, for a review) incorporating a number of macroeconomic variables and current investigations using traditional accounting variables. Macroeconomic variable forecasts enhanced the power of explanatory models of security returns and volatility during the 1970s, while modern forecasting methodologies rely more on forecasts of earnings per share, which in turn, determine security returns. Fundamental accounting variables such as accounts receivable, inventory, and capital expenditures (Abarbanell & Bushee, 1997) are included in many multivariate models of earnings per share. The current literature on financial forecasting is leaning toward such multivariate forecasting models as providing improved explanatory power of security returns over univariate models using a single fundamental accounting variable (Zhang, Cao, & Schneiderjans, 2004). Further, Zhang et al. (2004) observed the superiority of neural network forecasting models, which incorporate nonlinearity in their functional forms in accordance with the nonlinearity in quarterly earnings per share data. Yet another body of literature recognizes that investors differ and such differences explain corporate earnings beyond a strictly balance sheet-oriented accounting approach. In normal market activity, there are investors of varying levels of rationality. Certain investors are optimistic while others are more risk-averse. In general, financial markets are composed of investors with heterogeneous expectations as recognized in a series of theoretical models in which heterogeneous expectations have been shown to affect security prices (Chen, Hong, & Stein, 2001; Diamond & Verrecchia, 1987; Jarrow, 1980; Mayshar, 1987). Empirical studies require a proxy for heterogeneous expectations since expectations cannot be measured directly. Proxies have been limited heretofore to trading volume (Lee & Swaminathan, 2000) and dispersion of analysts’ earnings forecasts (Diether, Malloy, & Scherbina, 2002). Both studies found evidence of higher security returns for stocks with low trading volume and greater consensus among analysts’ earnings forecasts. Conversely, negative security returns were observed for stocks with excessively optimistic valuations. This study proposes the use of a new proxy for heterogeneous expectations, namely, the differential between earnings whisper forecasts and analysts consensus forecasts and demonstrates its use in predicting security returns.

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REVIEW OF THE LITERATURE This study uses the differential between whisper forecasts of earnings and analysts’ consensus forecasts as the measure of heterogeneous expectations. Message boards of Internet sites devoted to investments receive a multitude of postings wherein individuals attempt to forecast the earnings per share of a particular stock. These unofficial forecasts of earnings provided by individuals are termed whisper forecasts. Consensus among whisper forecasts is achieved by sites such as earningswhispers.com and whispernumber.com, which synthesize the whisper forecasts into a single whisper number. Analysts forecasts of earnings differ from whispers in that they originate from institutional forecasters employed by large brokerage houses throughout the country armed with sophisticated analytical tools, econometric software, corporate annual reports and SEC filings, as opposed to the relatively simple analytical tools and publicly available documents of individual investors. Since earnings whispers differ from analysts earnings forecasts, the difference between the two forecasts may act as a proxy for heterogeneous expectations. Empirical support for this thesis may be found in the Bagnoli, Beneish, and Watts’ (1999) comparison of whisper forecasts and analysts’ consensus forecasts generated by the First Call Corporation. Whisper forecasts were found to be significantly different from First Call forecasts with trading strategies based on whisper forecasts earning significantly different market- and size-adjusted returns than a strategy based on First Call consensus forecasts. Opening positions five, three, and two days prior to the earnings announcement and closing them at the end of the trading day on the day of announcement, they found significantly different market- and size-adjusted returns for all three holding periods. The literature on individual versus institutional investors suggests that individual investors are likely to be optimists, while institutional investors are the more rational. Individual investors who lack access to a range of news reports and analyses are, most certainly, not professional investment managers. Brennan (1995) observed that only about 27 percent of households hold stocks and even with high levels of ownership of assets, the percentage is only 48 percent. Individuals are frequently misled by the exceptional returns offered by commodity funds (Elton & Gruber, 1989). Brennan’s review (1995) cites studies in which new issues of closed-end funds and Real Estate Investment Trusts (REIT), which are dominated by individual ownership are overpriced with higher underwriting fees than competing initial offerings (Peavey, 1990; Wang, Chan, & Gau, 1992; Weiss,

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1989). Given that individuals are less likely to be in a position to conduct rigorous fundamental analyses of financial statements, they are overwhelmingly influenced by the recent past returns in making purchase decisions (Patel, Zeckerhauser, & Hendricks, 1991) and fail to make accurate predictions about the direction of price movements following events such as earnings announcements. Welker and Sparks (2001) demonstrate that in the pre-earnings announcement period, individuals were unable to predict the content of forthcoming news. This effect was exacerbated in the postearnings announcement period, in which individuals reacted in an opposing direction to the expected price movement following the announcement. Simply, individuals were significantly inclined to purchase following negative news and sell following positive news. In contrast, the direction of trading volume for institutions was consistent with the expected price movement to the news. Welker and Sparks (2001) conjecture that the opposing position of individuals and institutions in the post-announcement period suggests that the two groups either have different sources of information or vary in their interpretation of the content of information. Institutions have been shown to improve the efficiency of setting security prices, with securities tracked by multiple analysts responding rapidly to new information (Brennan, Jegadeesh, & Swaminathan, 1993). The price response of a stock to trades increases with the number of analysts tracking it, and in turn results in the more rational pricing of securities (Brennan & Subrahmanyam, 1994a, b). Lakonishok, Shleifer, and Vishny (1992) demonstrated that institutional managers failed to destabilize prices for over 700 pension funds managed by over 300 money managers. Institutional investors did not engage in herding (acting in concert) in their trades of large stocks and even though there was some evidence of herding in small stocks the magnitude was limited. Given the greater rationality of institutional investors, if the forecasts of individual investor or whispers are sufficiently close to those of the institutional investors, such forecasts are more likely to be accurate estimates of earnings. Such stocks are termed low-differential stocks. Stocks with considerable divergence in earnings estimates between individuals and institutions are termed high-differential stocks. The divergence for highdifferential stocks arises from a lack of agreement between individual and institutional investors. Since analysts are more rational, the departure from their estimates results in overvaluation or excessive optimism. All current and previous estimates of whisper forecasts have shown that they lie above analysts consensus forecasts suggesting that the whispers are more optimistic. Following the earnings announcement, the rationality of

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low-differential stocks may be translated into higher security returns, while the optimism of high-differential stocks may result in lower security returns as the dissemination of actual earnings information during the announcement rewards low-differential stocks and negates the optimism of highdifferential stocks. The next issue is whether heterogeneous expectations of earnings have an impact on stock prices. The existing arguments are largely theoretical with the Williams (1977), Goetzmann and Massa (2001) models relating heterogeneous expectations to stock returns, while a host of others use heterogeneous expectations to obtain predictions of stock prices (Jarrow, 1980; Mayshar, 1987; see Diether et al., 2002, for a review). The Miller (1977) model argues that prices reflect the views of optimistic investors if pessimistic (and more rational) investors are excluded from the market by high short-sale costs. The Miller model is one of a few price-optimism models (see Diamond & Verrecchia, 1987; Hong & Stein, 2000) that holds that excessively optimistic investors purchase stock with the highest valuations suffering losses in expectations. The larger the disparity between optimistic and true valuations of stock prices, the greater the market price forecasted by the optimistic investor relative to the true value of the stock and the lower its future returns. In other words, an investor who is sufficiently optimistic about a stock will invest in it resulting in overvaluation of its price. As expectations of a stock’s value fail to materialize, losses in expectations result followed by lower future returns. It follows that the larger the discrepancy between optimistic and true valuations of stock prices or the greater the heterogeneity of expectations about stock prices, the higher the market price forecasted by the optimistic investor and the lower the future returns. Therefore, we set forth that any differential of earnings forecasts acting as a proxy of heterogeneous expectations will be empirically related to lower future stock returns. Support for the Miller (1977) model comes from a series of investigations using proxies for heterogeneous expectations. Chen et al. (2001) drew on the Hong and Stein (1999) model to test the impact of heterogeneous expectations on security returns. This model is based on the assumption that there are only two investors in the market, A and B, both of whom receive signals about future returns. Given a moderately binding short-sale constraint, both investors may be thought of as mutual fund holders whose charters prohibit them from short selling. The rest of the market may be thought of as rational arbitrageurs or hedge funds, which are not affected by short-sale constraints. In a departure from full Bayesian rationality, each investor only pays attention to his or her own signals even if the signals of

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the other are apparent. Model asymmetries occur at two points in time. At the first point in time, B receives a pessimistic signal and leaves the market, while A receives a positive signal raising expectations even further. However, if A receives a negative signal, he or she exits the market, while B may enter buying the stock at a reduced price. Negative skewness of returns is expected to result with A’s departure and B’s purchase at a substantially reduced price reversing the optimism in prices obtained at the first point in time. Chen et al. (2001) used breadth of mutual fund ownership as their proxy for the heterogeneity of opinions finding that reductions in breadth of ownership lead to lower future returns. As A and B are both considered to be mutual fund investors, their convergence of opinion that stock prices will continue to fall at the second point in time leads to reduction in their purchases of mutual funds. Therefore, the reduced breadth of ownership of diverse investors in mutual funds depresses future stock returns. Diether et al. (2002) used dispersion in analysts’ earnings forecasts as their proxy for heterogeneous expectations observing that a portfolio of stocks in the highest quintile of dispersion underperformed their lowest dispersion counterparts by a full 9.48 percent per year. However, dispersion in analysts’ earnings forecasts is frequently viewed as a proxy for risk. The basic argument is that nondiversified investors will demand restitution for the individual risk of their security holdings. As dispersion in analysts’ forecasts is a predictor of uncertain future returns, stocks with higher dispersion in earnings forecasts should be required to earn higher future returns so that dispersion becomes not only a proxy for heterogeneous expectations but for risk. Diether et al. (2002) found, however, that the relationship between dispersion and future returns was significantly negative, leading to their rejection of the concept that dispersion is a proxy for risk. These results remained robust across: (a) size differentials failing to reveal the commonly observed large return differentials between the lowest and highest dispersion quintiles for small stocks, (b) book-to-market ratios as low book-to-market stocks have higher levels of market capitalization and consequently, lower return differentials, (c) momentum to exclude the likelihood that highmomentum stocks have lower future returns, and (d) three-factor model variables. In accordance with Fama and French (1996), who demonstrated that the book to market ratio, earnings to price ratio, and cash flow to price ratio produce positive returns regardless of size, Gibbons, Ross and Shanken (1989) proved empirically that the Fama and French (1996) threefactor model captured average returns. Accordingly, Diether et al. (2002)

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conducted three- and four-factor time series regressions of the excess return over the market portfolio on size and value premiums finding that a large portion of a return on portfolios in the highest dispersion quintile was left unexplained. As they rejected the Gibbons, Ross, and Shanken test, the three-factor model failed to significantly explain portfolio returns, the regression returning nonsignificant coefficients for the highest and lowest dispersion quintiles. Lee and Swaminathan (2000) observed that higher trading volume predicted lower future returns. Optimistic investors are the architects of higher trading given the Miller assumption that rational investors are prohibited from selling due to higher short-sale costs. They draw on prior research (Datar, Naik, & Radcliffe, 1988) which revealed a link between high turnover and lower-future stock returns to demonstrate that high-volume stocks earn lower average returns in the immediate aftermath of earnings announcements for both a general sample of stocks and the largest 50 percent of the NYSE and AMEX firms. They postulate that the lower returns of such high-volume stocks may be attributable to investor misconceptions about future earnings. Excessive optimism is aroused by analyst forecasts that predict higher long-term growth and return on equity for high volume glamour stocks only to be dashed with inferior future operating performance. Unrealistic expectations originate from analysts’ extrapolation of five years of superior operating performance prior to the earnings announcement for high volume stocks. However, this position reverses during the short period prior to the earnings announcements with significantly negative returns for high-volume stocks as optimism fades with the market’s growing realization of the lower future earnings of these firms. In a subsequent review, Hong and Stein (2002) reversed their initial position of lower future returns by demonstrating that rational arbitrageurs intervene to reverse any price movements by the actions of investors A and B so that the market clears at a price that is equal to the stock’s true value. Diamond and Verrecchia (1987) concur with their modeling of a rational market maker, who correctly infers that certain investors with low valuations are constrained from selling their stock; as the rational market maker refuses to be swayed by the actions of overconfident investors, market prices will remain unbiased regardless of the degree of differences of opinion about the stock’s final value. Thus, we have two initial hypotheses. Our first hypothesis is that stocks with high differentials between whisper and analyst forecasts of earnings will have significantly lower future returns than their counterparts with low differentials. Conversely, we have the alternate

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hypothesis based on the rational market-maker argument, that there is no relation between observed differences between whisper and analyst forecasts of earnings and future returns. Can we assume that excessive optimism will drive earnings estimates upwards? The answer is in the affirmative if we accept that the conditions of the Miller (1977) model prevail. Miller (1977) stated that optimists predominate in markets in which rational investors are precluded from selling by high short-sale costs. Therefore, optimists will raise the estimates of earnings, which will result in lower security returns after the true earnings are revealed during the earnings announcement. The foregoing discussion suggests the following hypothesis: Hypothesis 1. The lower the differential between whisper forecasts and analysts’ consensus forecasts of earnings, the higher will be their future returns. Specifically, stocks with low differentials will have future returns that are significantly higher than their high differential counterparts. Hypothesis 1A. (Alternate Hypothesis). There is no relationship between earnings whisper forecast differentials and security returns as any price optimism will be reversed by rational arbitrageurs so that final security prices will clear at equilibrium levels. By definition, value stocks have weaker operating performance, larger declines in past operating performance and higher book-to-market ratios (Lee & Swaminathan, 2000). In prior studies, low dispersion stocks for Diether et al. (2002) and low-volume stocks for Lee and Swaminathan (2000) behaved like value stocks, and high dispersion or high-volume stocks behaved like glamour stocks. It follows that low-differential stocks may resemble value stocks, while high-differential stocks may find similarity with glamour stocks. By virtue of greater conformity between the expectations of optimistic and pessimistic investors, low-differential stocks are less likely to be subjected to irrationally optimistic expectations. This may be due to the fact that they are less well known, have had weaker past operating performance, and stronger fundamentals, and consequently, are more likely to have hitherto unknown price potential. Conversely, the greater the divergence in opinion between excessively optimistic investors and pessimists along with the dominance of optimists assures that high-differential stocks are more likely to be subject to the hype and hysteria commonly associated

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with glamour stocks so that there is overconfidence in expectations of their performance, with subsequent declines in returns. Hypothesis 2. High-differential stocks act as value stocks while lowdifferential stocks act as glamour stocks.

METHODOLOGY Data and Sample Characteristics Earnings whisper forecasts were collected daily from whispernumber.com and on the reporting dates from earningswhispers.com from the inception of their reporting (January 1999–February 2003) yielding observations for 457 stocks. While both sites are official repositories of whisper numbers, their method of data collection differs. Earningswhispers.com presents narrative summaries of whisper and earnings information along with numbers for a limited number of stocks on certain dates. Whisper forecasts are obtained by the site through scanning of electronic message boards and electronic mail. In contrast, whispernumber.com solicits whisper forecasts from its subscribers. Whisper and earnings forecasts are reported daily until the earnings release for a broad range of stocks. On a single day, February 11, 2003, both whisper and analyst forecasts were reported for 20 stocks, though the usable number of forecasts was 15, given that forecasts remained unchanged on the other stocks. Therefore, although daily data is available through whispernumber.com, only about 60–75% of it is usable, due to the repetition of data values. Finally, whisper numbers are created from the whisper forecasts so that each stock has a unique whisper number that represents the consensus of the whisper forecasts. Hypothesis 1 was tested using pair wise t tests of the differences between high- and low-differential stocks. It was further tested using an econometric model in which security returns are predicted by differentials, book to market ratios, market capitalization, price, volatility, and momentum. Rj ¼ N i+b1 Di+b2 BEME+b3ME+b4P+b5V+b6Mo+xi Rj ¼ Stock Return, 1 month after portfolio formation Di ¼ Whisper-Analyst Forecast Differential measured as Earnings Whisper Number
Analysts Consensus Forecast BEME ¼ Book to Market Ratio measured as (Book value of Stockholder’s Equity + Balance Sheet Deferred Taxes
Value of Preferred Stock)/ Market Capitalization

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ME ¼ Market Capitalization measured as Market Price Per Share  Number of Shares Outstanding P ¼ Market Price at the time of Portfolio Formation V ¼ Volatility measured as the Standard Deviation of Returns from t
12 to t
2 Mo ¼ Momentum measured as returns from t – 12 to t – 2 (12–2 months prior to the current period), with positive returns indicating winners and negative returns indicating losers. Why were the above predictors (book to market ratio, market capitalization, price, volatility, and momentum) included in this model? The literature (see Fama & French, 1996, for a review) has established the relationship between each of these variables and security returns. Stocks with high book-to-market ratios, small stocks (low market capitalizations), and low returns over the past year have experienced higher returns chiefly due to undervaluation. Stoll (1978) theorized that in order to induce dealers to move off the efficient frontier with minimal risk for their portfolios, i.e., for dealers to acquire higher risk portfolios, additional compensation must be paid to the dealers, which could take the form of higher returns. We expect negative relationships between differentials, market capitalization, price, and momentum with stock returns, and positive relationships for price, book-to-market ratios, and volatility with the criterion. With the exception of the whisper forecasts obtained from whispernumber.com and earningswhispers.com, data was obtained from The Center for Research in Security Prices (CRSP) and Thomson’s First Call Reports. Since whisper and earnings forecasts for each stock were obtained on different dates, so were the returns. For example, IBM’s whisper and analyst consensus forecasts were obtained on September 5, 2001, and AOL’s were on September 18, 2001, then a portfolio of IBM, AOL, and another stock was created (most portfolios consisted of three stocks, although there were a few two stock portfolios), held for a month, and returns measured during the following month to produce portfolios in five differential quintiles. The differential quintiles were formed with differentialso0.02 being the lowest quintile, followed by 0.03–0.04, 0.05–0.06, 0.07–0.08, and 40.09 being the highest quintile. A perusal of the differentials revealed a dearth of negative differentials so that all whisper–analyst consensus differentials were uniformly positive. It follows that the whisper forecasts were overestimates of earnings while the analyst forecasts were underestimates as observed in the earlier Bagnoli et al. (1999) study.

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RESULTS Hypothesis 1 stated that the lower the differential between whisper forecasts and analysts’ consensus forecasts of earnings, the higher would be their future returns. Table 1 shows that this hypothesis was supported as mean portfolio returns were significantly lower for the highest differential stocks over the lowest differential quintile. To test if our results are robust to size, each month, we assigned stocks to five market capitalization quintiles. Within each size quintile, stocks were ranked into five quintiles based on earnings differentials as of the previous month. The average monthly return differential significant across size indicating that our results are immune to size differences.

Table 1.

Mean Portfolio Returns by Size and Average Differentials on Sorts by Size. Differential Quintile 1

Size Quintile 2

3

4

5

Panel A: Mean portfolio returns by size and whisper–analyst forecast differentials 1 1.1079 2.010 0.3808 1.0100 1.2100 2 0.1534 0.7080 0.3200 0.1800 0.3213 3 0.1460 0.1340 0.5042 0.1344 0.1892 4
0.4338
0.1358
0.2600
0.1800
0.0763 5
0.6642
0.3318
0.4044
0.6142
0.9050 Low-high t statistics





t ¼ 2.7041

T ¼ 9.4712

Panel B: Mean differential by size category 1 0.0156 0.0144 2 0.0300 0.0350 3 0.0543 0.0544 4 0.0720 0.0740 5 0.1168 0.3140

t ¼ 1.3110 0.0143 0.0314 0.0550 0.0725 0.2430





t ¼ 1.69

t ¼ 2.78

0.0143 0.0350 0.0535 0.0750 0.1533

0.01316 0.0341 0.0617 0.0700 0.6700

Note: Panel A reports t testing the mean differential between high- and low-differential portfolios sorted by size. Each month stocks were sorted into five categories based on the current level of market capitalization (o $ 9 billion–small cap, $ 10–49 billion–mid cap, and4$ 50 billion–large cap). Panel B reports the mean differential for all categories by size.  po0.05,  po0.01,  po0.001

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We triple sorted on size, book-to-market (BE/ME) ratio and earnings forecast differentials to determine if our results were robust to book-tomarket effects. Since low book-to-market stocks have relatively higher levels of market capitalization, we attempted to control for the fact that large return differences between low- and high-differential quintiles for small stocks may be due to book-to-market effects. As there were an insufficient number of stocks to separate into several size or book-to-market or earnings differential quintiles, we first sorted the stocks into three categories based on the current level of market capitalization with those with capitalizations o$ 9billion designated as small cap, o$ 49 billion mid cap, and 4$ 50 billion large cap. Each size group was further sorted into three categories in terms of book-to-market ratio, and then into three-differential groups, formed by merging the differential quintiles used earlier (differentials o0.02 were designated as low, 0.03–0.06 were medium, and 40.07 were high). Table 2 presents the returns on the resulting portfolios. The return differential on low and high whispers-earnings forecast differentials is significant for seven out of nine differential categories with one category not reporting any results due to insufficient data. This indicates that low-differential stocks produce significantly higher returns across size and book-to-market or that we are simply not capturing book-to-market effects. The final portfolio strategy involved three-way cuts on size, momentum, and whisper-analyst earnings differentials to eliminate the possibility of a momentum effect, whereby high-momentum stocks have significantly lower returns than others (Jegadeesh & Titman, 1993). Stocks were first sorted into three categories based on market capitalization. Within each size category, the stocks were sorted into two groups based on past returns from t
12 to t
2 to capture momentum effects. The groups with positive returns were designated ‘‘winners’’ or high-momentum stocks, while those with negative returns were dubbed ‘‘losers’’ or low-momentum stocks. Finally, stocks were sorted into high and low whisper–analyst earnings differential groups based on differentials in earnings forecasts for the next month. Table 3 presents the returns on the resulting portfolios. For all size categories, low-differential stocks have significantly higher returns than high-differential stocks with strongly significant differences (t values ranging from 5.1 to 8.79 all of which are significant at the 0.001 level) indicating that the differential effect is robust across momentum categories, or that we are simply not capturing the momentum effect. Mid-cap losers (low momentum) stocks have the highest return differentials with high-differential stocks earning a negative 1.0383% return over a one-month period versus positive 1.2491% return for low-differential stocks.

Mean Portfolio Returns and Mean Differentials on Sorts by Size and Book-to-Market. Low Book-to-Market

Differential

Small

Medium

Medium Book-to-Market Large

Small

Panel A: Mean portfolio returns by size and book-to-market Low 0.67 0.84 0.54 0.86 Medium 0.24 Insufficient data
0.34 High 1.66
1.10
1.26
1.12    Low-high t statistic 2.095 3.7918 3.8815 12.82 Panel B: Mean differential by size and book-to-market Low 0.01514 0.0223 0.0177 Medium 0.1800 Insufficient data 0.0300 High 0.275 0.2344 0.1692

0.0206 0.0492 0.1200

High Book-to-Market

Medium

Large

0.68
0.27 Insufficient data

1.07 Insufficient data
0.69





7.85 0.025 0.0531 Insufficient data

0.01938 Insufficient data 0.22625

Small

Medium

Large

1.15 0.25
1.79

1.12 0.80
1.01

0.75
0.03
1.12

5.00

3.49

3.11

0.0198 0.034 0.1000

0.01895 0.0600 0.1067

0.01577 0.04600 0.0900





Forecasting Security Returns

Table 2.



Note: Panel A of this table reports t testing the mean differential between high- and low-differential stocks sorted by size and book-to-market. Each month, stocks were sorted into three categories based on the current level of market capitalization (o $ 9 billion–small cap, $ 10–49 billion–mid cap, and4$ 50 billion–large cap). Each size group was further sorted into three categories in terms of book-to-market ratio, and then into three differential groups followed by computation of the return differential on low- and high-differential categories. Panel B shows mean differentials by size and book-to-market category.  po0.05,  po0.01, po0.001

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Table 3. Differential

Stock Returns and Differentials by Sorts on Size, and Momentum. Losers (Low Momentum)

Small cap

Mid cap

Large cap

Panel A: Mean returns by momentum and size Low 1.2695 1.2491 0.7799 High
1.2548
1.0383
0.9400    Low-High t statistic 6.0890 8.7900 5.17 Panel B: Mean differentials by momentum and size Low 0.0578 0.01545 0.021 High 0.1836 0.09000 0.130

Winners (High Momentum) Small cap

Mid cap

Large cap

1.5842
1.3635

1.1058
1.3917

1.1069
0.5609







7.5770

5.86

5.1000

0.021 0.1218

0.0227 0.2000

0.027 0.173

Note: Panel A reports the results of three-way cuts on size, momentum, and whisper–analyst forecast differentials commencing with sorting stocks into the three-size categories used in Table 2. Within each size category, the stocks were sorted into two groups based on past returns from t – 12 to t – 2 to capture momentum effects. Groups with positive returns were termed winners and those with negative returns were termed losers. Finally, stocks were sorted into low- and high-differential groups based on differentials in ratios for the past month.  po0.001

Tables 1–3 provide preliminary evidence that supports the first hypothesis that high-differential stocks earn significantly lower returns than lowdifferential stocks, this finding being robust across size, book-to-market, and momentum effects. As shown in Table 4, in a linear model, all relationships were observed to be in the theorized direction with differentials explaining a significant 1.4% of the variance in security returns (t ¼
1:91; po0:05). As this predictor increased, security returns decreased, so that higher differentials lowered security returns and lower differentials increased security returns, in accordance with Hypothesis 1. Can we be certain that whisper forecast differentials capture a unique amount of the variance in security returns? The traditional model of estimation of security returns is the Capital Asset Pricing Model (CAPM). CAPM states that the return on any security j depends on the market risk premium, or the additional amount that investors are willing to pay to invest in stocks over Treasury bills. According to Fama and French (1996), if an independent variable explains a significant amount of the variance in security returns over and above that explained by the market risk premium in the CAPM, it is indeed a valid predictor of security returns. If whisper forecast differentials explain variance in security returns beyond that of the

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Table 4. Results of Cross-Sectional Regressions of Stock Returns on Whisper–Analyst Forecast Differentials. Variable Differential Book to market Market capitalization Price Volatility Momentum

Coefficient
0.0238536 (
2.122) 0.0003826 (2.961)
123.5303725 (
4.060)
0.0051328 (
2.070) 9.5645356 (3.484)
0.3155796 (
1.70)

N ¼ 457 R2 ¼ 1.9325% Note: This table reports the results of a cross-sectional regression to test if whisper–analyst consensus forecast differentials significantly influenced stock returns 1 month after portfolio creation. Stock returns in the 1 month post-portfolio formation period were regressed on differentials, the book-to-market ratio measured as the book-value-to-market capitalization, the logarithm of market capitalization, stock prices, and volatility measured as the standard deviation of daily returns over the previous 12 months, and momentum, obtained by a variable designating stocks as winners or losers based on their returns over the past 12 months. With the exception of momentum, all predictors were highly significant in explaining variance in the criterion. T ratios are reported in parentheses following the regression coefficients.  po0.05,  po0.01,  po0.001, po0.1

market risk premium, they explain security returns. Table 5 shows that for a quadratic functional form, whisper forecasts explain a unique amount of the variance in the expected or forecasted return. The negativity in whisper forecast differentials may be explained by the fact that the lower the differential the higher the security returns. In conclusion, we accept the first hypothesis that low-differential stocks have significantly higher returns than high-differential stocks. We reject the contention of the alternate hypothesis that rational arbitrageurs reverse price optimism so that final stock prices clear at equilibrium levels. The second hypothesis maintains that low-differential stocks act like value stocks, while high-differential stocks act like glamour stocks. This

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Table 5. Results of Cross-Sectional Regressions of Stock Returns on Whisper–Analyst Forecast Differentials and the Market-Risk Premium. Variable

Coefficient

Differentials


0.2334 (
0.9034) 

2

Differential

Market-risk premium R2 N


3.0995 

13.9774 0.071 457

Note: This table reports the results of a cross-sectional regression employing a quadratic functional form to test if whisper–analyst consensus forecast differentials significantly influenced stock returns 1 month after portfolio creation. Stock returns in the 1-month post-portfolio formation period were regressed on differentials and the market-risk premium. The market-risk premium is the difference between the return on the market, proxied by the monthly return on the S & P 500 index and a 1-month Treasury bill. Both whisper forecasts and the market-risk premium were significant in explaining security returns. T tests are listed in parentheses following the regression coefficients.  po0.05,  po0.01

hypothesis is supported with low- (high-) differential stocks displaying many of the characteristics associated with value (glamour) investing. Low-momentum stocks show lower returns during the past year than high-momentum stocks earning them the title of losers. Given that value investing requires small market capitalizations, small market cap losers should display higher future returns than high market cap winners. Table 3 shows us that this is indeed the case for the mean future return for small cap losers is 1.2695 (SD ¼ 0.0178) which is significantly higher than the mean future return for large market cap winners of
0.5609 (SD ¼ 0.5765) t ¼ 7.0987, po0.001. By the same token, small cap stocks with high book-to-market ratios have significantly higher mean future returns 1.15 (SD ¼ 0.865) versus
1.26 (SD ¼ 1.61) t ¼ 4.7077, po0.001) their low book-to-market large cap counterparts (glamour) stocks. In both cases, the glamour stocks have negative future returns as the high prices fueled by superior past performance and the buildup of high past returns fails to materialize in higher earnings (in the wake of an earnings announcement) so that prices and returns adjust downwards rapidly. The next question is whether the return differential is higher for low book-to-market (value stocks) than it is for the high book-to-market counterparts (glamour stocks). Value stocks exhibit

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higher return differentials in the large size category. For stocks with market capitalizations above $ 50 billion, i.e., the largest stocks, a value pattern exists, or the third hypothesis that glamour stocks exhibit lower returns than value stocks is supported for large stocks.

CONCLUSIONS, RECOMMENDATIONS AND DIRECTIONS FOR FUTURE RESEARCH We have introduced the concept of the whisper–analyst consensus forecast differential as a proxy for heterogeneous expectations and predictor of security returns. Our principal finding is that low-differential stocks (differentials of o0.02) with greater agreement between whisper forecasts and analysts consensus forecasts are not subject to excessive optimism in predicting security returns, and may therefore, be considered as credible predictors of security returns. Such stocks are less well-known value stocks. This article also provides support for the Miller model’s contention that negative returns result whenever rational investors are excluded from short selling or any other source of friction. However, it must be accepted that our results pertain solely to ultra-short time horizons with excessively high expectations by individual investors (as represented by the whisper forecast) immediately prior to the earnings announcement resulting in a market overreaction which is corrected in the month following the earnings announcement. This study and the Diether et al. (2002) study are the only two of the numerous empirical investigations of heterogenous expectations that are based on the same premise that heterogenous expectations may be proxied by the consensus or lack thereof of earnings forecasts. Diether et al. (2002) viewed heterogeneous expectations as the dispersion (or varying levels of consensus) of earnings forecasts. High dispersion stocks showed a divergence of expectations regarding earnings in a market in which optimists predominate so that they (as predicted by the Miller model) were optimistic and therefore posted lower future returns. Low dispersion stocks showed greater conformity of expectations among analysts with more rational or higher future returns. Likewise, this study has found that high-differential stocks, with greater divergence between optimists and pessimists in a market in which optimists predominate post lower future returns than their lowdifferential counterparts. Both studies, may therefore, be particularly useful to determine the impact on stock prices of the predictors in the wake of earnings announcements. Over the same time horizon, a month following the observation of the predictor, the whisper analyst consensus earnings

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forecast differential showed negative returns for high-differential stocks and positive returns for their low-differential counterparts suggesting an immediate price correction in the wake of the earnings announcement made within a few days of the measurement of the earnings differentials. It rejects the hypothesis that rational market makers react to stabilize prices so that no abnormal returns are earned after earnings announcements (Diamond & Verrecchia, 1987; Hong & Stein, 1999). The other principal study that supports our evidence is that of Lee and Swaminathan (2000) whose proxy for heterogeneous expectations of trading volume, yielded superior operating performance for low volume (akin to our low-differential stocks) over high-volume stocks for up to eight quarters subsequent to the earnings announcement. If we place our results within the framework of an intertemporal sequence, we posit that following the eight quarters of abnormal returns, there is a reversal in the intermediate-term time horizon with high-volume stocks outperforming low-volume stocks suggesting that the optimism that was dashed by the earnings announcement is rejuvenated, so that optimism prevails and high-volume (glamour) stocks continue to outperform their low-volume (value) counterparts. From 3 to 5 years, the long time horizon reversal occurs again as optimism fades and negative returns are experienced by high-volume stocks. In other words, earnings differential effects predominate for the first month following the earnings announcement, only to be dominated by momentum effects for the next year with earnings differential effects for the next year with earnings differential effects strengthening over the long-term 3–5 year period. Our results find a theoretical basis in Daniel, Hirshleifer, and Subrahmanyam (1998), who observed that certain stocks are subject to an overconfidence bias. Daniel et al. (1998) examined samples of prominent (glamour) stocks and undervalued (value) stocks. Glamour stocks were the subject of much speculation by analysts as to their earnings; most of whom predicted excessively high earnings. Their overconfidence was fueled by the higher momentum displayed by these stocks during the previous year. Such stocks suffered declines in future portfolio returns. In this context, the highdifferential stocks in this study are well known and the subject of much speculation as to their final price, their past favorable performance fueling momentum leading traders to buy on good news. Such overconfidence results in inferior earnings leading to weak stock returns. Our results corroborate De Long et al.’s (1987) contention that well-known stocks (highdifferential stocks in this study) are the subject of much speculation as to their final price, their favorable past performance attracting momentum traders. Momentum traders confine themselves to purchases of stocks with

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rising prices and sales of stocks with declining prices. Assuming that the favorable past performance of high-differential traders will continue, such traders purchase them, their initial optimism fading with the announcement of weak earnings and lower future returns. At this point, we may create an intertemporal model of stock returns over various time periods. There are two effects on future security returns, the first, due to whisper-earnings differentials and the dispersion in forecasted earnings termed the earnings effect, and the second, due to momentum or past security returns termed the momentum effect. The two effects operate in diametrically opposite directions. The earnings effect, which is most pronounced in the immediate one-month aftermath of earnings announcements, provides a correction to excessive optimism so that stocks that show high differentials between whisper forecasts and analysts consensus forecasts in our study will have lower future returns following the earnings announcement for up to one month after the earnings announcement. The earnings effect is also apparent in high-volume stocks though it may last longer as Lee and Swaminathan (2000) report that abnormally low returns are earned by these stocks for up to eight quarters following the earnings announcement with abnormally high returns earned by low-volume stocks for the same time period. This is the first reversal of stock returns as excessive optimism is corrected by stocks failing to live up to their promise with the announcement of weak earnings. During the third year, the intermediate-term horizon emerges with momentum effects predominating as winners (stocks with high returns during the three years prior to the earnings announcement) display higher returns than losers. Therefore, earnings effects decline as the time period from the initial earnings announcement lengthens, while past performance in terms of the relative strength of security returns becomes the principal predictor of future stock returns. Jegadeesh and Titman (1993) provide evidence of the continuation of stock returns from the period prior to the earnings announcement over this intermediate time horizon. In a direct test of earnings and momentum effects, Chan, Jegadeesh, and Lakonishok (1996) demonstrated that stock returns during the intermediate-term horizon underreacted to earnings news as the higher momentum of winners subsumed the earnings effect. Factors other than earnings and momentum do not appear to affect security returns during this time period as Fama and French (1996) observed that the three-factor model failed to explain intermediate-term price momentum suggesting that other effects determined security prices. As security returns are higher for winners, there is a second form of overconfidence, i.e., that due to excessive optimism from superior past operating performance.

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This correction takes place in the third phase, the long-term time horizon, from the end of the third year to the fifth year, when the winners underperform the losers, or high-momentum stocks with superior past performance underperform low-momentum stocks with inferior past performance. We have expanded the literature on whisper forecasts of earnings. Whereas the Bagnoli et al. (1999) study favored the use of whisper forecasts in that they were found to be more accurate than First Call forecasts in predicting earnings, we consider whisper forecasts to be symptomatic of excessive optimism as they overestimate the prices of our sample of stocks. To reconcile the two positions, we assume that whisper forecasts in the period preceding earnings announcements exhibit optimism, although this may not be continued into the intermediate-term of the following 3–12 months. Further, their results may have been due to the nature of data collection with greater inaccuracy among whisper forecasts generated through the monitoring of message boards rather than the use of published whisper numbers as in our study. Their study should be replicated to determine if the results hold with more reliable measures of whisper forecasts. The final hypothesis that high-differential stocks exhibit some of the characteristics of glamour stocks, while low-differential stocks behave like value stocks are supported. Individual investors, who make up the population predicting whisper forecasts are lured by the high momentum and past operating performance of high-differential, high-volume stocks commonly known as glamour stocks to expect excessively high returns from these stocks. In contrast, they underestimate the future prices of low volume, less common value stocks with weaker momentum (dubbing them losers) and weaker operating performance. Subsequent to the earnings announcement, the market’s overreaction is corrected in the form of negative returns on the glamour stocks and positive returns on the value stocks. Although future testing is needed, we can state that high-differential stocks are indeed glamour stocks, while low-differential stocks are value stocks. Lee and Swaminathan shed further light on the issue with their finding that high-volume stocks that underperform in the short-term, do outperform in the long-term or the 3–5 year time horizon, whereas the situation is reversed with lowvolume stocks that outperform in the short-term and underperform in the long-term. Future research should determine if such reversals occur in the long-term for high- and low-differential stocks. Future research should test if the Lee and Swaminathan results may be replicated for whisper–analyst forecast differentials and dispersion in earnings forecasts. Lee and Swaminathan used trading volume as their proxy for heterogeneous expectations of earnings; although our measure was more

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direct, as was that of Diether et al. (2002) with dispersion in earnings forecasts. Could it be that there is information contained in trading volume that extends beyond the period surrounding earnings announcements that is not contained in dispersion and whisper–analyst forecast differentials so that our results cannot be sustained over the intermediate-term when momentum effects predominate (Jegadeesh & Titman, 1993)? What about the long-term, i.e., the 3–5 year period into the future; do whisper–analyst forecast differentials have any impact on such a long-time horizon? We may speculate that they would not, unless the fact that they are either glamour or value stocks results in inferior or superior performance. Regressions must be performed with forecast differentials and glamour/value as different predictors in order to be able to determine the relative importance of each variable in predicting future stock returns. The practical implications of this study surround the predictability of stock returns following earnings announcements. We can assume that stocks that are widely followed, generate much media hype and speculation between individuals and analysts about their future performance or excessive optimism about future prices will suffer the greatest corrections in the wake of earnings announcements, though such effects may reverse in the next 3– 12 months so that investors should be willing to hold them regardless of their poor immediate performance. Likewise, value stocks may exhibit positive returns during the first month following the earnings announcement; however, in the intermediate-term, i.e., the 3–12 month time horizon this situation may revert to weaker performance as momentum effects outweigh earnings effects. However, it does not pay to hold glamour stocks over the 3–5 year time horizon as they may then suffer declines in returns exhibited in the first month following earnings announcements. This suggests considerable volatility in stock prices so that the investors who are most likely to profit are those who clearly define the time horizon for holding their particular assets. Those who are ultra-short-term investors should hold value stocks, intermediate 3–12 month investors should hold glamour stocks, and long-term investors should hold value stocks.

REFERENCES Abarbanell, J. S., & Bushee, B. J. (1997). Fundamental analysis, future EPS, and stock prices. Journal of Accounting Research, 35(1), 1–24. Bagnoli, M., Beneish, M. D., & Watts, S. G. (1999). Whisper forecasts of quarterly earnings per share. Journal of Accounting and Economics, 28, 27–50. Brennan, M. (1995). The individual investor. Journal of Financial Research, 18, 59–74.

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Brennan, M., Jegadeesh, N., & Swaminathan, B. (1993). Investment analysis and the adjustment of stock prices to new information. Review of Financial Studies, 6, 799–824. Brennan, M., & Subrahmanyam, A. (1994a). Investment analysis and price formation in securities markets. UCLA Working Paper no. 23–93. Brennan, M., & Subrahmanyam, A. (1994b). Market microstructure and asset pricing: On the compensation for market illiquidity in stock returns. UCLA Working Paper no. 7–94. Chan, J., Jegadeesh, N., & Lakonishok, J. (1996). Momentum strategies. Journal of Finance, 51, 1681–1713. Chen, J., Hong, H., & Stein, J. C. (2001). Forecasting crashes: Trading volume, past returns, and skewness in security prices. Journal of Financial Economics, 61, 345–381. Daniel, K., Hirshleifer, D., & Subrahmanyam, A. (1998). A theory of overconfidence, self-attribution, and security market under- and overreactions. Journal of Finance, 53, 1839–1888. Datar, V., Naik, N., & Radcliffe, R. (1988). Liquidity and asset returns: An alternative test. Journal of Financial Markets, 1, 203–220. Diamond, D. W., & Verrecchia, R. E. (1987). Constraints on short-selling and asset price adjustments to private information. Journal of Financial Economics, 18, 277–311. DeLong, J., Bradford, A., Shleifer, A., Summers, L. H., & Waldemann, R. T. (1987). Noise trader risk in financial markets. NBER Working Paper, 2385. Diether, K. B., Malloy, C. J., & Scherbina, A. (2002). Differences of opinion and cross section of stock returns. Journal of Finance, 57, 2114–2141. http://www.Earningswhispers.com. Elton, E. J., & Gruber, M. J. (1989). New public offerings, information, and investor rationality: The case of publicly offered commodity funds. Journal of Business, 62, 1–16. Fama, E. F., & French, K. R. (1996). Multifactor explanations of asset pricing anomalies. Journal of Finance, 51, 55–84. Gibbons, M. R., Ross, S. A., & Shanken, J. (1989). A test of efficiency of a given portfolio. Econometrica, 57, 1121–1152. Goetzmann, W. H. & Massa, M. (2001). Heterogeneity of trade and stock returns: Evidence from index fund investors. ICF Working Paper No. 00–28, Yale. Hong, H., & Stein, J. C. (2000). Differences of opinion, rational arbitrage, and market crashes. Working Paper, Stanford University. Hong, H., & Stein, J. C. (1999). Differences of opinion, rational arbitrage, and market crashes. NBER Working Paper. Jarrow, R. (1980). Heterogenous expectations, restrictions on short sales and equilibrium asset prices. Journal of Finance, 35, 1105–1114. Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48, 65–395. Lakonishok, J., Shleifer, A., & Vishny, R. (1992). What do money managers do? Harvard University Working Paper. Lee, C. M., & Swaminathan, B. (2000). Price momentum and trading volume. Journal of Finance, 55, 2017–2069. Mayshar, J. (1987). On divergence of opinion and imperfections in capital markets. American Economic Review, 73, 114–128. Miller, E. (1977). Risk, uncertainty, and divergence of opinion. Journal of Finance, 32, 1151–1168. Patel, J., Zeckhauser, R., & Hendricks, D. (1991). The rationality struggle: Illustrations from financial markets. American Economic Review, 81, 311–383.

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Peavey, J. W. (1990). Returns on initial public offerings of closed-end funds. Review of Financial Studies, 3, 695–708. Stoll, H. (1978). The supply of dealer services in securities markets. Journal of Finance, 33, 1133–1151. Stollis, R. (2005). Predicting returns and volatility with macroeconomic variables: Evidence from tests of encompassing. Journal of Forecasting, 4(3), 221–232. Wang, K. S., Chan, H., & Gau, G. W. (1992). Initial public offerings of equity securities: Anomalous evidence using REITS. Journal of Financial Economics, 31, 381–410. Weiss, K. (1989). The post-offering price performance of closed-end funds. Financial Management, 18, 57–67. Welker, M., & Sparks, H. C. (2001). Trade patterns. Journal of Financial Research, 24, 261–287. Williams, J. J. (1977). Capital asset prices with heterogeneous opinion. Journal of Financial Economics, 5, 219–241. Zhang, W., Cao, Q., & Schneiderjans, M. J. (2004). Neural network earnings per share forecasting models: A comparative analysis of alternative methods. Decision Sciences, 35(2), 205–238.

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COMBINING MOVING AVERAGES WITH EXPONENTIAL SMOOTHING TO PRODUCE MORE STABLE SALES FORECASTS Tej S. Dhakar, Charles P. Schmidt and David M. Miller ABSTRACT When Winters’ exponential smoothing was applied to forecasting apparel sales at a large apparel manufacturing firm, it produced good 30-day ahead forecasts but fared poorly on long-term forecasts. It was observed that sales for many of the items exhibited extreme volatility from one month to the next, even though the moving average for these items displayed a fairly stable underlying trend. Because of extreme volatility in sales, the trend computed from individual data points experienced constant fluctuation, frequently changing direction from positive to negative and vice versa. As a result, for many key items with relatively stable long-term trends, the long-term forecasts produced by Winters’ exponential smoothing were not accurate. In some cases with significant negative trend, the long-term forecasts turned negative – something that was not acceptable. To overcome these problems, the paper proposes a forecasting model that uses centered moving average to estimate the base level. While the Advances in Business and Management Forecasting, Volume 4, 119–131 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04007-9

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base level fluctuates constantly when Winters’ exponential smoothing is used, the base level shifts gradually and smoothly when the centered moving average is used. As a result, the new system not only produces good 30-day ahead forecasts but also leads to more accurate long-term forecasts. While Winters’ exponential smoothing performs very well on some items and very poorly on others, the new forecasting model performs more consistently over all items.

INTRODUCTION The smoothing technique known as Winters’ Method (Winters, 1960) has been widely discussed and used in time series forecasting. In some applications, it can lead to accurate estimates while falling short in others. One such situation encountered by the authors led to the modification presented in this paper. In particular, a major apparel manufacturer was utilizing Winters’ method along with several other techniques to forecast sales for a wide variety of women’s garments. The firm engaged a team of faculty and graduate students from the University of Alabama’s outreach program, the Alabama Productivity Center, to investigate possible improvements in the forecasting strategies and techniques currently being used by their Forecast Analysts. The team’s experience was that Winters’ exponential smoothing produced good 30-day ahead forecasts but fared poorly on longer-term forecasts. It was observed that many of the items exhibited extreme volatility from one month to the next, even though the moving average for those items displayed a fairly stable underlying trend. Because of extreme volatility in sales from one month to the next, the trend based on individual data points experienced constant fluctuation, frequently changing direction from positive to negative and vice versa. As a result, for many items with relatively stable long-term trend, the longer-term forecasts from Winters’ exponential smoothing were not accurate. In some cases with significant negative trends, the longer-term forecasts turned negative – something that was not acceptable. To overcome these difficulties we developed a forecasting model that uses centered moving average to estimate the base level. While the base level fluctuated constantly when Winters’ exponential smoothing was used, the base level shifted gradually and smoothly when the centered moving average was used. As a result, the new system not only produced good 30-day ahead forecasts but also led to more accurate longer-term forecasts. It was also observed that while Winters’ exponential smoothing performed very well on

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many items and very poorly on others, the new forecasting model performed more consistently over all items. The next section states a real example that is used throughout the paper to explain the forecasting model. We then present the notation used in the paper followed by the proposed forecasting model. The following sections discuss the computation of the base level, the trend, the seasonal factors, the projected base levels and the forecasts under the proposed forecasting model. This is followed by a discussion of model fitting and application. Finally, we present the results of the application of the forecasting model at a large apparel manufacturing company and conclude the paper.

EXAMPLE We will use the following example of a basic women’s robe sold through several retail outlets. The example consists of data for about seven years. The sales data are ‘‘booked for shipment’’ data, i.e., sales are classified into months according to shipment dates (Table 1).

NOTATION The following basic notations will be used throughout this paper. Special notation will be introduced when required. t ¼ Any month i ¼ Calendar month corresponding to month t (i ¼ 1, 2, y, 12) Table 1.

Sales for Robes.

Month

1997

1998

1999

2000

2001

2002

2003

1 2 3 4 5 6 7 8 9 10 11 12

12,705 17,919 26,275 10,830 7,731 13,249 13,193 20,673 32,330 17,570 14,803 24,221

12,850 28,136 24,946 21,070 21,458 33,149 25,625 26,774 24,134 19,994 10,407 13,281

18,577 23,468 26,493 23,179 19,110 25,319 19,423 21,605 24,131 12,185 11,072 10,382

13,720 20,833 25,943 19,251 18,605 22,743 20,726 19,983 19,485 11,883 8,185 11,660

13,802 19,407 20,173 18,485 13,602 22,069 16,212 15,494 23,107 9,365 7,492 9,997

9,811 20,743 15,699 12,287 11,995 16,463 11,058 14,553 14,300 7,965 5,747 5,780

7,117 6,125 9,016 6,801 6,883 5,981 5,605 4,892 8,035 4,146

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Ft ¼ Forecast for month t At ¼ Sales for month t Mt ¼ Moving average centered on month t Bt ¼ Base level in month t Tt ¼ Trend in month t Si ¼ Most recent smoothed value of seasonal factor for calendar month i 0 Si ¼ Previous smoothed value of seasonal factor for calendar month i b ¼ Smoothing constant for trend d ¼ Damping constant for trend g ¼ Smoothing constant for seasonal factors

PROPOSED FORECASTING MODEL The proposed forecasting model is based upon the premise that the centered moving average is a good indicator of the underlying base level even when sales are trending up or down. Computation of the moving average is in itself a smoothing process and each moving average, by definition, is related to the value of the previous moving average. Therefore, no further smoothing of the base level is considered necessary. Trend is computed as the change in moving average and smoothed with prior estimates. The seasonal factors too are computed with respect to the centered moving average and are smoothed once in each calendar year with prior estimates. The base level (centered moving average) is projected into the future and the individual forecasts are computed by applying the most recent values of seasonal factors to the projected values of base level. If t is the latest month for which data are available and i the calendar month (1, 2, y, 12) corresponding to t, then the complete forecasting model can be stated as follows: M t
5:5 ¼ M t
6:5 þ

At
At
12 12

T t
5:5 ¼ bðM t
5:5
M t
6:5 Þ þ ð1
bÞT t
6:5 S i
6 ¼ g

Y t
6 þ ð1
gÞS 0i
6 M t
6

Btþx ¼ M t
5:5 þ

ðd5:5þx
1ÞT t
5:5 ln d

F tþx ¼ Btþx Siþx

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Data for the first 13 months are used in computing the initial values of the base level and trend. Smoothing of the base level and trend begins from month 14. Data for the first 18 months are used in computing initial values of the seasonal factors and are smoothed beginning from month 19. As a result, a minimum of 18 months of data are required before the first forecast can be produced using the initial values of seasonal factors.

COMPUTATION OF BASE LEVEL Centered moving average is used to estimate the base level. Initial (first) value of the base level is computed as M 6:5 ¼ ðA1 þ A2 þ    þ A12 Þ=12: Thereafter at the end of each month t, a new value of moving average, centered on month t
5.5, is computed by the following equation: M t
5:5 ¼ M t
6:5 þ ðAt
At
12 Þ=12: Since computing the centered moving average is itself a smoothing process, no additional smoothing is carried out. For the robes example, the initial base level M6.5 is computed at the end of period 12 as M 6:5 ¼ ðA1 þ A2 þ    þ A12 Þ=12 ¼ ð12; 705 þ 17; 919 þ    þ 24; 221Þ=12 ¼ 17; 625: At the end of period 13, M7.5 can be computed as M7.5 ¼ M6.5+(A13
A1)/12 ¼ 17,625+(12,850
12,705)/12 ¼ 17,637.

COMPUTATION OF TREND The initial value of the trend T7.5 is computed as T7.5 ¼ M7.5
M6.5. Thereafter, given the smoothing constant b, trend is updated through the following equation: T t
5:5 ¼ bðM t
5:5
M t
6:5 Þ þ ð1
bÞT t
6:5 : For the robes example, the initial value of trend T7.5 is computed as T7.5 ¼ M7.5
M6.5 ¼ 17,637
17,625 ¼ 12. Assuming b ¼ 0.4, the second value of trend T8.5 is computed as T8.5 ¼ b(M8.5
M7.5)+(1
b) T7.5 ¼ 0.4 (18,488
17,637)+(1
0.4) 12 ¼ 348.

COMPUTATION OF SEASONAL FACTORS The initial values of the seasonal factors are computed beginning from period 13, when the first moving average centered on a month M7 becomes available as M7 ¼ (M6.5+M7.5)/2. Assuming that first data in the time series belongs to January (calendar month 1) as in the robes example, initial value of the seasonal factor S7 is computed by the ratio: S7 ¼ A7/M7. Initial value

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of the seasonal factor S8 is computed by the ratio: S8 ¼ A8/M8. This process is continued until initial values of all the seasonal factors have been computed by the end of period 24. Initial values of the seasonal factors are smoothed beginning from period 25. After each month t, moving average centered on month t
6, denoted Mt
6, is computed by the following equation: Mt
6 ¼ (Mt
5.5+Mt
6.5)/2. 0 Given Si as the previous value of seasonal factor for calendar month i and smoothing constant g, the new smoothed value, denoted Si is determined by 0 the following smoothing equation: Si ¼ g (At
6/Mt
6)+(1—g) Si . For the robes example, at the end of month 13, the moving average centered on month 7 is computed as M7 ¼ (M6.5+M7.5)/ 2 ¼ (17,625+17,637)/2 ¼ 17,631 and initial value of the seasonal factor S7 is computed by the ratio: S7 ¼ A7/M7 ¼ 13,193/17,631 ¼ 0.74828. The second value of S7 is computed at the end of month 25 (January 1999). Thus, 0 assuming g ¼ 0.6, S7 ¼ g (A19/M19)+(1
g) Si ¼ 0.6 (25,625/ 22,057)+(1
0.6) 0.74828 ¼ 0.9964.

COMPUTATION OF PROJECTED BASE LEVELS The most recent moving average Mt
5.5 is centered on the month t
5.5 and therefore, the moving average line ends at the month t
5:5: The most recent updated trend Tt
5.5 is used to project from this moving average in order to compute the projected base levels. Projecting 6.5 periods into the future from this moving average results in the base level centered on month t+1, which is used to compute the 30-day ahead forecast. Projecting 7.5 periods into the future from this moving average results in the base level centered on month t+2, which is used to compute the 60-day ahead forecast. In general, projecting x+5.5 periods into the future from this moving average results in the base level centered on the month t+x, which is used to compute the x
month ahead forecast. Given uncertainties of the future and the fact that trend rarely stays the same, the trend is dampened by the damping constant d (0.9 r d r 1) such that trend at time t+1 is d times the trend at time t. Therefore, the base level in period t can be computed as Bt+1 ¼ Mt
5.5+(d6.5
1) Tt
5.5)/ ln d. In general, the base level for period t+x can be computed as Bt+x ¼ Mt
5.5+(d5.5+x
1) Tt
5.5)/ ln d (see Appendix A for derivation). For the robes example, the last month for which sales data are available is October 2003 (t ¼ 82). The last moving average can be computed as M76.5 ¼ 6,344. The most recent smoothed trend can be determined to be

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T76.5 ¼
491.9. Assuming d equal to 0.95, the base level for period 83 (November 2003) can be computed as B83 ¼ M76.5+(d6.5
1) T76.5 /ln d ¼ 6344+(0.956.5
1) (
491.9)/ln 0.95 ¼ 3,625. Similarly, the base level for period 84 (December 2003) can be computed as B84 ¼ M76.5+(d7.5
1) T76.5/ln d ¼ 6344+(0.957.5
1) (
491.9)/ln 0.95 ¼ 3,281.

COMPUTATION OF FORECASTS Once the x-step ahead projected base level Bt+x has been computed and the relevant seasonal factor Si is available, then the forecast is computed as Ft+x ¼ Bt+x Si. For the robes example, the 30-day ahead (November 2003) base level B83 was computed as 3,625. The smoothed value of the seasonal factor for November S11 is 0.5694. Therefore, the forecast for November 2003 can be computed as F83 ¼ B83 S11 ¼ 3,625 * 0.5694 ¼ 2,064. The 60day ahead (December 2003) base level B84 was computed as 3,281. The smoothed seasonal factor for December S12 is 0.6684. Therefore, the forecast for December 2003 can be computed as F84 ¼ B84 S12 ¼ 3,281 * 0.6684 ¼ 2,193.

MODEL FITTING AND APPLICATION After computing the initial base level from the first 12 data points, the forecasting model is applied starting from the month 13 until the latest period for which data are available. Application of the model requires numerical values for the smoothing constants, i.e., b for trend and g for seasonal factors and d for damping constant. Since the trend is updated every month, the smoothing constant for trend b is allowed to vary between 0.1 and 0.5 in steps of 0.1. The seasonal factors are updated once in a calendar year and therefore, the smoothing constant for seasonal factors g is allowed to vary over a wider range between 0.1 and 0.9 in steps of 0.1. We restrict the damping constant d to values between 0.9 and 1 in steps of 0.01 so that the trend would be dampened by a minimum of 0% and a maximum of 10% per month. Since the forecasts are dependent on values of the smoothing constants and the damping constant, the model is executed with different combinations of values of smoothing constants and the damping constant. Each time the model is executed, the mean absolute deviation (MAD) for the 30-day ahead sales forecasts for the last 12 months is recorded. The combination of

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values of the smoothing constants and the damping constant that results in the least MAD for the last 12 months is selected as the ‘‘best’’ combination of smoothing constants and the damping constant. It may be noted that minimizing MAD minimizes MAPE as defined in the next section. After the ‘‘best’’ smoothing constants and the damping constant have been determined, the model is re-executed with the ‘‘best’’ smoothing constants and the damping constant to compute the final forecasts. A spreadsheet implementation of the forecasting model for the robes example can be found in Appendix B.

MODEL PERFORMANCE The model was tested on 80 items at a large apparel manufacturing company for a period of six months. Forecast error for the 30-day ahead forecasts was computed as follows: Mean Absolute Percent Error ðMAPEÞ ¼

Total Absolute Error  100 Total Sales

MAPE for the model forecasts was compared with MAPE for the forecasts produced by Forecast Analysts at the apparel manufacturing company. MAPE for the Analysts for the 80 items for the six-month period was computed to be 24.5%. The forecasting model proposed in the paper reduced the MAPE for the same items for the same period to 22.2% – a reduction of 9.4%. This improved level of forecast error was particularly attractive in view of the fact that it was not aided by the extraneous sales information available to Forecast Analysts and incorporated into their forecasts. A computerized system based on the proposed forecasting system was implemented at the apparel manufacturing company. It was decided to provide the Analysts with forecasts from the new forecasting system, which they could further improve based on additional information available to them and their experience and judgment.

CONCLUSION When applying Winters’ exponential smoothing technique to forecasting apparel sales for a major company, we noticed extreme volatility in monthly

Combining Moving Averages with Exponential Smoothing

127

sales figures. The extreme volatility made the model parameters swing wildly resulting in a high degree of instability in the forecasts. We, therefore, decided to use centered moving averages in combination with Winters’ exponential smoothing to bring about stability in the performance of the forecasting model. Our objective was to produce not only good short-term but good longer-term forecasts as well. We found that the centered moving average was a good indicator of the underlying base level. In addition, it is based entirely on actual data and therefore, avoids the assumptions required in the process of computing the smoothed values of the base level in Winters’ exponential smoothing model. The forecasting model was tested on 80 items for a period of six months at a large apparel manufacturing company. The results from the forecasting model were compared with forecasts developed by the Forecast Analysts. The forecasting model reduced the forecast error by an average of 9.4% compared to the Analysts without the benefit of extraneous information available to the Analysts.

REFERENCE Winters, P. (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6, 324–342.

APPENDIX A: DERIVATION OF FORMULA FOR PROJECTED BASE LEVEL Starting with a base level M0 and smoothed trend T0 and given a damping constant d such that the trend in period t+1 equals the trend in period t times the damping constant d, the base level at time t can be computed as follows:
t t Z t d ðdt
1ÞT 0 t Bt ¼ M 0 þ T 0 d dt ¼ M 0 þ T 0 ¼ M0 þ ln d 0 ln d 0 Changing the origin to t
5.5 and extending the base level to t+x, the above can be rewritten as Btþx ¼ M t
5:5 þ

ðd5:5þx
1ÞT t
5:5 ln d

128

APPENDIX B: SPREADSHEET IMPLEMENTATION OF THE FORECASTING MODEL Beta ¼ 0.4 Gamma ¼ 0.6 Delta ¼ 0.95 Sales Yt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

12,705 17,919 26,275 10,830 7,731 13,249 13,193 20,673 32,330 17,570 14,803 24,221 12,850 28,136 24,946 21,070 21,458 33,149 25,625 26,774

Period Mov Avg T Mt

6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.5

17,625 17,637 18,488 18,378 19,231 20,375 22,033 23,069 23,578 22,895 23,097 22,730 21,819 22,296 21,907

Trend Tt

12.1 347.8 164.4 440.0 721.5 1,096.3 1,072.2 846.7 234.8 221.7
13.5
372.8
32.8
175.3

30-day 60-day Period Mov Avg Seasonal 30-day 60-day Bt Bt t Mt Si Ft Ft

17,704 20,411 19,286 21,663 24,363 28,093 28,996

17,712 20,654 19,401 21,970 24,867 28,858

7 8 9 10 11 12 13 14 15 16 17 18 19 20

17,631 18,063 18,433 18,804 19,803 21,204 22,551 23,323 23,236 22,996 22,914 22,275 22,057 22,101

0.7483 1.1445 1.7539 0.9344 0.7475 1.1423 0.5698 1.2063 1.0736 0.9163 0.9365 1.4882 0.9964 1.1847

21,021 18,608 33,186 33,029

Error Et

4603.565 6411.768

TEJ S. DHAKAR ET AL.

Period T

21.5 22.5 23.5 24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5 32.5 33.5 34.5 35.5 36.5 37.5 38.5 39.5 40.5 41.5 42.5 43.5 44.5 45.5 46.5 47.5 48.5 49.5

22,036 22,212 22,016 21,363 20,847 20,416 20,416 19,765 19,820 19,579 19,174 18,954 18,909 18,581 18,539 18,324 18,433 18,298 17,911 17,886 17,645 17,751 17,758 17,639 17,159 17,095 16,678 16,622 16,246


53.6 38.1
55.4
294.2
383.3
402.3
241.5
405.2
220.9
229.2
299.4
267.5
178.8
238.2
159.8
181.7
65.6
93.4
210.9
136.6
178.2
64.3
35.9
69.0
233.8
165.8
266.2
182.2
259.8

28,258 24,192 24,322 22,656 19,758 22,115 20,938 21,740 22,422 21,710 19,737 18,728 18,192 19,081 17,525 18,599 18,312 17,519 17,476 17,920 17,264 17,656 17,320 18,070 17,781 16,745 17,130 16,660 17,396

29,744 28,849 24,356 24,477 22,646 19,498 22,092 20,816 21,702 22,449 21,671 19,532 18,460 17,911 18,912 17,242 18,445 18,152 17,310 17,289 17,795 17,098 17,544 17,193 18,025 17,716 16,597 17,035 16,535

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

21,971 22,124 22,114 21,690 21,105 20,631 20,416 20,090 19,793 19,699 19,376 19,064 18,931 18,745 18,560 18,432 18,379 18,365 18,104 17,898 17,765 17,698 17,755 17,699 17,399 17,127 16,886 16,650 16,434

1.3606 0.9160 0.5814 0.8243 0.7561 1.1650 1.2080 1.0588 0.9539 1.3664 1.0000 1.1538 1.3090 0.7564 0.5905 0.6677 0.7503 1.1466 1.3430 1.0689 1.0099 1.3176 1.1004 1.1390 1.1956 0.7189 0.5270 0.6873 0.8041

49,561 22,604 18,181 25,879 11,258 26,678 22,479 19,919 20,998 32,309 19,665 22,186 24,753 17,478 10,189 15,331 13,845 20,410 21,112 18,973 16,468 24,126 17,320 20,850 23,277 12,666 10,115 11,124 13,053

52,169 25427.34 26,955 2610.475 18,207 7774.034 27,959 12598 12,904 7318.557 23,521 3209.93 23,718 4014.099 19,073 3259.869 20,324 1888.126 33,409 6989.628 21,592 242.3018 23,138 581.3018 25,118 621.9457 16,407 5292.865 10,995 883.2886 14,213 4949.244 13,945 124.7451 21,147 422.8556 20,911 4831.493 18,305 278.1081 16,975 2136.503 23,363 1382.812 17,544 3406.226 19,838 867.1453 23,595 3791.607 13,401 783.1255 9,800 1930.083 11,374 536.4944 12,407 749.3206

129

24,134 19,994 10,407 13,281 18,577 23,468 26,493 23,179 19,110 25,319 19,423 21,605 24,131 12,185 11,072 10,382 13,720 20,833 25,943 19,251 18,605 22,743 20,726 19,983 19,485 11,883 8,185 11,660 13,802

Combining Moving Averages with Exponential Smoothing

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

130

APPENDIX B.

(Continued )

Period Sales Period Mov Avg Trend 30-day 60-day Period Mov Avg Seasonal 30-day 60-day T Yt T Mt Tt Bt Bt t Mt Si Ft Ft 19,407 20,173 18,485 13,602 22,069 16,212 15,494 23,107 9,365 7,492 9,997 9,811 20,743 15,699 12,287 11,995 16,463 11,058 14,553 14,300 7,965 5,747 5,780

50.5 51.5 52.5 53.5 54.5 55.5 56.5 57.5 58.5 59.5 60.5 61.5 62.5 63.5 64.5 65.5 66.5 67.5 68.5 69.5 70.5 71.5 72.5

15,871 16,173 15,963 15,906 15,767 15,435 15,546 15,173 14,657 14,523 14,055 13,626 13,548 12,814 12,697 12,552 12,200 11,976 10,757 10,201 9,743 9,317 8,444


305.5
62.6
121.5
96.0
113.0
200.8
76.0
194.7
323.4
247.6
335.4
373.1
255.2
446.7
314.7
247.0
288.8
263.1
645.1
609.8
548.8
499.7
649.2

17,560 17,258 15,866 16,178 15,206 15,615 14,810 14,183 15,827 15,292 15,375 15,142 14,324 15,126 14,097 12,869 13,154 12,201 11,564 12,137 10,345 10,958 11,186

17,351 17,535 17,210 15,703 16,063 15,020 15,487 14,628 13,969 15,784 15,207 15,308 15,063 14,184 15,073 13,961 12,643 12,981 11,967 11,303 11,959 10,033 10,738

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

16,058 16,022 16,068 15,935 15,836 15,601 15,490 15,359 14,915 14,590 14,289 13,841 13,587 13,181 12,755 12,624 12,376 12,088 11,367 10,479 9,972 9,530 8,881

1.1838 1.2926 1.1178 0.9161 1.3632 1.0637 1.0557 1.3809 0.6643 0.5189 0.6947 0.7469 1.3895 1.2317 1.0251 0.9366 1.3434 0.9743 1.1905 1.3711 0.7450 0.5694 0.6684

20,135 23,177 16,959 16,339 20,036 17,182 16,868 16,956 11,378 8,059 10,567 12,175 16,957 19,552 15,757 11,790 17,931 12,978 12,208 16,759 6,872 5,686 7,771

19,895 23,550 18,394 15,859 21,164 16,528 17,639 17,489 10,042 8,318 10,451 12,308 17,832 18,335 16,848 12,790 17,234 13,807 12,634 15,608 7,944 5,206 7,459

727.8666 3004.21 1526.051 2736.889 2033.239 970.2455 1373.529 6150.812 2012.782 567.1501 569.6846 2364.234 3786.364 3853.234 3470.032 205.4274 1468.065 1920.11 2344.671 2459.457 1093.256 60.93097 1990.952

TEJ S. DHAKAR ET AL.

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

Error Et

7,117 6,125 9,016 6,801 6,883 5,981 5,605 4,892 8,035 4,146

73.5 74.5 75.5 76.5

7,989 7,184 6,662 6,344


571.3 10,604 11,014
664.8 10,522 10,402
607.7 7,192 10,338
491.9 6,830 6,741 6,710 6,404 6,556 6,327 4,855 6,207 4,832 4,402 3,510 4,433 3,303 3,045 3,625 2,879 3,281

73 74 75 76

8,217 7,587 6,923 6,503

0.8185 1.0402 1.2740 1.0375

7,920 8,227 14,620 14,454 8,858 12,733 7,001 6,910 6,284 5,997 8,807 8,500 4,731 6,047 5,752 5,241 4,812 6,078 2,461 2,269 2,064 1,639 2,193

803.4918 8495.094 158.0281 200.0563 598.6666 2825.843 874.098 860.1073 3222.781 1685.293

Combining Moving Averages with Exponential Smoothing

73 74 75 76 77 78 79 80 81 82 83 84

131

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132

IMPROVED EXPONENTIAL SMOOTHING WITH APPLICATIONS TO SALES FORECASTING Tej S. Dhakar, Charles P. Schmidt and David M. Miller ABSTRACT When the number of observations and smoothing constant are small in the exponential smoothing procedure, the oldest observation receives a lopsided weight compared to other observations. This paper proposes a correction to the smoothing equation so that weights decrease continuously with a constant ratio between any two successive weights. The new smoothing equation is applied to various forecasting models.

INTRODUCTION Exponential smoothing is used to compute the weighted average of observations collected over a number of periods in the past. Each new observation is given a higher weight compared to the previous observation. This procedure is used in a variety of applications including forecasting.

Advances in Business and Management Forecasting, Volume 4, 133–138 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04008-0

133

134

TEJ S. DHAKAR ET AL.

The underlying smoothing equation used for exponential smoothing can be stated as given below X^ t ¼ aX t þ ð1
aÞX^ t
1 where Xt are the actual observations, X^ t the smoothed value, and a the smoothing constant. This is a recursive equation, which can be written in the expanded form as follows. X^ t ¼ aX t þ að1
aÞX t
1 þ að1
aÞ2 X t
2 þ . . . þ að1
aÞt
2 X 2 þ ð1
aÞt
1 X 1 As can be seen in the above equation, exponential smoothing maintains a constant ratio of 1/(1
a) between any two successive weights except for the last two weights in the equation, where the ratio becomes a/(1
a). When ao0.5, the ratio a/(1
a) between the first two observations (X1 and X2) becomes smaller than 1 and subsequently, results in a higher weight for the first observation compared to the second observation. For example, with t ¼ 4 and a ¼ 0.2, X^ 4 ¼ 0:2X 4 þ 0:16X 3 þ 0:128X 2 þ 0:512X 1 This would not be a problem, if there were a large number of observations or the smoothing constant was large as the weight assigned to the first observation would be very small. But when the number of observations and the smoothing constant are small, the first observation has a lopsided weight compared to the rest of the observations in the equation, such as in the example above. Assuming there is no particular reason why the first observation should be given a lopsided weight, the paper proposes a correction to the smoothing equation so that a constant ratio is maintained between any two successive weights including the weights for the first two observations in the equation.

PROPOSED SMOOTHING EQUATION Let k be a constant ratio between any two successive weights, such that the weight given to any observation is k times the weight given to the previous observation. Then, kt
1 X t þ kt
2 X t
1 þ    þ kX 2 þ X 1
t  X^ t ¼ k
1 k
1

where the denominator is the sum of the geometric series kt
1, kt
2, y, k, 1.

Improved Exponential Smoothing

135

The above equation can be rewritten as   kt
1 ðk
1Þ k
1 ^ Xt ¼ Xt þ ðkt
2 X t
1 þ    þ kX 2 þ X 1 Þ ðkt
1Þ kt
1 Noting the recursive nature of the equation 0 1  t
1  t
2 t
1 k ðk
1Þ k
1 @k X t
1 þ    þ kX 2 þ X 1 A
t
1  X^ t ¼ Xt þ k
1 ðkt
1Þ kt
1 k
1

Thus, the final equation can be written as  t
1  kt
1 ðk
1Þ k
1 ^ ^ X t
1 Xt þ Xt ¼ ðkt
1Þ kt
1 where k in the above equation is the constant ratio between any two successive weights. k ¼ 1 implies equal weights for all observations and the smoothed value would be a simple average of all the observations. k ¼ 2 implies that a new observation would be given twice the weight as given to the previous observation. k in the above equation is equivalent to the constant ratio of 1/(1
a) between any two successive weights (except for the last two) in standard exponential smoothing. A comparison between k and a for a few values is given in Table 1. As noted before with t ¼ 4 and a ¼ 0.2, the standard exponential smoothing resulted into the following expanded equation with a lopsided weight for the first observation: X^ 4 ¼ 0:2X 4 þ 0:16X 3 þ 0:128X 2 þ 0:512X 1 Using the equivalent value of k ¼ 1/(1
a) ¼ 1.25 and the new exponential smoothing equation, we get the following expanded equation. In this equation, a constant ratio of 1.25 is maintained between all successive observations. X^ 4 ¼ 0:339X 4 þ 0:271X 3 þ 0:217X 2 þ 0:173X 1 The new exponential smoothing equation requires carrying only the serial number of the observation (the number t) in addition to the information carried forward from the previous period by standard exponential smoothing. Table 1. k a

1.1 0.081

1.2 0.167

Comparison between k and a. 1.5 0.333

2 0.5

3 0.667

5 0.8

10 0.9

136

TEJ S. DHAKAR ET AL.

Just as in case of standard exponential smoothing, no information from any periods prior to the last needs to be carried forward. Even though the new smoothing equation requires slightly more computation, the equation can be incorporated easily into any spreadsheet or computer program. Given the computing speeds available today, the additional computation time is negligible even on a personal computer.

APPLICATIONS TO SALES FORECASTING Sales forecasting is one of the important areas of application of the exponential smoothing concept. The new exponential smoothing equation can be incorporated into different sales forecasting models as presented below. Weighted moving average: The weighted moving average is used for shortterm forecasting when trend and seasonal variations are not important. The weighted moving average equation can be written as follows: F tþ1 ¼ wp At þ wp
1 At
1 þ    þ w2 At
pþ2 þ w1 At
pþ1 where At is the actual sales for period t, Ft the forecast for period t, and wt are the weights such that all weights add up to 1. The new exponential smoothing equation can be used to develop weights such that a constant ratio k is maintained between successive weights. The following equation used recursively p
1 times with A^ t
pþ1 ¼ At
pþ1 would result in a p-period weighted moving average with a constant ratio between successive weights:  p
1  kp
1 ðk
1Þ k
1 ^ F tþ1 ¼ A^ t ¼ A þ At
1 t ðkp
1Þ kp
1 Simple exponential smoothing: Simple exponential smoothing too is used for short-term forecasting when trend and seasonal variations are not important. Simple exponential smoothing uses the following equation: F tþ1 ¼ aAt þ ð1
aÞF t where At is the actual sales for period t, Ft the forecast for period t, and a the smoothing constant. The revised equation after incorporating the proposed exponential smoothing method with k as the constant ratio between successive weights and F0 as the starting forecast can be written as follows:  t  kt ðk
1Þ k
1 At þ F tþ1 ¼ tþ1 Ft ðk
1Þ ktþ1
1

Improved Exponential Smoothing

137

Exponential smoothing with trend: Holt (1957) extended simple exponential smoothing model to forecasting when demand pattern includes significant underlying trend. He suggested the following equations, where in addition to the symbols used above Bt is the base level in period t, Tt the trend in period t, and a and b are the smoothing constants for base level and trend, respectively. F tþ1 ¼ Bt þ T t Bt ¼ aAt þ ð1
aÞðBt
1 þ T t
1 Þ T t ¼ bðBt
Bt
1 Þ þ ð1
bÞT t
1 The new smoothing equations with B0 as the starting base level, T0 the starting value of trend, and k and g the new smoothing constants for the base level and trend respectively can be written as follows: F tþ1 ¼ Bt þ T t  t  kt ðk
1Þ k
1 A þ ðBt
1 þ T t
1 Þ t ðktþ1
1Þ ktþ1
1  t  gt ðg
1Þ g
1 ðBt
Bt
1 Þ þ tþ1 T t ¼ tþ1 T t
1 ðg
1Þ g
1

Bt ¼

Exponential smoothing with trend and seasonal variation: Winters (1960) suggested an exponential smoothing procedure for items that exhibit significant trend and seasonal variation. The equations comprising this procedure are given below, where in addition to the symbols used before, Si,j is the value of seasonal index for seasonal period i in the jth seasonal cycle and a, b, and g are the smoothing constants for base level, trend, and seasonal components, respectively. F tþ1 ¼ ðBt þ T t ÞS iþ1; j
1 Bt ¼ aðAt =Si; j
1 Þ þ ð1
aÞðBt
1 þ T t
1 Þ T t ¼ bðBt
Bt
1 Þ þ ð1
bÞT t
1 S i; j ¼ gðAt =Bt Þ þ ð1
gÞSi; j
1 The new exponential smoothing equations with B0 as the starting base level, T0 the starting value of trend, Si,0 the starting value of seasonal index for seasonal period i, and k, g, and h the new smoothing constants for the

138

TEJ S. DHAKAR ET AL.

base level, trend, and seasonal indices, respectively can be written as follows: F tþ1 ¼ ðBt þ T t ÞS iþ1; j
1  t  kt ðk
1Þ k
1 ðAt =S i; j
1 Þ þ Bt ¼ tþ1 ðBt
1 þ T t
1 Þ ðk
1Þ ktþ1
1  t  gt ðg
1Þ g
1 ðBt
Bt
1 Þ þ tþ1 T t ¼ tþ1 T t
1 ðg
1Þ g
1 S i; j

 j  h j ðj
1Þ h
1 ðAt =Bt Þ þ ¼ jþ1 Si; j
1 ðh h jþ1
1
1Þ

CONCLUSION Exponential smoothing procedure is used in a variety of applications including forecasting to compute the weighted average of observations collected over a number of periods in the past. When the number of observations and the smoothing constant are small, the first observation receives a lopsided weight compared to the rest of the observations in the equation. The paper has presented a correction to the smoothing equation so that a constant ratio is maintained between any two successive weights including weights for the first two observations in the equation. The new equation requires carrying only the serial number of the observation (the number t) in addition to the information carried forward from the previous period when using standard exponential smoothing. Even though the new smoothing equation is slightly more complex than the standard equation, it can be incorporated easily into any spreadsheet or computer program. Given the computing speeds available today, the additional computation time is negligible even on a personal computer.

REFERENCES Holt, C. C. (1957). Forecasting seasonals and trends by exponentially weighted moving averages. Pittsburgh, PA: Carnegie Institute of Technology (O.N.R. Memorandum, No. 52).. Winters, P. F. (1960). Forecasting sales by exponentially weighted moving averages. Management Science, 6, 324–342.

USING FLOW-THROUGH AND DIFFUSION MODELS TO FORECAST NEW PRODUCT SALES Michael D. Geurts and David B. Whitlark FLOW-THROUGH MODELS The forecasting of new products can often be done with a combination of flow-through models and diffusion models. Flow-through models are some times called a funnel model or economic analysis. The process is to identify the total potential market based on demographics and then to reduce that number by the factors that might influence the availability of the product, the consumer awareness, the cost of the product, or relative advantage of the product. For example, there is a potential new product of non-injected insulin. The insulin would be administered orally or as a nasal spray. The initial process in a flow-through model would be to estimate the demographics of potential users. In the case of insulin, it would be primarily people with diabetes. However, not everyone with diabetes would use the product. Suppose the estimated population in the US and Canada that has diabetes is 30,000,000 people. Suppose also that an additional 2,000,000 people get diabetes each year and that 1,000,000 with diabetes die every year. That would mean that over a 10-year period there would be a potential market of 40,000,000 users. The first limitation that would keep some of the 40,000,000 people from using the product would be that they have diabetes but it has not been Advances in Business and Management Forecasting, Volume 4, 139–143 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04009-2

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MICHAEL D. GEURTS AND DAVID B. WHITLARK

diagnosed. If this were 25% of the diabetics, this would reduce the potential users to 30,000,000 potential buyers. A second reduction factor might be those diabetics controlling the disease by diet and exercise. This might be 20% of diabetics; this would reduce the market potential to 24,000,000 buyers. An additional adjustment needs to be made because some diabetics control with medication rather than insulin injections; suppose this is 50% of those still in the potential users group. Some of these individuals may not go to the oral/nasal insulin. Suppose that of the 12,000,000 medicationcontrolled diabetics 4,000,000 prefer to not adopt the new insulin technology. That results in the potential buyers being reduced to 20,000,000. The next consideration might be price suppose because the cost for the new insulation is higher than injected and some insurance companies will not pay for the oral/nasal insulin. As a result, another 5,000,000 potential users will not adopt resulting in the potential buyers being reduced to 15,000,000. An additional issue is diabetics knowing of the product. If the product is marketed only to doctors, there may be some individuals who rarely go to doctors that do not find out about the product. Others may be in a geographic area that is not able to be serviced with the new medication. Also, some individuals may be very reluctant to change to a new process. Assume that 1,000,000 people are in the above situations reducing the potential buyers to 14,000,000 people. As a result, the forecast of adopters is 14,000,000 individuals in the next 10 years. The remaining forecast is to estimate how many adopters there may be in each year. To do this a diffusion model might be used.

DIFFUSION MODELS The diffusion process is the idea that new products flow through a population with a predictable rate. Individuals are classified into groups based on the time from which the product is introduced until it is adopted. The first groups to adopt are called innovators and early adopters. Subsequent adopters are called imitators. The imitators can be further divided into groups such as early majority, late majority, and laggards. The rate at which the adoption of the product goes from the innovator group to the laggard group is called the adoption or diffusion rate. The rate depends on the nature of the new product and how it is marketed. The analogy is often used that new products flow through population like new diseases flow through a population. Forecasters who work on disease forecasting can generally forecast the spread of a disease like a new strain of

Using Flow-Through and Diffusion Models

141

flu with a high amount of accuracy. The adoption process is sometimes called epidemic theory. The diffusion forecasting model classifies adopters into two groups, innovators and imitators. Innovators adopt when they hear about the product. Imitators adopt based on knowing others are hearing about others who have adopted the product. The rate of adoption will be increased if the product has the following characteristics: A. B. C. D. E. F. G. H.

Can be tried in small samples Consistent with existing technology/products Simplicity Easily recognized advantage Costs Industrial product Medical product Is marketed to a population with a large number of innovators.

The diffusion model requires three inputs: 1. p, the coefficient of innovation 2. q, the coefficient of imitation 3. m, the market potential. p and q can be calculated from an article written by Sultan, Farley, and Lehmann (1990). This article is a summary of several diffusion studies and suggests what the values for p and q should be for different product characteristics, market types, and geographic locations. A forecaster must estimate the m value. This is the total market potential that is expected to be sold. A flow-through model could be used to estimate m. In the oral/nasal insulin example given above the m value is 14,000,000. The coefficients imitation and innovation change from product to product. The process of estimating p and q in diffusion studies uses OLS regression: SðT Þ ¼ a þ bY T
1 SðTÞ ¼ a þ bY T
1 þ cY 2T
1 for T ¼ 2 and 3ST ¼ sales at time T and Y T
1 ¼ St : These are the accumulated sales up to time T. The coefficient ‘‘a’’ estimates pm, ‘‘b’’ estimates q–p, and ‘‘c’’ estimates
q/m. When the model was first developed, it required p, q, and m to be determined by using at least the first 3 years of sales. Because of this, it could not be used to forecast new product sales. However, now the model can be used to forecast new product sales because many studies have been conducted to determine p and q for different products. Sultan et al. (1990) have collected the values for innovation and imitation rates from 213 applications of the diffusion model. The rates are

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classified by product type, country, type of innovation, and estimation procedure. These data are very useful for forecasters who are involved in new product forecasting that uses the diffusion model or a modified diffusion model. The, overall model can be modified to include a rate of repurchase for products that are not durables. Diffusion Model Software Software for the diffusion model is available from many sources. For example, two books; Urban and Star (1991) as well as Lilien (1993), both have software packages that contain the diffusion model as a spread sheet. The procedure is also discussed in most books on marketing models. The Oral/Nasal Insulin Example Using the Sultan et al. (1990) article, the estimate for p is 0.03. The q value is 0.55. From the flow-through forecast the m value is 14,000,000. Oral/Nasal Insulin

S in OOO

Parameter Annual sales S(t) Cumulative sales S(t) Year (t)

p 0.03 q 0.55 m 14000

471.61 704.40 1019.56 1406.67 1806.69 2092.39 2102.76 1766.62 1215.67 694.12 344.24 156.68 68.23 29.11 12.31

100.00 571.61 1276.01 2295.57 3702.24 5508.93 7601.32 9704.07 11470.69 12686.37 13380.48 13724.72 13881.41 13949.64 13978.75 13991.06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Peak time 7.0

Peak magnitude 2140.73

After 7 years nearly 10 million have adopted and after 10 years nearly all have adopted the product. Below is plots of the cumulative and yearly adoptions (Figs. 1 and 2).

Using Flow-Through and Diffusion Models

143

16000.00 14000.00 12000.00 10000.00 Series1

8000.00 6000.00 4000.00 2000.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fig. 1.

Cumulative Adoptions

2500.00 2000.00 1500.00 Series1 1000.00 500.00 0.00 1 2 3 4 5 6 7

Fig. 2.

8 9 10 11 12 13 14 15

Yearly Adoptions

REFERENCES Lilien, G. L. (1993). Marketing management. South San Francisco, CA: The Scientific Press. Sultan, F., Farley, J. V., & Lehmann, D. (1990). A meta-analysis of applications of diffusion models. Journal of Marketing Research, XXVII, 70–77. Urban, G. L., & Star, S. H. (1991). Advanced marketing strategy. Englewood Cliffs, NJ: Prentice-Hall.

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AN APPLICATION OF A REPEAT PURCHASE DIFFUSION MODEL TO THE PHARMACEUTICAL INDUSTRY Franklin J. Carter, Carol M. Motley, Alphonso O. Ogbuehi and Jacqueline A. Williams ABSTRACT It is crucial for managers in detail-intensive industries to find the optimal mix of detailing (personal selling) and other promotional efforts to forecast sales, meet sales projections, and avoid costly marketplace failures. To that end, we develop and test a repeat purchase diffusion model that allows for the individual and joint impacts of detailing, pricing, and other promotional activities (advertising and sampling) on the adoption of a new product. This model recognizes manager-defined segments of users and incorporates manager’s judgments on the allocations of resources among these various segments. Users are categorized as potential new users and repeat users, based upon salescall attractiveness ratings provided by the sales organization. Market potential is then a function of sales call activity. Our results provide the marketing manager in a detail-intensive industry more feedback on the impact that variations in marketing activities could have on new product sales than previously available. Advances in Business and Management Forecasting, Volume 4, 145–173 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04010-9

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1. INTRODUCTION New product failures are common and expensive in the consumer packaged goods industry. They are even more costly, however, in detail-intensive industries (DIIs) where the selling process is complex and relies heavily on personal selling. For example, in the promotion of pharmaceuticals, the ratio of field-force expenditures to promotional expenditures for most major products is well above unity (70% for detailing versus 30% for other promotion). Ethical pharmaceutical firms (i.e., firms that market products available only by a physician’s prescription) rely primarily on the sales force to market their products (Lodish, Curtis, Ness, & Simpson, 1988). It is difficult, however, to isolate the effect of the sales force from all other effects that might cause a fluctuation in sales. These effects include pricing, advertising, changes in distribution, customer needs, and competitive behavior. Advertising is often used by a firm to ‘‘open the door’’ for its sales people. For the sales person to be effective in such instances, the firm and products that the sales person represents should already be familiar to the potential buyer. In this case, advertising complements and enhances the direct selling effort, yet direct contact is usually essential to obtain an order. Although advertising is a far more visible form of promotion, in reality firms allocate considerably more resources for personal selling, indicating the critical role personal selling assumes in promotional activities. In modeling the diffusion process for detail-intensive firms, there must be an element of personal selling available at each level for the selling effort to be successful. All models must make provisions for the concept that if there is no personal selling, sales decrease dramatically. These models should take into account the interaction between the detailing component of the marketing mix and the supportive component. In addition, the models must include some degree of heterogeneity to resemble the various ways companies allocate their limited resources to attain maximum sales effort. The term ‘‘allocation’’ in this instance means how managers segment their markets of potential calls and divide their budgets accordingly. Managers in DIIs continually seek the appropriate mix of marketing activities that will maximize sales of existing offerings. Additionally, models that incorporate such information are sought to accurately forecast the performance of new products and services (Mahajan & Wind, 1988). Specifically, models of the diffusion process for firms in DIIs are required to make provisions for the interaction between detailing and the supportive components of the marketing mix because the presence, or possibility of, a significant level of interaction among these variables could affect

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performance (Zeithaml, 1992). However, these interactions have not been widely examined in marketing resource allocation models. Consequently, it is not clear how the response function should be estimated to include interactions between and among elements of the marketing mix. In addition, there does not appear to be any models that include some degree of heterogeneity in resource allocation to resemble the various ways limited resources could be distributed to attain maximum sales effort. We develop a repeat purchase diffusion model that specifically considers the impact of the various promotional activities on new product adoption in a DII. We extend the Mahajan, Wind, and Sharma (MWS, 1983) model of repeat purchase diffusion in a nonuniform influence environment. We model the ability of promotional activities (e.g., personal selling, journal advertising, and sampling) to diffuse information to decision makers. We also include managers’ judgments of how to allocate resources to targeted user groups. Product demand is modeled as a function of total product call attraction, which is based on the probability of convincing potential users to become new and eventually, repeat product users. We evaluate the characteristics of resource allocation and how that allocation affects the adoption of a new product in a detail-intensive environment, i.e., the pharmaceutical industry. We discuss diffusion and repeat purchase models in Section 2. The allocation model (AM) is formulated in Section 3. In Section 4, we present and discuss the results of AM and compare it to that of MWS (1983), and concluding remarks are in Section 5.

2. DIFFUSION AND REPEAT PURCHASE Mahajan, Muller, and Bass (1993) indicate that a diffusion model represents the spread of an innovation among a given set of prospective adopters over time. The diffusion model is designed to depict the successive increase in the number of adopters and to predict the continued development of a diffusion process already in progress. Diffusion models describe both the adoption rate and sales of new products, but focus primarily on the adoption process. Most applications have represented the adoption of new durable goods, with each adopter representing one unit sold. In addition, there have been a few applications for nondurables, in which each adopter generates a stream of sales. Diffusion models can be used for either forecasting or normative purposes (i.e., to resolve issues such as the price of the new product over time) (Kalish & Sen, 1986). In addition, these models provide guidelines for targeting marketing efforts as potential segments of the market are identified.

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Noting that the successful increase in the number of users may include firsttime product adopters and repeat buyers, a number of repeat purchase models have been developed (MWS, 1983). Since the majority of these models were designed to forecast sales in consumer products where word of mouth has little effect on purchase, few repeat purchase models explicitly include word of mouth in their formulations (Midgley, 1978; Dodson & Muller, 1978; Lilien, Rao, & Kalish, LRK, 1981; MWS, 1983). We briefly review three models that allow for both repeat purchase and the effects of word of mouth. In addition, these models were calibrated using data from the pharmaceutical industry. In formulating the adoption component, we follow the general framework of Bass (1969) (see Mahajan, Muller, & Wind, 2000, for a comprehensive review). Repeat buyers can be incorporated into the Bass (1969) model by adding the portion of users that are repeat users to the number of new adopters, yielding:
 Nt N tþ1 ¼ p þ q ðm
N t Þ þ rN t (1) m where ðp þ qN t =mÞðm
N t Þ ¼ total number of adopters at time t, p ¼ coefficient of innovation, q ¼ coefficient of imitation, r ¼ coefficient of retention, N ¼ number of repeat buyers, Nt+1 ¼ total number of users at time t+1, and m ¼ market potential. LRK (1981) developed a discrete time model using this form for repeat purchase of ethical pharmaceutical products. LRK (1981) include the efforts of the sales force (detailing) in an examination of the adoption of drugs by physicians. They substitute marketing efforts for p, q, and r and make the following simplifying assumptions: (1) the number of doctors in the class is fixed, (2) all physicians are in the same class, and (3) advertising effectiveness is not related to the current number of prescribing physicians. Although the model is structured in terms of prescribing physicians, the total number of prescriptions written is observed. This method of tabulating adoption of a drug appears to overestimate the total number of prescribing physicians because both new prescriptions and refills of existing prescriptions are counted. LRK (1981) handle market feedback differently than Bass (1969). Positive sales increases are accelerators and declines in sales (e.g., bad product experience) lead to negative feedback. While the LRK model assumes that the influence of word of mouth is constant throughout the diffusion process, there does not appear to be substantial theoretical support for this assumption (Easingwood, Mahajan, & Muller, 1983). On the contrary, for most innovations, the imitation effect is likely to change over time. For

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example, Hernes (1976) suggests that the coefficient of imitation should decline, rather than remain stable, because the remaining potential adopters are less responsive to the product and associated communications. To address this issue, MWS (1983) use the nonuniform influence coefficient of imitation suggested by Easingwood et al., 1983, and substitute qðN t =mÞd (time-varying coefficient of imitation with d a constant) for qðN t =mÞ in Eq. (1). The MWS model could accommodate the assumption that diffusion proceeds in a uniform manner by setting d ¼ 1: The presence of the nonuniform effect would be indicated by d6¼1. Values of d between 0 and 1 cause an acceleration of influence leading to an earlier and higher peak in the level of adoptions, which would cause a high initial coefficient of imitation to decrease with penetration. Values of d greater than 1 delay influence, causing a later and lower peak. In addition, consistent with the nature of the pharmaceutical industry (The Quaestus Group, 1993), the MWS model does not allow for the effects of negative word of mouth. Unlike the LRK model, the MWS model allows the diffusion curve to attain its maximum rate of adoption at any stage of the diffusion process and the diffusion curve is independent of the potential market captured by the innovation (Easingwood et al., 1983). Hahn, Park, Krishnanmurthi, and Zoltners (HPKZ, 1994) extend the diffusion framework by incorporating repeat purchase behavior and the effect of competitive marketing efforts by entering and defending firms. It is a four-segment trial and repeat model that can be calibrated using aggregate data for frequently purchased products in the early stage of the life cycle. Like the previous model, the framework is simplified before the model is calibrated on aggregate data. Similar to LRK and MWS, HPKZ assume a constant repeat purchase rate and coefficient of imitation. That is, they assume that neither rate is affected by the marketing mix. However, marketing activity did affect the coefficient of innovation in the form of total marketing effort (marketing communication). In summary, these models suggest that marketing activities influence both sales forecasts and actual sales of new products. However, none of these models demonstrate the separate or interactive effects of detailing, pricing, and the supportive promotion so common in the industry examined. Additionally, these models do not include target marketing in which managers allocate differential resources to different target segments. Moreover, in all the models, market potential, is estimated by the model and not by the marketing effort of each company. Our objective is to address the gaps in the literature and develop a model for allocating resources and forecasting sales in DIIs in which managers rely

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heavily more on personal selling than other promotional activities. Our model makes provisions for the concept that if there is no personal selling then sales decrease proportionally. Although a large amount of the marketing budget is allocated to detailing, it is accompanied by supportive promotion to ensure trial, secure repeat usage, and remind users of product attributes between sales calls. Therefore, our model of DIIs takes into account the interaction between and among the detailing component of the marketing mix and other supportive components. Given the importance of detailing, it is our contention that the number of calls to potential customers determines market potential. In other words, if there is no personal contact between the consumer and the salesperson, then the probability that the customer will purchase is small. In addition, our model includes some degree of heterogeneity to approximate the various ways managers allot limited resources to attain maximum sales effort.

3. THE MODEL 3.1. Marketing/Sales Organization/Customer Relationship in a Detail-Intensive Industry In this section, we suggest a framework of the industry under review, develop a conceptual model that includes the marketing decision variables substituted for the coefficients p, q, and r in the MWS model (as recommended by Bass, Krishnan, & Jain, 1994), and discuss how to calculate the allocation of the limited resources across three heterogeneous groups. We also indicate how to estimate the market potential in a detailintensive environment. We must first understand the business transactions and the resource allocation in the industry we are examining to develop a model that incorporates the allocation of those resources. As suggested by Mahajan and Wind (1988), we propose a model of the pharmaceutical industry to demonstrate the application of AM in a DII. We use management and expert judgment, and information on analogous products (i.e., products similar to the new product under consideration) to forecast the likely performance of the new product. The relationship between the marketing organization and its customer in a DII is depicted in Fig. 1. This relationship is captured by agency theory where one party (the principal) depends on another party (the agent) to undertake some action on the principal’s behalf (Bergen, Dutta, & Walker,

Direct to Consumer Promotion Marketing Organization

Supportive Promotion Promo Material Resources ($) Incentives

Direct to Decision-Maker Promotion

Demand

Decision Maker (Physician)

Consumer (Patient) Prescription (NRX)

A Repeat Purchase Diffusion Model

ENVIRONMENTAL FACTORS Pharmaceutical Company

Feedback Promotional Activities Information

Sales Organization

NRX

Competition

Fig. 1.

Third Party Payer (HMO/PPO/ Medicare)

Refill

Pharmacy

Information Flow in the Pharmaceutical Industry. 151

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1992). Most marketing departments consist of multiple organizations at various functional levels (product managers, salespeople, etc.). Developing and implementing marketing strategies and programs necessarily involves managing agency relationships. In a DII, the ultimate customer can be viewed as engaging in an agency relationship as he or she attempts to gain accurate product information and desired product benefits from a supplier, or agent (Coughlan, 1988; Devinney, 1988). In industries of this type, an individual can shift roles from principal to agent across different contexts. For example, the decision maker in DII acts primarily as an agent whose function is to represent the needs and interests of the potential end user when buying goods and services from other suppliers. However, when seeking information from a given supplier on which to base a purchase decision, the decision maker acts as a principal and the supplier’s sales person is the agent. The marketing organization as principal also relies on the sales person to act as its agent. Here, the marketing manager knows the sales person’s characteristics and abilities, but knowledge of the sales person’s actions on the job is neither perfect nor complete. In contrast, the sales person has information that the marketing manager wants. For example, although a manager may know how many calls a salesperson made in one week, that sales person has additional information about the preparation required for each call, the quality of the sales presentations made, and the chances of completing a sale. The sales person’s additional knowledge becomes the basis for our first proposition about selling activity in a DII: Proposition 1. The probability that a decision maker will use a product is a function of the end users’ needs, relationship with company/sales person, location in territory, time constraints, accessibility, etc. Proposition 1 uses the sales organization’s subjective judgment. The marketing organization relies on the sales team to segment the population into homogeneous groups based on usage potential. The sales team is then directed to spend limited detailing time with as many potential users as possible. Marketing, serving in a support function through direct decisionmaker promotion (DDMP), can effect product demand through direct-toend-user promotion (DEUP); however, marketing cannot ensure product usage and eventual adoption with only supportive promotion. Only the sales person can secure that kind of commitment. Hence: Proposition 2. Without personal selling the probability that the decision maker would recommend use of the company’s products is very low.

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In a DII, the sales person serves as both the agent for the marketing organization and the decision maker. The sales person rarely has direct contact with the end user. All efforts focus on the decision maker. A major portion of the sales person’s success is contingent upon the effectiveness of personal contact with decision makers whose product use is highly probable. The following are four critical stages for decision makers: 1. Pre-launch: Prior to the product launch the sales person is given a list of all potential users (M*) to ‘‘prospect and profile’’ and eventually to segment into four groups based on potential receptability to the new product. Proposition 3. Membership in each segment is determined by the probability that the decision maker would respond favorably to this form of promotion. 2. Launch/Introduction: Given the four segments and the new product in hand, the sales person attempts to secure trial from influential users and to verify the pre-launch segmentation. 3. Postlaunch evaluation: The sales person develops a call-cycle strategy and reports the ‘‘permanent typing’’ of potential users to the marketing organization. 4. Permanent classification: The marketing organization adjusts its resource allocation plan to reflect both the size and strategy of the potential enduser groups. Proposition 4. Marketing management considers the size of the market, the total number of expected calls to each segment, and the potential effectiveness of its controllable resources (i.e., advertising and pricing) before it allocates noncontrollable resources (i.e., handouts and samples). The relationship between the marketing organization and the customer in a DII is depicted in Fig. 1, which allows for some interesting observations. Pharmaceutical products are similar to nondurables in that they are repeat purchases for the decision maker prescribing the drugs and durables for patients as long as they (the patients) have the disease state (The Quaestus Group, 1993). Pharmaceutical company managers are more interested in the repurchase pattern than in the initial trial rate. All things being equal, a low trial rate usually suggests poor coverage by the field force and/or less than optimal promotional campaigns, therefore new product trial can be stimulated by various marketing activities. However, a low repurchase rate could suggest that a product does not meet the consumer’s expectations, a condition that is more difficult to correct.

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The physician acts primarily as a representative (decision maker) for the patient (end user) when prescribing drugs. The physician searches for prescription drugs based on his or her knowledge of each patient’s needs, and then prescribes products that most closely approximate the solution to the patient’s problem. Although the physician is engaged in this relatively active and educated search, he or she could be unaware of a product until its existence is formally communicated (Smith, 2000). The role of a pharmaceutical company’s sales force is to make physicians aware of the product, provide additional information about the product, and encourage physicians to prescribe that drug. Other promotional efforts such as advertisements in medical journal, direct mail, free product samples, and other specialized forms of product promotion (symposia and convention booths) supplement the efforts of the field sales force. The results of a survey conducted by Scott-Levin Associates (2000) suggests that physician’s primary sources of information about pharmaceutical products were sales representatives (81.9%), articles in medical journals (87.2%), other doctors (67.3%), materials from pharmaceutical companies (50.4%), and medical journal advertising (36.3%). While the percentages of physicians who receive information from each of these sources is known, it is not known how each of these affects product usage. This framework describes the basic interactions in a DII. Therefore, if we define market potential as the total number of personal contacts (mt), then we know that this total number can be divided as follows in an n segment model: mt ¼

n X

mi;t

(2)

i¼1

where mt ¼ total potential, mi,t ¼ potential users that are Type i in the call mix, and n ¼ number of segments. Substituting into the MWS model, we obtain a model with dynamic market potential (DM): !
d Nt N tþ1 ¼ pt m1;t þ qt m2;t þ rt m3;t (3) mt This formulation sets a limit on the potential number of users in each segment. By definition, m1;t  mt and with 0  pt  1; pt m1;t must be  mt : The same applies for rt. For the time-varying coefficient of imitation, 0  qt ðN t =mÞd  1; and qt ðNt=mÞd m2;t  mt : By definition, heterogeneity is preserved because no one physician can belong to two groups. Therefore, pt and qt are the likelihoods that each type of potential user will become a user

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in the next time period and rt is the probability of retention. As defined by MWS (1983), this probability can be effected by the physicians’ exposure to marketing activities and can be defined to include the marketing decision variables specifically responsible for that likelihood. That is, the allocation of marketing resources to each of these heterogeneous segments can be handled with the following allocation variables: n X

ai;t ¼ 1

(4)

i¼1

where ai,t is the fraction of marketing resources allocated to target segment i potential users. The marketing decision variables allocated to a segment are xjit ¼ ai;t X jt

(5)

with Xjt is the market decision variable j (e.g., advertising, personal selling, other supportive promotions) and xjit is the portion allocated to segment i. We now define lt ¼

n X k X

  li;t bij ; xjit

(6)

i¼1 j¼1

as the total marketing budget, where bij ¼ model parameters, and k ¼ number of decision variables to be allocated. The allocation of noncontrollable resources is based on the percentage of total calls made to each segment. Sales persons distribute their noncontrollable resources to decision makers who demonstrate interest in the products. This scenario describes the fundamental interaction in a DII. The following formula can forecast users in a detail-intensive environment in which managers determine resource allocation (AM):
 NðtÞ d Nðt þ 1Þ ¼ ðl1 ðtÞÞma1 ðtÞ þ l2 ðtÞ ma2 ðtÞ þ rðtÞNðtÞ (7) mðtÞ Firms target their resources toward physicians who have maximum probability of using the firm’s products. That is, in a DII, marketing resources are allocated to convert the maximum number of potential users into actual users. For this analysis, we assume that the firm allocates the same percentage across the entire marketing mix. The next step is to determine the functional form of lt and the allocation percentages as defined in Eq. (4).

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3.2. Allocation Percentages and the Effects of the Marketing Mix We now discuss the results of a survey of marketing managers, sales managers, and sales representatives in the pharmaceutical industry. The study was designed to understand new product introduction in the industry, including how resources are allocated, markets segmented, and market potential estimated. The results provide the information needed to complete the formulation of the AM (The Quaestus Group, 1993). Some interesting and useful managerial observations from this research follow: O1. Pharmaceutical firms segment physicians into five groups. Physicians, the target market for most firms, are segmented in many ways, most frequently by call attractiveness. Call-attraction is based on probable usage, projected total number of class prescription for each number of class prescriptions for each period, and the type of promotion to which each is likely to respond. This information is gathered by sales representatives at pre-launch and continually updated. The ratings are given in Table 1. When a permanent classification is assigned, potential users are segmented into Types p and q based on whether marketing activity or a combination of marketing activity and information from current users influences them. Current users are categorized as Type r and given priority for follow-up sales calls. All others are classified as Type n and designated nonusers/no call. Table 1. User Segmentation and Typing. Segment

User Type

Description

Call Activity

1

p

10–20% of calls

2

q

3

r

4

n

Innovators: potential users; varying prescribing levels; influential physicians affected by marketing mix Imitators: potential users; high prescribers, high potential; influenced by marketing mix and innovators Current prescribers: varying prescribing levels; influenced by sales representatives. Objective is retention Varying prescribing levels, low potential

10–30% of calls

60–80% of calls

No calls

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O2. Without detailing, trial and adoption will probably not occur. Representatives call upon physicians from numerous organizations; therefore, it is nearly impossible to gain adoption or obtain repeat purchase of a new product without contact from a sales person. As a result, the largest determinant of potential market size (mt) is the number of sales contacts. The maximum number of potential users and current users that can be contacted over a given period could estimate the total number of potential users. However, in most instances, firms have other products to include in the call mix and therefore must allocate only a portion of their resources to the new product. Because of the importance of repeat purchase, priority is given to the retention of current users when allocating sales calls. The sales representatives surveyed stated that current users generally received 1–2 calls per cycle (2–4 weeks), including both successfully completed calls (presentations to decision makers) and administrative calls (sample drops). Therefore, the total potential for the retained segment (m3,t) is equal to the total number of current users (Nt) plus the total number of administrative calls (ACt). The remainder of the calls are allocated based call attractiveness, as previously described.   m2;t ¼ mt
m3;t  f t (8)     (9) m1;t ¼ mt
m3;t  1
f t m3;t ¼ mt  a3;t ¼ N t þ ACt

(10)

where ft ¼ portion of potential users that are imitators, and 1
ft ¼ portion of potential users that are innovators. O3. The allocation of promotional expenditures is based on previous period performances and the expected number of calls to each target group. The sales force must balance time allocated to selling existing products and time spent on the new offering. In addition, management allocates promotional resources based on prior effect and future objectives. The relationship between resource allocation and the assignment of promotional expenditures to each of the respective tasks can be represented as: a1;t ¼ ð1
a3;t Þ  ð1
f t Þ

(11)

a2;t ¼ ð1
a3;t Þ  f t

(12)

a3;t ¼ m3;t =mt

(13)

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O4. Sampling signals the physician’s commitment to the product. Because of the limited supply, samples are left with the physician only when the representative believes the commitment is genuine. Therefore, acceptance of samples suggests a commitment to use the product. Again, as with detailing, sampling influence usage at each stage and their importance varies with user type. For example, while an innovator who relies heavily on the marketing organization for information might consider samples important, an imitator may write a prescription without ever receiving a sample. A current user will generally rely more on samples and less on other marketing mix. Rather than model the number of samples distributed, we observe the cumulative number of samples available for distribution because the latter is more indicative of the sales call activity or lack thereof. SCit ¼ SCit
1 þ Sit
nit

(14)

where SCit ¼ number of samples available for distribution, Sit ¼ number of samples distributed, and nit ¼ new prescriptions written. It is assumed that only one sample is distributed for each new prescription written O5. Pricing becomes an influential factor for patients. Recently, end-user cost has become a major issue in this DII. Consequently, sales representatives are either selling low cost as a product feature to gain additional usage or defending the high prices of their products when attacked by competition. It is important to note that product price is rarely mentioned as a feature in the pre-adoption stage; it only becomes important in the presence of competitive detailing or patient feedback. Therefore, the more positive the difference between the price of the product and the price of an alternative competitive product, the smaller the demand for the focal product. As a result, we include the price difference between the product analyzed and the product offered by the competition and note its effects on retention of new product users. Pd t ¼ Pt
PCt

(15)

where Pdt ¼ price difference, Pt ¼ price of the focal product, and PCt ¼ competitive price. O6. The generally positive and nonuniform effect of word of mouth becomes less effective over time. Both the number of adopters and the length of time the product has been on the market influence the physicians affected by this factor. For example,

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physicians in a Scott-Levin study stated that they wait at least three years after a product is introduced to see how others receive the product. When the word-of-mouth effect is present, it will exert a positive influence on adoption and will serve to augment detailing activities for that group. We denote the word-of-mouth influence by ðm3;t =mtÞd ; which represents the non uniform influence (NUI) factor or the number of current users who affect nonusers ðm2;t Þ:1 This treatment of the word-of-mouth effect is an extension of the flexible influence model offered by Easingwood et al. (1983). O7. Competitors will not counter-detail until they find use for the new product. Generally, managers in pharmaceutical firms will not counter-detail (i.e., use selling efforts to demarket the competitive offerings) unless the new competitive product is being prescribed by the physician. Therefore, the competitive response to new product introduction can be modeled as a function of the number of calls completed by the competition in which a direct comparison of the focal product (D2t) and the competitor’s offering is presented. These counter-detailing calls are limited to those physicians who are current users of the focal product; therefore, the upper bound of the counter-detailing calls is the market share of the focal product. ! 3 X Dit D2t  yt Dt
(16) i¼1

where D2t ¼ competitive detailing, yt ¼ market share for the focal product, Dt ¼ total detailing in product category, and Dit ¼ total detailing for the focal product to segment i. 3.3. The Full Allocation Model As previously suggested, we examine the adoption of a new product in the pharmaceutical industry, a DII, where M* represents the total number of physicians in the population. However, not all of the physicians are potential users. We estimate the number of prescribing physicians (Nt) by taking the total number of prescriptions written (nt) and dividing by the average number of prescriptions written per physician (At) (MWS, 1983). We use new prescriptions, rather than total prescriptions filled, because new prescriptions represent a more accurate measure of physician activity. We examine three segments: innovators, imitators, and current users.

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The AM specification derived from the preceding conceptual framework suggests a system of three equations in which the dependent variables are the number of adopters who are innovators, imitators, and current users from the retention stage. The AM theory was derived from the observation that the marketing mix and its allocation affect both the adoption and retention processes. The probability of product use can be affected by varying the allocation of marketing mix activities to each segment, allowing for the following definition of the marketing response functions:    l1 ðtÞ ¼ a1 ðtÞ bp þ d 1 ðtÞupðtÞ b1 þ b2 scðtÞ þ b3 j ðtÞ    d l2 ðtÞ ¼ a2 ðtÞ bq þ d 1 ðtÞupðtÞ b4 þ b5 scðtÞ þ b6 j ðtÞ N ðtÞ=m rðtÞ ¼ ar ðtÞ



   br þ d 1 ðtÞupðtÞ b7 þ b8 scðtÞ þ b9 j ðtÞ
b10 OðtÞ

(17) (18) (19)

where bi (i ¼ 1y10) represents each parameter of the model, and OðtÞ comprises the exogenous effects such as competitive detailing, poor experience by users, poor experience by patients, negative word of mouth, increased usage of new products, etc. For this analysis, we use only d 2 ðtÞ as exogenous factors with other outside factors captured in the constant br (Note that the true value of b10 d 2 ðtÞ þ br is equal to 1
rðtÞ:) bp, bq, and br are constants such that if there is no marketing-mix information, the model reduces to the Bass model as a special case (Bass et al., 1994). Using substitution into Eq. (7), we get the full AM to be estimated:    NðtÞ  Nðt þ 1Þ ¼ ð1
f ðtÞÞ  1
bp þ d 1 ðtÞ upðtÞ b1 þ b2 scðtÞ mðtÞ  þb3 jðtÞ ðmðtÞ
NðtÞÞnð1
f ðtÞÞ      NðtÞ  þ f ðtÞ 1
bq þ d 1 ðtÞ upðtÞ b4 þ b5 scðtÞ þ b6 jðtÞ mðtÞ
 NðtÞ d ðmðtÞ
NðtÞÞnf ðtÞ  mðtÞ     NðtÞ  b þ d 1 ðtÞupðtÞ b8 scðtÞ þ b9 jðtÞ
b10 d 1 ðtÞ þ NðtÞ mðtÞ r ð20Þ

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4. EMPIRICAL APPLICATION 4.1. The Data To test the hypothesis that the effective allocation of the marketing mix has an impact on a product’s adoption and retention process, we use data from three pharmaceutical products. For comparison, we estimate parameters of the (1) MSW model (Eq. (1)); (2) Dynamic Model (DM)(Eq. (3)), which is nested in AM when l1(t), l2(t), and r are constant; and (3) Full AM(Eq. (20)). Sets of monthly data for several ethical drugs were obtained from IMS America, a pharmaceutical market research firm, for the period from September 1980 to December 1993. The sample consists of new ethical pharmaceutical products sold to primary care office-based physicians. Data elements reported include the number of new prescriptions written (nrx(t)), company detailing (d1(t)), journal advertising (j(t)), sampling (s(t)), total detailing (dt), unit price (up(t)), and average number of prescriptions written per physician (ave(t)). From these data, we could determine each drug’s standing in comparison to competitive prescriptions, and the decay curve for average new prescriptions written per physician. It is important to note that all data were multiplied by an arbitrary constant to maintain confidentiality. Three products that have been on the market for different lengths of time were selected randomly (72, 94, and 150 months) for comparison. A portion of the data (48, 70, and 126 months, respectively) was used to calibrate the models with the holdout sample (24 months) that was used to test the forecasting abilities the each model. We treat the lag of decision variables similar to Berndt, Bui, Reiley, and Urban (1995) that note that marketing effects provide long-term information. Therefore, the cumulative efforts of the decision variables are indicated. If, for example, Ct, is the cumulative information at the end of month t, then C t ¼ ð1
oÞC t
1 þ F t

(21)

where Ft ¼ flow of new information efforts during month t, and o ¼ monthly depreciation rate. Since o is unknown, we estimate it econometrically for both detailing and journal advertising, and use the cumulative decision variables in the model. Given the industry we are studying and the models to be estimated, we hypothesize the following: H1. Promotional activities would have positive or no effects on the usage of the new products. We do note, however, that the combined effects of

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certain marketing variables could signal negative activity. For example, a negative sign for detail/pricing/sampling signals that too many samples of this product (oversampling) are being distributed, and that this variable indicates a negative correlation between the number of cumulative samples and the number of current users. This finding is consistent with industry expectations (The Quaestus Group, 1993). H2. We also expect l1(t) and l2(t) to decrease over time. That is, the longer the product has been on the market, the less likely it is that sales will be generated by new users. H3. We expect to see the retention rate reach 100%, then fluctuate never dropping below 80% after the peak. The retention rate is determined by the percentage of resources allocated for that purpose as overall resources allocated for the product decreases. Also, the reduction in retention could be attributed to first-time users moving back into the potential users group. This action will be evident by a corresponding increase in l2(t). H4. Consistent with previous diffusion models, we expect a very good fit with data for all models. However, we do expect, that when measuring the mean-squared error (MSE) and sum of squared error (SSE) the AM will outperform the DM (the nested model) and MSW. 4.2. Calibration Results We estimate the parameters of the models using Nonlinear Least Squares to minimize the SSE that is reported as the loss value. Regression analysis is used primarily for two purposes: (1) for estimating and testing hypotheses about coefficients of interest and (2) for forecasting. We are initially interested in the relationship of attributes in explaining market behavior. Specifically, because coefficient magnitudes are not instructive, we examine both the signs and the significance of the parameter. Table 2 presents the parameter estimates and asymptotic standard errors estimated. 4.3. Fit Statistics (H4) First, we examined the models in general using estimates based on the three data sets. We could not reject the null hypothesis that the parameters are equal to zero, with F values (p value ¼ 0.0001) for all nine estimated models far exceeding the critical values. A priori, there is no reason to

Product 1 Parameter l1 l2 r d RSS Variance N (cases) R2 F bp b1(up(t) d1(t)) b2(up(t) d1(t)*sc(t)) b3(up(t) d1(t)*j(t))

Dynamic

0.058 0.134 (0.0127) (0.0256) 0.135 0.155 (0.120) (0.070) 0.539 0.652 (0.4377) (0.083) 1.599 1.178 (0.210) (0.149) 64.078 64.037 1.335 1.334 48 48 0.979 0.979 4905.659 4907.460

Product 3

Allocation

MWS

Dynamic

Allocation

MWS

Dynamic

Allocation

0.061 a 0.105 a 0.708 a 1.396 (0.187) 60.758 1.266 48 0.980 1142.130 0.758215 (0.720) 0.032700 (0.0369)
0.000497

0.005 (0.004) 0.118 (0.031) 0.715 (0.088) 1.292 (0.088) 958.210 7.605 126.000 0.985 4254.199

0.022 (0.010) 0.177 (0.046) 0.910 (0.017) 1.063 (0.094) 974.968 7.738 126.000 0.923 7028.130

0.043 a 0.119 a 0.529 a 1.310 (0.018) 844.350 6.701 126.000 0.941 2132.140
0.096348 (0.12624) 0.037400 (0.00801)
0.000153

0.1235 (0.0108) 0.2700 (0.0900) 0.4871 (0.1124) 2.1339 (0.0791) 994.6100 14.6266 70.0000 0.9225 3290.44

0.3159 (0.0242) 1.2319 (0.1852) 0.6028 (0.0388) 1.6581 (0659) 994.1990 14.6206 70.0000 0.9230 3291.97

0.1107 a 0.1722 a 0.7271 a 1.4989 (0.1266) 777.2670 11.4304 70.0000 0.9395 1080.97
0.182922 (0.15583) 0.261188 (0.03027)
0.002351

(0.000285) 0.000248 (0.000153) 3.847171 (2.7532)

(0.000106) 0.000011 (0.000035) 2.183400 (0.12099)

(0.000325)
0.000189 (0.000152) 4.961820 (0.79493)

163

bq

MWS

Product 2

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Table 2. Parameter Estimates with Calibration Data Sets.

164

Table 2. (Continued ) Product 1 Parameter b4 (up(t) d1(t)) b5(up(t) d1(t)sc(t)) b6(up(t) d1(t)j(t)) br b7(up(t) d1(t)) b8(up(t) d1(t)sc(t)) b9(up(t) d1(t)j(t))

Dynamic

Allocation 0.038128 (0.0625)
0.000446 (0.00034)
0.000733 (0.000344) 1.060420 (0.0515)
0.002867 (0.004) 0.000007 (0.000006) 0.000020 (0.000025) 0.006380 (0.00226)

MWS

Dynamic

Product 3 Allocation 0.030080 (003345)
0.000112 (0.000010) 0.000015 (0.000010) 0.919060 (0.012778) 0.002410 (0.000522)
0.000004 (0.000001) 0.000009 (0.000001) 0.000804 (000147)

MWS

Dynamic

Allocation 0.048224 (0.01493) 0.000087 (0.000014) 0.000053 (0.000035) 0.959560 (0.01983) 0.000457 (0.00143) 0.000000 (0.000001) 0.000021 (0.000003) 0.000381 (0.000149)

Notes: (a) no SE values given because these are mean values for the three parameters.(b) The MWS model is as in Eq. (1). The Dynamic model is as in Eq. (3). The allocation model is as in Eq. (20).

FRANKLIN J. CARTER ET AL.

b10(d2(t))

MWS

Product 2

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believe that the marketing-mix parameters would be 0. As expected with diffusion models that include the peak in the data sets, the mean corrected R2 values were high (0.96 for MSW, 0.96 for DM, and 0.97 for AM). As we hypothesized (H4), the SSE and MSE were improved 13% and 8%, respectively, when comparing MSW to AM. Also, the measures were improved by 12.2% and 7.6%, respectively, when comparing DM to AM. When comparing MSW to DM for products 1 and 3, we found only slight improvements in both measures. The measures did not improve, however, for product 2 (the longer time series). Managers offered two possible explanations for these findings: (1) we examined more than 12 years of monthly data, and (2) the product examined had been subject to other factors not captured in our model, such as patent expiration followed by heavy competition from generic products, other new product introductions and a shift in company priorities. The AM model predicts a better fit than the other two models do for the three products as a whole; however, for product 1 the MSE increased slightly. This increase resulted because the improvement in the SSE was insufficient to compensate for the loss in degrees of freedom (from 4 to 14 parameters with only 48 cases). This was not the case with the other two products. We can conclude, therefore, that with decision variables, the AM model will forecast deviations in users (Bass et al., 1994). 4.4. Parameter Estimates and Standard Errors (H1) A total of 42 parameters were estimated from the three models. There were 30 parameters that tested the interaction of the marketing mix. Of the 30 parameters (excluding d and constants) estimated, seven had negative signs. Six of the seven negative signs were from detailing/pricing/sampling, which as mentioned earlier may suggest over-sampling. Managers must interpret findings since many companies have different sampling strategies. We used nine parameters to test the interaction of each of the three combinations of resources allocated (detailing/pricing, detailing/pricing/ sampling, and detailing/pricing/journals). For detailing/pricing, six of the parameters were significant (a ¼ 0:1), with one negative sign (not significant). For product 2, all detailing/pricing parameters were significant. For product 3, detailing/pricing was significant for segments 1 (Type 1) and 2 (Type 2) and none were significant for product 1. For detailing/pricing/ sampling, seven of nine parameters had negative signs that could indicate of over-sampling. Eight proved to be significant including Type 1 for product 1; Types 2 and 3 for product 3; and Types 1 and 2 for product 3. For

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detailing/pricing/journals there were two negative signs, which was not expected. Competitive detailing was significant for all three products. The constants p0, q0, and r0 proved to be significant in eight of nine cases with negative signs indicated in two instances (both cases for p0). The retention constant (r0) was highly significant for each product, which indicates that the likelihood of retention is captured by some marketing factors and exogenous factors. Only three parameters were negative and could not be explained. Because they were not significant, we could re-estimate the model and exclude them. The fitted value of d (the nonuniform influence factor) was greater than 1 for all three products, indicating a low initial influence that peaks later in the life cycle. Parameter estimates were 1.4 for product 1, 1.31 for product 2, and 1.5 for product 3. We can conclude that peak usage is experienced later in the product’s life cycle. 4.5. l1, l2, and r(t) values (H2 and H3) Estimates for the parameters l1, l2, and r for the MSW and DM models, and mean l1, l2, and r values for the AM are shown in the tables. The more dynamic l1(t), l2(t) and r(t) for the AM model are plotted in Figs. 2–4. We draw the following conclusions. Plots of the time-varying values for the likelihood of trial for Type 1 [l1(t)], Type 2 [l2(t)], and retention [r(t)] users clearly indicate that the likelihood of trial by Types 1 and 2 declines over time as company’s allocate the majority of their resources to retaining users and combating competitive activities. This, finding is consistent with industry experience that user 1.2 1 0.8 0.6 0.4 0.2 0 1

12 23

34

45

56 67

78

89 100 111 122 133 144

Time pt

Fig. 2.

qt

rt

Product 1 Allocation Model Estimates of Reparameterized Coefficients for Calibration and Holdout Samples.

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1.5 1 0.5 0 1

7

13 19 25 31 37 43 49 55 61 67 73 79 Time pt

Fig. 3.

qt

rt

Product 2 Allocation Model Estimates of Reparameterized Coefficients for Calibration and Holdout Samples. 1.2 1 0.8 0.6 0.4 0.2 0 1

7

13

19

25

31 37 Time pt

Fig. 4.

43 qt

49

55

61

67

rt

Product 3 Allocation Model Estimates of Reparameterized Coefficients for Calibration and Holdout Samples.

retention becomes more significant the closer the product gets to patent expiration. As expected, for all three products, the l1(t) was consistently much lower than l2(t) or r(t). Thus, we conclude that the conversion rate of Type 1 users will fall far below that of Type 2 users. The slope at mean values (first derivative of n(t+1) with respect to d1(t)) decreases (negative values) for products 1 and 3 and increases (positive value) for product 2. Likewise, the first derivative of l1(t) and l2(t) with respect to d1(t) is positive (increasing) for all three products, whereas, the first derivative of r(t) with respect to d1(t) for products 1 and 3 is negative at the mean values and positive at the mean values for product 2.

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In the long run, retention rates tend to decline. One would expect them to remain at 80% or better. This drop in retention rates can be attributed to competitive activities, which were found to be highly significant in each case. Industry experts note that as market and product mature, user retention becomes more difficult because new products compete for company resources. Retention declines were proportional to the increases in Type 2 trial. As users move from potential to trial, they may require more detailing time to be established as retained users. Our model does not differentiate between types of retention. 4.6. Forecasting The holdout samples for each product provide an opportunity to challenge how well the model predicts the number of users during the time periods subsequent to the calibrated period. We also observe how close the AM comes to predicting the actual values by noting the ratio of predicted to actual users. Figs. 5–7 display the actual and corresponding predicted values for the respective products. Comparing the predicted and actual values for the AM, we find that the model performed well. Specifically, the ratio of predicted users to actual users was 0.973 for product 1, 0.944 for product 2, and 0.956 for product 3. Various measures for assessing prediction accuracy among the three models are shown in Table 3 using a 24-month (holdout sample) forecast. The measures are designed to evaluate ex post forecasts; that is, forecasts for which the exogenous variables do not have to be forecasted. We choose to show all four measures because each uses different criteria. For example, two of the 60.00

Users (000)

50.00 40.00 30.00 20.00 10.00 0.00 1

3

5

7

Actual

Fig. 5.

9

11 13 15 17 19 21 23 Time

Allocation

MWS

Product 1 Holdout Samples Forecast.

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80 60 40 20 0 1

3

5

7

9 11 13 15 17 19 21 23 Time Allocation

Actual

MSW

Fig. 6.

Product 2 Holdout Samples Forecast.

1

5

Users (000)

50 40 30 20 10 0 3

7

9

11 13 Time

Allocation

Fig. 7.

17

19

Actual

21

23

MSW

Product 3 Holdout Samples.

Table 3.

Forecasting Measures.

Product 1

RMSE MAE Theil U Theil UD

15

Product 2

Product 3

MSW

DM

AM

MSW

DM

AM

MSW

DM

AM

2.24 29.40 0.16 3.03

1.26 3.38 0.05 1.04

1.35 4.16 0.06 1.12

1.64 11.51 0.06 1.60

1.46 7.96 0.05 1.64

1.66 13.58 0.07 1.73

10.02 10.70 0.15 1.57

3.36 4.14 0.06 1.59

6.77 7.63 0.11 1.77

measures are based on the residuals from the forecasts (RMSE (root meansquared error) and MAE (mean absolute error)). The Theil U statistic is related to R2, but is not bounded by 0 and 1 (Large values indicate poor forecast performance.). The Theil UD compares measures in terms of the

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changes in N(t+1). These measures will reflect the model’s ability to track turning points. For product 1, the DM and AM were superior to MSW in forecasting for all measures. The DM was better than the AM but there we found no discernible difference in the forecast values (Fig. 5). For product 3, the DM and AM remained superior to the MSW, although the gap began to narrow (Fig. 7). For product 2 (Fig. 6), all models were very close to the actual values, with the AM performing slightly worse on all measures. The AM seems a worse predictor of data changes over time than the other models. All models seem to be very close when calibrated with 70–80 periods of data. However, the AM offers more diagnostic information. The question is whether managers are willing to trade forecast accuracy for useful diagnostic information.

5. CONCLUSION We suggest a diffusion model for forecasting sales and allocating marketing resources in an industry heavily reliant upon personal selling. Our model includes elements of the promotional mix (i.e., journal advertisements, price, and samples), as well as detailing to estimate the probability of new product use by the various segments. The proposed model is a three-segment model that estimates the effects these marketing activities have on two types of tiers and repeat purchasers. In a repeat purchase environment, the manner in which marketing resources are allocated is as indicative of product success as is the total amount of resources allocated. Proper segmentation of potential users determines the success of a new product, especially in a DII. Proper allocation of resources to each segment influences the launch of a new product. Unlike other repeat purchase diffusion models, the framework is not simplified before the model is calibrated. The number of personal contacts to potential or current users determines market potential, which is often the case in DIIs. The priority of personal contacts and in turn the segmentation is determined by call attractiveness, which is an indication of the sales organizations estimation of the probability of usage. We modeled the effects of resource allocation, both variables within (i.e., detailing, pricing, journal advertising, and sampling) and outside the control of management (i.e., competitive detailing, other exogenous factors) on each segment. The sales-call cycle, which helps determine sales-call frequency, and supportive promotion at each stage in the new product introduction

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process are crucial. We include information from managers in the industries under examination to demonstrate how to include the allocation of each decision variable. Examination of the allocation of resources allows us to determine the effect each marketing activity has on the likelihood of use by a segment. We only include price in the retention phase because managers indicate it generally only becomes an issue then, when there is increased competition and a variance in price. Marketing managers can use the AM model not only to forecast sales, but also to provide diagnostic information on how potential end users will respond to marketing stimuli. The parameter signs indicate marketing strategy, such as over sampling, as well as how each segment responds to certain marketing activities. Other models appear to provide limited diagnostic information regarding marketing-mix variables and competitor’s actions. We suggest that managers use the AM model before introducing a new product to adjust both their percentages of calls and the resources allocated to each segment to optimize sales. With unlimited resources the product could reach maximum potential (M*), however constraints do limit sales, including: (1) (2) (3) (4) (5)

the budget allocated for marketing and sales expenditures, the percentage of the total sales force allocated to the product, the number of sales representatives, the potential number of calls per day for each sales representative, and revenue goals and managerial priorities.

Our model is not without shortcomings and will benefit from modifications and extensions. This paper provides a basic framework for expansion. Areas for further investigation include the development of a more normative model to establish optimal levels of sales force activities, such as the number of sales calls needed to ensure adoption and retention, the optimal number of sales representatives needed to achieve maximum sales, and the optimal call cycle at each stage of the product introduction process. In addition, similar to Chatterjee and Eliashberg (1990), one could design AM that uses a micro-modeling approach and examines different levels of repeat purchase to help managers determine when a user has reached maximum potential. The influence of management’s allocation of resources must be represented in extensions to the current model. It is vital to develop approaches that are normative by design and result in actual use in industries for which they are developed. Our model should provide a first step in achieving these objectives.

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NOTES 1. It should be noted that the word-of-mouth effect reflects local individuals’ influence on nonusers, and does not include company-sponsored information, such as clinical studies in medical journals otherwise known as supportive promotion.

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Mahajan, V., & Wind, Y. (1988). New product forecasting models: Directions for research and implementation. International Journal of Forecasting, 4, 341–358. Mahajan, V., Wind, Y., & Sharma, S. (MWS). (1983). An approach to repeat-purchase diffusion analysis. AMA proceedings (Series 49, pp. 442–446). Chicago, IL: American Marketing Association. Midgley, D. (1978). Innovativeness: The concept and its measurement. Journal of Consumer Research, 4(March), 229–247. The Quaestus Group. (1993). Introducing new products in the pharmaceutical industry. Unpublished report, Belmont, CA: The Quaestus Group. Scott-Levin Associates. (2000). Pharmaceutical company images. Newtown, PA: Scott-Levin Associates. Smith, M. C. (2000). Pharmaceutical marketing: Strategy and cases. Binghamton, NY: Pharmaceutical Products Press. Zeithaml, V. (1992). Review of marketing, Vol. 4. Chicago, IL: American Marketing Association.

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FORECASTING PRODUCT SALES WITH CONJOINT ANALYSIS DATA David B. Whitlark Conjoint analysis is a tool for assessing the feasibility and forecasting potential sales of new or re-engineered products. It does this in the context of key competitors and current market conditions. It is a two-step process consisting of utility estimation and preference simulation. The preference simulation outputs market share estimates. Obtaining market share estimates, however, is only a starting point for forecasting product sales. Market share estimates obtained with conjoint analysis assume no competitive reaction, 100 percent product awareness, and 100 percent product availability. An accurate sales forecast will also take into account possible competitive reactions and changes in product awareness and product availability that unfold over time. The purpose of the article is to describe (1) how to evaluate the potential impact of competitive reaction using decision trees and (2) how to integrate market share estimates with the impact of changes in product awareness and product availability using ‘‘funnel analysis.’’

CONJOINT ANALYSIS Conjoint analysis was introduced starting in the 1970s (Green & Rao, 1971) and logged more than 1,000 commercial applications by 1980 (Cattin & Wittink, 1982). During the 1980s usage increased tenfold (Wittink & Cattin, 1989). Today it may be the most widely used quantitative product Advances in Business and Management Forecasting, Volume 4, 175–182 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04011-0

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development tool in the US and Europe (Wittink, Vriens, & Burhenne, 1994). In conjoint analysis, competing products are broken apart and expressed as sets of features and feature levels. For example, one feature of an automobile is horsepower and horsepower can have different levels such as 100, 150, and 200. In addition to horsepower, examples of other feature categories are fuel efficiency, interior styling, exterior styling, warranty, service record, dealership reputation, cargo space, passenger headroom, driver legroom, safety equipment, seat upholstery, power equipment, and so forth. Automobiles are complex products and consist of many feature categories and feature levels, but even simple products such as credit cards and coffee can be expressed as sets of features and feature levels. Conjoint analysis measures the decision-making weight of each feature category and estimates utility scores for every feature level. Over the years many approaches have developed to collect and evaluate conjoint analysis data. Early conjoint applications asked respondents to compare product features two at a time and rank order all combinations of all feature levels. Respondent fatigue can be very high with this ‘‘trade-off’’ approach. For example, if the study requires six features and each feature consists of three levels each respondent must make 135 evaluations. The number of evaluations increases rapidly as product complexity increases. As a result the ‘‘trade-off’’ approach is now rarely used in commercial studies. Another early conjoint analysis approach asks respondents to make paired comparisons of competing product profiles. A product profile is a blend of features created with an experimental design with the goal of estimating an individual utility score for each feature level. The ‘‘paired comparison’’ approach does reduce the number of evaluations, but the task is still cumbersome. For example, when a study consists of only 10 product profiles a respondent must evaluate 45 paired comparisons. A third approach was developed to combat the issue of respondent fatigue and data reliability (Green & Srinivasan, 1978). This approach is called the full profile method. For many years it has been the most popular form of conjoint analysis. In this method respondents provide a preference rating, typically using a 1–7 or 1–10 scale, for each set of product profiles. Similar to the paired comparison method, product profiles are a blend of features created with an experimental design. The ‘‘full profile’’ approach greatly reduces the number of respondent evaluations. Unfortunately, the simplicity of the method encourages managers to divide products into finer and finer sets of feature categories and feature levels. This can again lead to problems with respondent fatigue. As a case in point, one automotive manufacturer wanting to redesign their dealership experience requested a study with a list of over

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20 features in which each feature category had between 5 and 12 levels resulting in well over 100 ‘‘full profile’’ evaluations. Respondents in the pilot test simply could not complete the task without suffering from fatigue. During the exercise, many started guessing, focusing on only one feature, or simply marking the middle of the preference scale after completing between just 20 and 30 evaluations. Today a fourth method called self-explicated conjoint analysis is growing in popularity for commercial studies (Srinivasan & Park, 1997). In this approach respondents provide a preference rating for each feature level then indicate the relative importance they assign to each feature category when making the buying decision. Feature level scores are combined with importance weights to compute respondent utilities. Selfexplicated conjoint analysis requires no experimental design and presents questions in a format that respondents find simple and familiar. The approach makes possible complex studies similar to the automotive dealership study described above. Without a doubt a great deal of effort has gone into designing different methods for collecting and evaluating conjoint analysis data. As a result simply deciding on which particular approach to use often becomes the research focus. This is unfortunate because other considerations such as product awareness, product availability, and competitive reaction may also have a profound impact on forecasting accuracy.

COMPETITIVE REACTION Thinking one move ahead can make the difference between success and failure when managing the marketing mix. Business managers all have stories about how a plan to gain competitive advantage turned into a financial disaster because their marketing ‘‘big idea’’ was matched by their competitors. A leading manufacturer of agricultural pesticides offers a good case in point. To gain competitive advantage, the company decided to add a warranty program to their marketing mix. The warranty promised to respray agricultural crops in which their pesticide did not meet certain efficacy criteria. Management was confident the company would gain many new customers and retain old customers because of the warranty program. Unfortunately management did not consider the potential impact of competitive reaction. Within a week of announcing their warranty program, the major competitors all announced similar warranty programs of their own. Incidentally, pesticide efficacy is heavily influenced by weather conditions. For the next two years unfavorable weather prevailed and pesticide manufacturers spent millions of dollars in

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respraying crops. None of the manufacturers wanted to be the first to withdraw their warranty program. But in the end, the company that first offered the program became the first to end the program. The company lost profits, customers, and brand equity because they did not consider the potential impact of competitive reaction. Fortunately, conjoint analysis can be supplemented with decision tree analysis to address the eventuality of competitive reaction. As an example, consider a leading packaged goods company that sells a premium line as well as a cost-conscious line of disposable diapers. The manufacturer believes their premium line of diapers is losing sales to competitors that have matched their premium features at a lower price point. The manufacturer does not want to reduce their own prices and risk losing sales on their costconscious line. They decide to reengineer their premium diaper by adding new high-value features. Using conjoint analysis they identify two product alternatives, i.e., Diaper A and Diaper B. Each alternative generates a different market share estimate, a different series of possible competitive reactions, and a different set of probabilities for each reaction. Exhibit 1 shows the situation. The preference simulation generates a unique market share estimate for each product alternative and depending on the blend of features it shows which competitors will be affected more than others by a product introduction. In this case, we see that it is believed that Diaper A provokes a high probability of reaction from Competitor 1 and no reaction from Competitor 3 and Competitor 4. On the other hand, Diaper B provokes a moderate chance of reaction from Competitor 3 and Competitor 4 and no reaction from Competitor 1 and Competitor 2.

Exhibit 1. New Product

Diaper A

Diaper B

Example Decision Tree for Diaper Manufacturer. Reaction

Probability

Both Competitor 1 and 2 Only Competitor 1 Only Competitor 2 No Competitor

10% 70% 10% 10%

Both Competitor 3 and 4 Only Competitor 3 Only Competitor 4 No Competitor

50% 15% 25% 10%

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The probabilities for each competitive reaction are usually assigned with the help of a Delphi session. In the Delphi approach managers are asked to consider a product introduction, review the projected changes in market share for all the key competitors, and then suggest possible competitive reactions. Competitive reactions can take the form of competitive price reductions, product modifications, and/or new product introductions. After discussing the possibility of a particular competitive reaction, managers write down their probability estimates of the reaction. The probabilities associated with the reaction are then revealed and managers with the most disparate views are asked to explain their opinions. The managers continue to discuss and write down probability estimates until the group comes to a consensus. It is a time consuming process, but the results are grounded in previous competitive behavior and the business experience of the managers. With the set of product alternatives, potential competitive reactions, and probabilities in place, an expected value is calculated for each alternative. The expected value equals ‘‘contribution to profit’’ multiplied by market size and weighted by market share and outcome probability as shown in Exhibit 2. The alternative generating the highest expected value on average will be the best alternative. The approach can consider a single product line as described above or may include competitive reactions across more than one product line. For example, considering competitive reactions that could impact sales of a cost-conscious line of diapers as well as a premium line of diapers. In this example, adjusting for the possibility of competitive reaction changes the best alternative from Diaper A to Diaper B.

Exhibit 2. New Product

Diaper A

Diaper B

Calculating Expected Value for Diaper Manufacturer. Probability

Market Share

Both Competitor 1 and 2 Only Competitor 1 Only Competitor 2 No Competitor

10% 70% 10% 10%

5% 8% 12% 15%

$50 $80 $120 $150

Both Competitor 3 and 4 Only Competitor 3 Only Competitor 4 No Competitor

50% 15% 25% 10%

10% 14% 14% 18%

$75 $105 $105 $135

Reaction

Profit EV (Millions) (Millions)

$88

$93

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ADDING CHANGE INTO THE FORECAST Expected value analysis is a good place to start when using conjoint analysis data for forecasting sales. However, the expected value analysis described above is static. History shows the chance of competitors reacting to a product introduction increases with time. We should try to account for an increasing probability of competitive reaction. In addition, product awareness and availability are not static, they develop over time. It takes time for people to learn about a new product and for the product to appear in retail stores. Marketing budget, media selection, advertising effectiveness, product uniqueness, product acceptance, promotional tactics, trade relationships, and choice of sales channels all can affect the growth of awareness and availability. Estimating how fast awareness and availability grows, similar to competitive reaction, requires a mixture of practical experience and managerial judgment. As an example, public awareness of the media campaign associated with one environmental issue in the US grew from 35 percent to 70 percent over a four year period after the expenditure of approximately $50 million. The particular campaign won national awards for advertising efficiency. One would expect to obtain different growth rates of awareness for different types of products, different levels of spending, and different kinds of advertising executions. Advertising agencies can help make these judgments. Account executives have access to historical records relating growth in awareness to spending rates and product category that when combined with the company’s own experience can yield reasonably accurate estimates. Product availability depends on the company’s ability to manufacture inventory, their relationship with the trade, promotional budgets, choice of marketing channel, and early sales results. Large, well-funded, and wellknown companies such as Procter & Gamble, Kimberly Clark, and Kraft Foods can expand product availability rapidly, particularly if initial product sales figures are favorable. Other companies with smaller budgets and limited market presence often expand availability very slowly. On the other hand, for the right type of products e-tailing can quickly provide nearly universal availability to anyone with a credit card who has access to the Internet.

FUNNEL ANALYSIS Funnel analysis has been used to forecast sales at companies such as General Motors. It is also described by Urban and Star (1991). As the name implies,

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it expresses product sales as an evernarrowing funnel. At the top of the funnel is the total number of households or consumers that could have any possible interest in buying the product, i.e., potential buyers. The number of potential buyers traveling down the funnel is whittled-down by (1) the percentage of people that become aware in the first year, (2) the percentage of people who have access to the product in the first year, and (3) the percentage of people that will choose to buy the product given the competitive situation modeled in the conjoint analysis simulation. The number flowing out of the bottom of the funnel represents first-time buyers. If the product will be purchased several times during the year, we will multiply the number of first-time buyers by an estimate of the percentage of repeat buyers and the average number of product units we expect repeat buyers to purchase during the year based on usage tests and/or historical data. We add together the units sold to first-time buyers with the units sold to repeat buyers to estimate total units sold for the first year. The process can be repeated for several years to complete a sales forecast for whatever time horizon is necessary for business planning. For each additional year we update the number of potential buyers, awareness percentage, availability percentage, and choice probability. The choice probability, that is the market share estimate output from the conjoint analysis model, may change slowly or rapidly depending on how quickly we feel that competitors will react to our product introduction. Funnel analysis brings together choice probabilities estimated using conjoint analysis data with many other elements necessary for forecasting sales. It allows us to integrate the effects of competitive reaction, development of the marketing mix, and rates of repeat purchase (Exhibit 3).

CHOICE PROBABILITIES: NECESSARY BUT INSUFFICIENT Conjoint analysis is a powerful tool for estimating utilities and choice probabilities, but choice probabilities are only one piece of the sales forecasting puzzle. These probabilities can change dramatically in the face of competitive reaction. Moreover, the speed and nature of competitive reaction can change based on the configuration of product we select. Different product configurations affect the sales of different competitors. Some competitors simply have more will and greater resources to react than others. Managers criticizing the accuracy of sales forecasts based on conjoint analysis data may want to consider integrating competitive reaction into their

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Exhibit 3. Example Funnel Analysis. Market Estimates Potential Buyers

US Households (Millions) 100

Buyers Aware Product Availability Choose Product

50% 70% 20%

50 35 7

Repeat Buyers One-Time Buyers

60% 40%

4.2 2.8

5.2

21.8

Total Units Sold

24.6

Average Units Purchased

models. Aside from competitive reaction, product awareness and availability will also profoundly effect sales. Accurate choice probabilities will not compensate for exaggerated levels of marketing support or misjudging growth in product awareness and access.

REFERENCES Cattin, P., & Wittink, D. R. (1982). Commercial use of conjoint analysis: A survey. Journal of Marketing, 46(Summer), 44–53. Green, P. E., & Rao, V. R. (1971). Conjoint measurement for quantifying judgmental data. Journal of Marketing Research, 8(August), 355–363. Green, P. E., & Srinivasan, V. (1978). Conjoint analysis in consumer research: Issues and outlook. Journal of Consumer Research, 5(September), 103–123. Srinivasan, V., & Park, C. S. (1997). Surprising robustness of the self-explicated approach to consumer preference structure measurement. Journal of Marketing Research, 34(May), 286–291. Urban, G. L., & Star, S. H. (1991). Advanced marketing strategy (pp. 110–114, 385–386). Englewood Cliffs, NJ: Prentice Hall. Wittink, D. R., & Cattin, P. (1989). Commercial use of conjoint analysis: An update. Journal of Marketing, 53(Summer), 91–96. Wittink, D. R., Vriens, M., & Burhenne, W. (1994). Commercial use of conjoint analysis in Europe: Results and critical reflections. International Journal of Research in Marketing, 11, 41–52.

IMPROVING SALES FORECASTS BY TESTING UNDERLYING HYPOTHESES ABOUT CONSUMER BEHAVIOR: A PROPOSED QUALITATIVE METHOD Eric D. DeRosia, Glenn L. Christensen and David B. Whitlark ABSTRACT Managers attempting to forecast sales frequently rely on assumptions and hypotheses about consumers and the underlying reasons for their behavior. For example, when forecasting industry sales in one product category, a forecaster may add industry sales for another product category to the forecasting model as an independent variable because the forecaster hypothesizes that consumers treat products in the second category as substitute products. To the extent that the forecasting model is influenced by such hypotheses, the hypotheses should be empirically tested. This article proposes and illustrates a qualitative method for empirically testing such a priori hypotheses.

Advances in Business and Management Forecasting, Volume 4, 183–197 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04012-2

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INTRODUCTION Sales forecasting and budgeting are important marketing management tasks. When sales do not live up to a forecast and the budget falls short, managers want to know why. They question whether the forecasting approach is flawed or if some unaccounted for influence has caused the forecast to go wrong. However, their questions often go unanswered. The most popular sales forecasting methods rely on time-series analyses and do not provide diagnostic information. Econometric forecasting methods that can help managers understand why sales differ from forecast are rarely used because (1) incremental time and cost of collecting econometric data, (2) concerns about model specification, and (3) inability of econometric forecasts to produce more accurate forecasts than simple projection methods such as ‘‘sales next period equal sales this period’’ (Brodie & De Kluyver, 1987). If managers could solve problems two and three, they would be willing to find ways to deal with the first problem. Understanding what is going wrong or right is very valuable information. This article describes a novel approach for addressing specification and accuracy concerns managers may have with econometric forecasting methods. We propose a qualitative research method that can be used to enhance model specification and consequently improve model accuracy. Consider, for example, an econometric forecasting model for in-home DVD recorders. A forecaster may hypothesize that consumers will treat digital personal video recorder (PVR) devices such as TiVoTM or ReplayTVTM as substitute products. As a result, the forecaster may consider adding PVR sales as an independent variable to the forecasting model. Similarly, a manager forecasting industry sales for sport utility vehicles may hypothesize that consumers are beginning to believe that such vehicles are unsafe and, therefore, may consider including an early indicator of this belief (e.g., the number of traffic fatalities involving sport utility vehicles) as an independent variable in the forecasting model. Likewise, if sales of landscaping services fall short of expectations after an unforeseen event such as a drought, the forecaster may hypothesize that it was the drought that caused the forecasting error by changing the behavior of customers. In these situations and in many others, forecasters make specific hypotheses about the underlying reasons for the behavior of consumers (i.e., the so-called ‘‘why?’’ questions of consumer behavior). When such assumptions and hypotheses about the ‘‘why?’’ questions of consumer behavior have a strong influence on the forecasting model, the hypotheses should not be left untested. To continue an earlier example, the

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forecaster could be wrong when she hypothesizes that consumers view PVRs as a substitute product for DVD recorders. If the hypothesis is incorrect, not only will an irrelevant and potentially misleading independent variable (i.e., PVR sales) be added to the forecasting model, but a more relevant independent variable (i.e., based on a more correct answer to the ‘‘why?’’ question) may be missing from the forecasting model. In general, to the extent that the forecasting method or forecasted values are influenced by the hypothesis, the hypothesis should be empirically tested. If the hypothesis is confirmed, the forecaster’s subjective confidence in future forecasts will be increased. If the hypothesis is disconfirmed, misspecifications in the forecasting model will be avoided.

QUANTITATIVE SURVEY METHODS Some methods for testing hypotheses are already well understood. Classical statistical testing is based on the notion of testing hypotheses with quantitative data. In the context of the ‘‘why?’’ questions of consumer behavior, quantitative data from survey research are frequently used for this purpose. A forecaster endeavoring to test the hypothesis that PVRs are used by consumers as substitutes for DVD recorders may rely on survey data. As part of such a survey, a sample of consumers may be asked questions about their shopping behavior, such as, ‘‘Before you purchased your Personal Video Recorder, did you consider purchasing a DVD recorder?’’ Similarly, survey participants may be asked questions about the importance of product attributes that are available with DVD recorders and not with PVRs, such as ‘‘How important is it to you that your video recording product allow you to share your recorded video with your friends?’’ Questions about participants’ beliefs and attitudes toward DVD recorders and PVRs may also be asked. Such surveys are designed to yield quantitative data that can be used to test the forecaster’s hypotheses. However, survey methods have known limitations that are particularly relevant to testing hypotheses related to the ‘‘why?’’ questions of consumer behavior. For example, survey methods are expensive and time-consuming, making hypothesis testing with survey research difficult in real-life settings. Perhaps even more important, some types of ‘‘why?’’ questions in consumer behavior may not be well suited to survey methods. There is a growing recognition (Fischoff, 1993; Simmons, Bickart, & Lynch, 1993; Wilson & Hodges, 1992) that survey methods may suffer from the ill effects of

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measure reactivity, that is, an influence of the questionnaire instrument on the very phenomena that the instrument is designed to measure. Questionnaire instruments inevitably provide a context that can bias the responses of participants. For example, reports of past behavior (e.g., shopping behavior for DVD recorders and PVRs) are strongly influenced by trivial changes in questionnaire construction such as question order (Schwarz, 1990). Asking questions about topics that participants have not previously considered may also cause measure reactivity in survey research, giving the researcher the illusion that the data are representative of the beliefs of people who have not encountered the questions (Simmons et al., 1993). For example, when participants are asked whether the ability to share DVD recordings with other people is important to them, most participants may respond that such sharing is indeed important to them. However, it may be the case that the question itself caused participants to consider for the first time the notion of sharing recordings with their friends and caused the participants to construct a new judgment regarding whether the ability to share is important. In this way, participants in a survey may report that the ability to share recordings with their friends is important even if a similar judgment has not been made by the customers in the larger population (i.e., those who have not participated in the study and have not encountered the survey question). It is very difficult to determine how the beliefs, attitudes, and behaviors reported by participants in a survey were influenced by the measurement effort itself. Because such beliefs, attitudes, and behaviors are frequently just the ones that are of interest when trying to answer the ‘‘why?’’ questions of consumer behavior, survey methods may not be well suited for testing such hypotheses. In addition to a susceptibility to measure reactivity, another inherent limitation of using survey methods for testing hypotheses is the inflexible nature of the method. A researcher attempting to use survey research to test a specific hypothesis will construct a questionnaire instrument for use in data collection. Except for unusual circumstances, the questionnaire instrument remains unchanged throughout the study, and the data are analyzed only after data collection is complete. Although this process is effective for many purposes, the structured nature of the process does not allow researchers to adjust the study during data collection and explore alternate hypotheses suggested by the study’s early results. As a consequence, the survey may offer evidence that disconfirms the hypothesis without offering any additional insights that suggest a more correct hypothesis. In this way, when a hypothesis is disconfirmed with survey research, the ‘‘why?’’ question regarding the behavior of customers may remain unanswered.

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QUALITATIVE METHODS As an alternative to survey methods designed to yield quantitative data, qualitative methods may be used. Qualitative methods are often less costly and less time-consuming than survey methods. Furthermore, because the best qualitative techniques provide participants with only minimal structure and guidance during measurement, participants only comment on concepts they find important, and such comments are in their own words. In this way, qualitative techniques reduce the problem of measure reactivity. In addition, qualitative techniques give the researcher the flexibility to alter the research plan during data collection. This flexibility allows the researcher to investigate a variety of possible answers to the research question and to explore alternate answers based on results that emerge during data collection. This article proposes the use of a new qualitative method that can be used to test hypotheses regarding the ‘‘why?’’ questions of consumer behavior that are relevant to forecasters. It should be noted that the qualitative method described here is different from the collection of methods sometimes known as qualitative forecasting, which includes historical analogy and the Delphi method for gathering expert opinions (for a review see Geurts, Lawrence, & Guerard, 1994, Chapter 14). The method described in this article is not designed to yield forecasts per se; instead, the method is designed to be used by forecasters to test the assumptions and hypotheses upon which their forecasts are based.

THE BIASING INFLUENCE OF A PRIORI HYPOTHESES Qualitative research methods are not new (see Denzin & Lincoln, 1994 for several examples), and the qualitative research methods that have been described in the literature differ from survey research methods in fundamental ways. Rather than employing a questionnaire instrument as a measurement tool, qualitative researchers rely on their own observations. Using a variety of techniques, including interpersonal interactions with research participants such as in one-on-one interviews, qualitative researchers immerse themselves in the context of interest to perform observations and generate data. In this way, qualitative researchers act as a ‘‘human measurement instrument.’’ In the data-analysis stage, qualitative researchers immerse themselves in the data to synthesize findings, identify emergent themes, and

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draw conclusions. In this way, qualitative researchers act as a ‘‘human analysis tool.’’ Theorists (e.g., Frey & Fontan, 1991) have long argued that because the researcher plays such a fundamental role in qualitative research (i.e., acting as human measurement instrument and analysis tool) the results of a qualitative study will be unavoidably influenced by the researcher’s expectations, biases, and prejudices. As a result, qualitative researchers have placed an emphasis on (1) acknowledging their a priori expectations and biases as clearly as possible, and (2) performing only research that is exploratory in nature. The former point is particularly relevant to this article. Researchers who have a priori hypotheses are unable to effectively and impartially act as the measurement instrument and analysis tool in a qualitative research setting. As a result, qualitative techniques have been thought to be well-suited for inductive research and ill-suited for deductive hypothesis testing (Patton, 1990). The qualitative researcher may begin with general subjects of inquiry and specific research questions, but specific hypotheses regarding the answers to those research questions have been thought to be incompatible with qualitative techniques.

OUTLINE OF PROPOSED METHOD This article proposes a new qualitative method that can be used to impartially test a priori hypotheses. In essence, the method is as follows. First, a forecaster with an a priori hypothesis engages another person to perform a qualitative research study. The forecaster informs the qualitative researcher regarding the research question and gives guidance concerning the context of interest. Importantly, however, the forecaster does not divulge her specific hypothesis to the qualitative researcher. Next, the researcher performs the qualitative research study by undertaking the necessary observations, synthesizing the data in analysis, and forming the logical conclusions – all without the biasing influence of any a priori hypotheses. After the qualitative researcher investigates the ‘‘why?’’ question and makes his conclusions, he and the forecaster meet to discuss the results of the study. Only at this meeting does the forecaster divulge her hypothesis to the qualitative researcher. If the forecaster’s hypothesis matches the researcher’s findings to a reasonable extent, the hypothesis is supported. If the forecaster’s hypothesis does not match the researcher’s findings, the hypothesis is not supported. Such a lack of support is not overly problematic because the researcher’s conclusion will provide an alternate answer to the research question that

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may be useful to the forecaster. Using the method outlined here, the biasing effect of an a priori hypothesis can be avoided while bringing the benefits of qualitative research (i.e., relatively quick and low in cost, fewer effects from measure reactivity, and flexible in the pursuit of answers) to bear on hypothesis testing.

EXAMPLE OF PROPOSED METHOD To demonstrate the proposed method, we will continue a previous example. A forecaster using an econometric model to forecast sales of home DVD recorders may hypothesize that consumers treat PVRs and home DVD recorders as substitute products. As a consequence, the forecaster may consider adding PVR sales to the forecasting model as an independent variable. Because this hypothesis has the potential to influence forecasted values, the hypothesis should be tested.

Qualitative Researcher The most important aspect of the method being proposed here is that the forecaster should not personally perform the qualitative data collection or analysis. Instead, the forecaster must secure the services of a researcher to conduct the qualitative study. It is imperative that the qualitative researcher remains blind to the hypothesis. The qualitative researcher should be given a general research question to guide the research – for example, ‘‘From the consumer’s perspective, what are the underlying drivers of digital recorder adoption, purchase, and use? More specifically, how do PVRs and in-home DVD recorders relate to each other in terms of the way consumers (1) think of them, (2) purchase them, and (3) use them?’’ As much detail as possible should be shared concerning the research question, thus preventing the qualitative researcher from inadvertently addressing immaterial issues and failing to adequately test the hypothesis. At the same time, any information that could bias the qualitative researcher should not be shared at this stage of the study. For example, as described here, the researcher was not told that the forecaster is affiliated with a maker of DVD recorders. When the researcher is not given any hints regarding the forecaster’s hypothesis, the researcher can be unbiased in his role as ‘‘human research instrument’’ and ‘‘human analysis tool.’’

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Zaltman Metaphor Elicitation Technique Although a wide variety of qualitative techniques for data collection and analysis are compatible with the qualitative method for hypothesis testing, we suggest that the Zaltman metaphor elicitation technique (ZMET) be used. ZMET is a relatively new qualitative technique that is ideally suited to answering the ‘‘why?’’ questions of consumer behavior. The technique is designed to explore the underlying sources of personal relevance that motivate consumer decision making and purchase. By employing ZMET, researchers can identify, catalog, and graphically display (Christensen & Olson, 2002) the relationship between PVRs and DVD recorders as seen by consumers. The ZMET and its role in the larger qualitative method is briefly described below; more detailed instructions for ZMET administration have been published by Zaltman (2003, 1997) and Zaltman and Coulter (1995).

ZMET Data Collection The data collection for a ZMET study is comprised of a series of one-on-one depth interviews. The qualitative researcher will recruit people who are willing to participate in an interview that can last 90 min or more. Typically, qualitative researchers employ non-probability sampling during the recruiting process. In the continuing example, the qualitative researcher might recruit 20 research participants, and to ensure a diverse set of viewpoints, the researcher might specifically recruit recent buyers of in-home DVD recorders and also owners of PVRs. Obviously, 20 observations is fewer than would typically be required for a survey research study. Fewer observations are necessary because the goal of a qualitative study is not to perform a statistical inference of a population parameter. Estimates of sampling error and other issues relating to statistical inference are irrelevant to the task at hand. Instead, the goal of a qualitative study is to achieve what is known as ‘‘saturation,’’ that is, collecting enough data that the researcher begins to observe themes repeated over and over again (Morse, 1994; Zaltman & Coulter, 1995). With this continual repetition of ideas, the researcher gains confidence that the themes are fundamental to understanding the topic of the study. Further, saturation implies the researcher gains little incremental insight with each additional interview. In fact, given the voluminous data collected over the course of a series of depth-interviews, very large sample sizes would make data analysis intractable. A study population of 15–20

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participants is generally sufficient in a ZMET study to achieve the important goal of saturation in the study (Zaltman, 1997). The qualitative researcher selected to perform the study must be experienced and comfortable with the ZMET methodology because the researcher is an active player in the gathering of data in the research interview. During the interviews, the researcher has the freedom to explore new ideas and undiscovered themes elicited through the ZMET process. These interviews should follow closely the several steps in the ZMET method as described by Zaltman (1997). Participants are required to prepare for the ZMET interview by following detailed instructions. First, participants are instructed to think about the things that are important to them about digital video recording and their recent DVD recorder (or PVR) purchase. Respondents are then instructed to find 10 pictures that represent their thoughts and feelings about DVD recorders (or PVRs) and bring the pictures to the interview. These photos can come from any source, but generally participants select them from catalogs, magazines, and newspapers. During the interview, each picture helps participants to express tacit cognitions and emotions that are associated with DVD recorders, PVRs, and in-home digital recording more generally. The interviewer asks the participant to explain what each picture represents. As the participant explains the meaning represented by the photo, interviewers are trained to identify important ideas mentioned by participants and then to ask followup questions to encourage the participants to elaborate on the idea further. To ensure understanding, the interviewer also uses reflexive interviewing techniques such as restating and summarizing participants’ comments (Athos & Gabarro, 1978; Rogers & Farson, 1984). Once the participant’s meaning is well understood, the interviewer uses probes to see how the ideas expressed are linked to other ideas already expressed by the participant. As an example of this process, one participant might bring a picture of a computer and comment, ‘‘I chose this picture because it is like my DVD recorder. This computer is compatible with other computers, and my DVD recorder is compatible. My favorite thing is how the DVDs I make are compatible with other people’s DVD players.’’ To understand this idea and its consequences, the interviewer may follow up with a question such as, ‘‘Why do you say that?’’ The participant may respond, ‘‘I love burning DVDs and giving them as little gifts to my friends. It’s so great that my machine is compatible with theirs.’’ This probing and questioning process is repeated until all the pictures brought by the participant have been explored. In this way, the ZMET interview identifies and explores the thoughts and feelings of participants concerning the topic at hand.

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The participant-driven nature of the ZMET interview process should be noted. Within the general topic of inquiry, participants are free to express the thoughts and feelings they judge to be important, and they do so in their own words. Thus, if a participant in a ZMET interview says ‘‘I love burning DVDs and giving them as little gifts to my friends,’’ the expression is fully voluntary and spontaneous on the part of the participant. Admittedly, such ZMET interviews may not fully eliminate measure reactivity. For example, describing a purchase process to an interviewer may bring thoughts and feelings into conscious awareness that otherwise would not have been the case. Nonetheless, the participant-driven ZMET interviews are less likely to lead to measure reactivity than questionnaire-driven survey research. Another aspect of the ZMET interview is the flexibility of the process. The researcher is free to explore new ideas as they emerge during the course of interviews. He can ask additional questions to understand the fullness of an idea the respondent is trying to convey. The interviewer can also investigate how these new ideas are interrelated with other ideas already encountered over the course of the study. Further, as new ideas begin to emerge during data collection, the researcher can ‘‘keep an eye out’’ for related and contradictory themes in subsequent interviews. This kind of flexibility is available with qualitative research but is not available with questionnaire instruments in survey research. ZMET Analysis After the interviews are complete, audio recordings of the interviews are transcribed into written form. The lengthy set of written transcripts becomes the data set used for analysis. At this point, the researcher changes hats and becomes the ‘‘human analysis tool’’ spoken of earlier. Content coding begins as the analyst reviews the transcripts in search of core ideas and repeated themes. Following the open coding techniques of grounded theory proposed by Strauss and Corbin (1990), all of the ideas expressed by the participants are identified. Specifically, as recurring concepts begin to emerge again and again across the several transcripts, the researcher creates codes to represent specific categories of meaning, and the researcher gives each code a name. From this process emerges a code list of the core ideas found in the data. Throughout the coding process, a rigorous questioning of each coding interpretation must be applied to verify that it is ‘‘grounded’’ in the actual verbatim statements of the respondents found in the transcripts. The analyst asks himself questions such as, ‘‘Does that code really reflect what the participant is saying? Is a different code required altogether?’’ By doing so, the voice of the consumer (Van Maanen, 1979) drives the analysis and

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interpretation. This ‘‘constant comparative method of analysis’’ (Glaser & Strauss, 1967, p. 101) adds rigor to the coding process by forcing the researcher to go back to the data again and again to ensure that each interpretation is well-supported by verbatim evidence in the transcripts. From this analysis emerges a collection of central themes regarding DVD recorders, PVRs, and in-home digital recording more generally. Furthermore, the interconnections between core ideas are revealed – particularly the relationship between DVD recorders and PVRs in terms of the purchase process. The researcher considers these themes and arrives at conclusions regarding the research questions.

The Hypothesis Test After the ZMET study is complete, the next step in the method proposed in this article is a meeting between the forecaster and the qualitative researcher. The researcher should present a written report to the forecaster that summarizes the study’s findings and provides specific examples from the interviews that support the findings. Then the forecaster should reveal the a priori hypothesis to the qualitative researcher. Finally, the extent of the match between the hypothesis and the findings of the study should be discussed. The key to the hypothesis test is an evaluation of how well the hypothesis matches the study’s findings. The researcher is in the best position to evaluate how closely the hypothesis matches the observed data. The forecaster is in the best position to evaluate how closely the researcher’s conclusions match the original hypothesis. Together, the forecaster and the researcher should judge how closely the a priori hypothesis matches the study’s findings. Because the research context is typically complex and there are usually a variety of ways to describe any single conclusion, an exact match between the hypothesis and the study conclusions is unlikely – even if the hypothesis is actually correct. If the hypothesis is indeed correct, however, the match between the hypothesis and the observed findings should be at least reasonably close. The forecaster and the researcher should not feel compelled to achieve a match; a lack of a match is not a failure. If the qualitative study suggests only a partial match (e.g., some consumers do treat PVRs and in-home DVD recorders as substitute goods, while other consumers treat them as complementary goods), then important insights have been gained. Even if the hypothesis is fully disconfirmed (e.g., consumers use in-home DVD

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recorders and PVRs for completely different purposes and therefore do not treat the two products as substitutes), valuable insight has been gained, and the forecasting model can be improved (e.g., because PVR sales will not be included as an independent variable). Thus, the forecaster and the qualitative researcher must avoid the temptation to declare the hypothesis to be confirmed when the study’s findings do not warrant such a conclusion.

LIMITATIONS OF THE PROPOSED METHOD All research methods have limitations (McGrath, 1982), and the method we propose here is no exception. The main limitation of the proposed method is that the findings of the qualitative study may have limited generalizability – an issue that stems largely from the sampling method. As described earlier, participants in qualitative studies are typically recruited with a non-probability sample. Indeed, instead of striving for a sample that is representative of a population of interest, a qualitative study is frequently improved if a diverse set of participants are interviewed (e.g., DVD-recorder buyers and PVR buyers). This diversity allows the qualitative researcher to experience a variety of viewpoints during data collection. Although such a sample is valuable from the qualitative researcher’s perspective, it certainly can be questioned whether the findings from such a study can be generalized to a broader population. Addressing the issue of generalizability in relation to hypothesis testing requires a consideration of the notion of the strength of a hypothesis test. Any test of a hypothesis can be evaluated in terms of how strongly it tests the hypothesis. A strong test presents a difficult challenge to the hypothesis, while a weak test presents a less difficult challenge (Calder, Phillips, & Tybout, 1981). In the current context, a study using a non-representative sample is weakened as a hypothesis test because the findings are not generalizable and could be idiosyncratic to the group of participants recruited for the study. One way of making the hypothesis stronger would be to perform the qualitative study with a sample that is fully representative of potential PVR buyers. The strength of the hypothesis test has important implications for evaluating the results of the test. When a hypothesis is exposed to a strong test and the hypothesis is not disconfirmed (i.e., when we cannot reject the hypothesis based on the results of the study), the forecaster’s subjective confidence in the truthfulness of the hypothesis is greatly increased. If, instead, the hypothesis is exposed to a weaker test and the hypothesis is not

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disconfirmed, the forecaster’s confidence in the truthfulness of the hypothesis is increased, but not so much as would have been the case if the hypothesis had passed a stronger test (Meehl, 1990). If a strong hypothesis test requires the same resources (e.g., time) as a weak hypothesis test, then the strong hypothesis test is superior. Frequently, however, stronger hypothesis tests are more difficult and require more resources to perform than weaker hypothesis tests. As a result, weaker hypothesis tests play a useful role in the process of a priori hypothesis testing (Calder et al., 1981). If a weak hypothesis test is employed and the hypothesis is not disconfirmed, the forecaster has at least some confidence that the hypothesis is correct, and the forecaster may find that subjective level of confidence to be sufficient. Even if that level of confidence is not sufficient (e.g., because the forecasting process is particularly sensitive to the outcome of the hypothesis test), exposing the hypothesis to a weak test before exposing it to a strong test is advantageous because the weaker test may be all that is necessary to disconfirm the hypothesis. That is, if the weaker test disconfirms the hypothesis, the forecaster can avoid the expense of a stronger test. For these reasons, it is useful to employ a weak hypothesis test as a first test; if the hypothesis survives the first hurdle and if greater confidence in the hypothesis is necessary, a second and stronger hypothesis test should then be employed. In summary, qualitative research techniques frequently use a non-representative sample, making the generalizability of the findings questionable. As a result, the method described in this article will yield hypothesis tests that are not typically as strong as tests that could be designed using other methods. This is not a serious drawback, however, because weak tests serve a useful purpose in the process of evaluating a priori hypotheses.

APPLICABILITY OF THE PROPOSED METHOD While the proposed method is relevant to forecasters who have untested assumptions and hypotheses about the ‘‘why?’’ questions of consumer behavior, the method may have more general applicability. Hypothesis testing is commonplace in theoretical marketing research, and the disadvantages of survey research methods – particularly the problem of reactive measures – are relevant to theoretical research. Academic marketing researchers can use the method proposed here to qualitatively test theory-based a priori hypotheses without concern for the biasing influence of such hypotheses on the test itself.

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CONCLUSIONS The qualitative method we have proposed in this article is particularly strong in the areas where survey methods have apparent limitations. As noted, a qualitative study can provide results much more quickly and cost effectively than a study using survey methods. The qualitative interview technique reduces the problem of measure reactivity by allowing participants to comment in their own words on issues they judge to be important. Further, a qualitative study is more flexible than a survey study, allowing the researcher to pursue unanticipated avenues during data collection based on the early results of a study. Despite these benefits, qualitative research has previously been thought to be incompatible with hypothesis testing because an a priori hypothesis held by a qualitative researcher inevitably biases the outcome of a qualitative study. This article has described a method that prevents such bias, allowing a priori hypotheses to be tested with qualitative techniques.

REFERENCES Athos, A. G., & Gabarro, J. G. (1978). Interpersonal behavior. Englewood Cliffs, NJ: PrenticeHall. Brodie, R., & De Kluyver, C. (1987). A comparison of the short-term forecasting accuracy of econometric and naı¨ ve extrapolation models of market share. International Journal of Forecasting, 3, 423–437. Calder, B. J., Phillips, L. W., & Tybout, A. M. (1981). Designing research for application. Journal of Consumer Research, 8(September), 197–207. Christensen, G. L., & Olson, J. C. (2002). Using the Zaltman metaphor elicitation technique (ZMET) to map consumers mental models. Psychology and Marketing, 19(June), 477–502. Denzin, N. K., & Lincoln, Y. S. (1994). Handbook of qualitative research. London: Sage. Fischoff, B. (1993). Value elicitation: Is there anything in there. In: M. Hechter, L. Nadel & R. E. Michod (Eds), The origin of values (pp. 187–214). New York: Aldine De Gruyter. Frey, J. H., & Fontan, A. (1991). The group interview in social research. Social Science Journal, 28(2), 175–187. Geurts, M., Lawrence, K. D., & Guerard, J. (1994). Forecasting sales. Greenwich, CT: JAI Press. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine De Gruyter. McGrath, J. E. (1982). Delemmatics. In: J. E. McGrath, J. Martin & R. A. Kulka (Eds), Judgement calls in research (pp. 69–101). Beverly Hills, CA: Sage. Meehl, P. E. (1990). Appraising and amending theories: The strategy of Lakotosian defense and two principles that warrant it. Psychological Inquiry, 1(2), 108–141.

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Morse, J. M. (1994). Designing funded qualitative research. In: N. K. Denzin & Y. S. Lincoln (Eds), Handbook of qualitative research (pp. 220–235). London: Sage. Patton, M. Q. (1990). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage. Rogers, C. R., & Farson, R. E. (1984). Active listening. In: D. A. Kolb, I. M. Rubin, & J. M. McIntyre (Eds), Organizational psychology: Readings on human behavior in organizations (4th ed., pp. 255–266). New York: Prentice-Hall. Schwarz, N. (1990). Assessing frequency reports of mundane behaviors: Contributions of cognitive psychology to questionnaire construction. In: C. Hendrick & M. S. Clark (Eds), Research methods in personality and social psychology (pp. 98–119). Newbury Park, CA: Sage. Simmons, C. J., Bickart, B. A., & Lynch, J. G., Jr. (1993). Capturing and creating public opinion in survey research. Journal of Consumer Research, 20(September), 316–329. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. Van Maanen, J. (1979). The fact of fiction in organizational ethnography. Administrative Science Quarterly, 24(December), 539–550. Wilson, T. D., & Hodges, S. D. (1992). Attitudes as temporary constructions. In: L. L. Martin & A. Tesser (Eds), The construction of social judgments (pp. 37–66). Hillsdale, NJ: Erlbaum. Zaltman, G. (1997). Rethinking market research: Putting people back in. Journal of Marketing Research, 34(November), 424–437. Zaltman, G. (2003). How customers think. Boston, MA: Harvard Business School Publishing. Zaltman, G., & Coulter, R. A. (1995). Seeing the voice of the customer: Metaphor-based advertising research. Journal of Advertising Research, 35(4), 35–51.

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PART D: FORECASTING METHODS AND ANALYSIS

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FORECASTING SALES OF COMPARABLE UNITS WITH DATA ENVELOPMENT ANALYSIS (DEA) Ronald K. Klimberg, Shelia M. Lawrence and Kenneth D. Lawrence ABSTRACT Regression analysis is the technique often used to provide forecast estimates of comparable units. The weights assigned to the predictor variables in a regression equation are based upon an average relationship. Nevertheless, realistically, the relative importance of each of the predictor variables will most likely vary from comparable unit to comparable unit. This results in the regression model’s providing forecast estimates that are sometimes too high or too low. In this paper, we will present a new methodology to incorporate into the regression forecasting analysis a new variable that captures the unique weighting of each comparable unit. This new variable is the relative efficiency of each comparable unit that will be generated by a technique called Data Envelopment Analysis (DEA). The results of applying this new regression forecasting methodology with the DEA variable to a sales territory data set will be presented.

Advances in Business and Management Forecasting, Volume 4, 201–213 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04013-4

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1. INTRODUCTION Forecasting, whether it is forecasting future demand, sales, or production, is important and unavoidable. ‘‘Every time we develop a plan of any type, we first make a forecast. This is true of individuals, profit and nonprofit companies, and government organizations; in fact, it is true of any entity that makes a plan’’ (Mentzer & Bienstock, 1998). In particular, sales forecast is a key input to a firm’s budgeting, operations, and financial planning process (Rahmlow & Klimberg, 2002). The accuracy of these forecasts is also important (Mentzer & Bienstock, 1998; Rahmlow & Klimberg, 2002). More accurate the forecasts, the more confidence users have in the forecast and the forecasting process (Rahmlow & Klimberg, 2002). As markets change and competition increases, the accuracy of these sales forecasts provides an important competitive advantage. Forecasting techniques can be broken down into two general categories – qualitative and quantitative. Qualitative forecasting techniques, such as executive judgment or Delphi technique, elicit managers’ opinions to provide a forecast. These executives or managers use their industry knowledge, past industry experience, and intuition to provide a forecast estimate. A major possible problem with these qualitative approaches is that the forecasts are subjective, that is, they are based on opinions which could be extremely bias. However, these qualitative forecasting techniques continue to be more popular than the quantitative approaches mainly due to the fact that these techniques are easily understood and inexpensive. Over the past several decades, more sophisticated and easy to use quantitative models have been developed. These quantitative forecasting techniques use historical data to predict the future. Most of the major quantitative forecasting techniques can be further categorized into either time series approaches or regression analysis. Time series forecasting techniques are forecasting techniques that only use the time series data itself and no other data to build the forecasting models. These time series approaches isolate and measure the impact of the trend, seasonal and cyclical time series components. On the other hand, regression analysis uses a set of predictor variables, possibly including the time series components, that are believe to influence the forecasted variable, e.g., sales. Regression techniques employ the statistical method of least squares to establish a statistical relationship between the forecasted variable and the set of predictor/causal variables. Overall, users of these quantitative approaches have been more satisfied

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with the results than users of qualitative forecasting techniques (Mentzer & Bienstock, 1998; Rahmlow & Klimberg, 2002). Many times the forecasting process is producing forecasts for comparable units. A comparable unit could be an individual, group of individuals, a department, a company, and so on. Each comparable unit should be performing somewhat the same set of activities. A few examples of producing forecasts for comparable units are providing forecasts for sales territories, sales representatives, or forecasts of production for factories. When applying regression analysis to a set of comparable units, the statistical relationship established is an average relationship using one set of weights assigned to the predictor variables. However, the relative importance of each of the predictor variables will most likely vary from comparable unit to comparable unit. As a result, in some cases, the regression model could provide forecast estimates that are too high or too low. In this paper, we will present a new methodology to incorporate into the regression forecasting analysis a new variable that captures the unique weighting of each comparable unit. This new variable is the relative efficiency of each comparable unit that will be generated by a technique called Data Envelopment Analysis (DEA). In the next section, we provide a brief introduction to DEA. Subsequently, we will present our new regression forecasting methodology and apply it to a data set. Finally, the conclusions and future extensions are discussed.

2. DATA ENVELOPMENT ANALYSIS (DEA) DEA utilizes linear programming to produce measures of the relative efficiency of comparable units that employ multiple inputs and outputs. DEA takes into account multiple inputs and outputs to produce a single aggregate measure of relative efficiency for each comparable unit. The technique can analyze these multiple inputs and outputs in their natural physical units without reducing or transforming them into some common measurement such as dollars. The Charnes, Cooper and Rhodes (CCR) DEA model (Charnes, Cooper, & Rhodes, 1978) is a linear program that compares the ratio of weighted outputs to weighed inputs, i.e., efficiency, for each comparable unit. The efficiency of the kth comparable unit (i.e., Ek) is obtained by solving the

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following linear formulation: MAX E k ¼

t P

ur Y rk

r¼1

s.t. m X

vi X ik ¼ 1

i¼1 t P r¼1

ur Y rj


m P

vi X ij  0

j ¼ 1; . . . ; n

i¼1

ur ; v i 


(1)

8 r; i

where Parameters Yrj ¼ amount of the rth output for the jth comparable unit; Xij ¼ amount of the ith input for the jth comparable unit; t ¼ the number of outputs; m ¼ the number of inputs; and n ¼ the number of comparable units; e ¼ is a small infinitesimal value; Decision Variables ur ¼ the weight assigned to the rth output; and vi ¼ the weight assigned to the ith input. The CCR DEA formulation determines objectively the set of weights, ur and vi, that maximizes the efficiency of the kth comparable unit, Ek. The constraints require the efficiency of each comparable unit, including the kth comparable unit, not to exceed 1, and the weights, ur and vi, must be positive. A similar DEA formulation must be solved for each comparable unit. A comparable unit is considered relatively inefficient (i.e., Eko1) if it is possible to increase its outputs without increasing inputs or decrease its inputs without decreasing outputs. A comparable unit identified as being efficient (i.e., Ek ¼ 1) does not necessarily imply absolute efficiency. It is only relatively efficient as compared to the other comparable units that are being considered. These efficiency ratings allow decision-makers to identify which comparable units need to improve and to what degree. Since the Charnes et al.’s 1978 paper, there have been thousands of theoretical contributions and practical applications in various fields using DEA

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(Seiford, 1996). DEA has been applied to many diverse areas such as: health care, military operations, criminal courts, university departments, banks, electric utilities mining operations, manufacturing productivity, and railroad property evaluation (Klimberg, 1998; Klimberg & Kern, 1992; Seiford & Thrall, 1990; Seiford, 1996).

3. NEW REGRESSION FORECASTING METHODOLOGY Our new regression forecasting methodology is designed to be applied to a historical data set of multiple inputs and outputs variables from a set of comparable units. Additionally, one output variable is assumed to be the principal/critical variable that will be needed to be forecasted, e.g., sales, production, or demand. The new regression forecasting methodology is the following three-step process: Step 1. Stepwise Regression Run a stepwise regression using the principal/critical output variable as the regression dependent variable, the variable to be forecasted, and all the input variables as regression independent variables. Only the input variables are included since these variables are controlled by the decision-maker, i.e., the decision-maker can decide the values of the input variables and henceforth can determine the forecasted value of the principal/critical value by substituting them into the regression equation. Stepwise regression is employed to decrease the number of input variables to be included to only those variables that are statistical significant and to improve the discriminating power of the DEA analysis. Step 2. DEA Analysis Given the set of statistically significant input variables from the stepwise regression in Step 1, we define the comparable unit efficiency as the ratio of weighted outputs to weighted inputs: E ¼ Efficiency ¼

weighted sum of outputs weighted sum of inputs

Run the DEA model for each comparable unit using the statistical significant input variables from (1) as inputs and in the numerator using all the output variables. Each efficiency score measures the relative efficiency of the comparable unit. These efficiency scores can be use to evaluate performance of the comparable units and provide benchmarks. Nevertheless, besides each efficiency score being comprised of a different set of inputs and outputs

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values, each comparable unit’s efficiency score includes a unique set of weights. The DEA process attempts to find objectively the set of weights which will maximize a comparable unit’s efficiency. Therefore, the DEA model has selected the best possible set of weights for each comparable unit. The variation of these weights from comparable unit to comparable unit allows each comparable unit to have its own unique freedom to emphasize the importance of each of these input and output variables in its own way. How well they do this is measured by the efficiency score. So, we will use these efficiency scores as surrogate measures of the unique emphasis of the variables and of performance and include them in Step 3. Step 3. Stepwise Regression Run a new stepwise regression. Again, use the principal/critical output variable as the regression dependent variable and all the input variables and the DEA efficiency score as regression independent variables. This stepwise regression model should be improved, i.e., should have a significantly lower standard error of the mean and increase R2.

4. EXAMPLE OF REGRESSION ANALYSIS OF COMPARABLE UNITS Cravens, Woodruff, and Stamper (1972) applied multiple regression to evaluate sales territory performance. The data set consists of the sales for 25 territories, each with the following eight predictor variables:        

Salesperson’s experience (Time) Advertising expense (Adver) Change in Market Share (Change) Workload per account (WkLoad) Market potential (MktPoten) Market Share (MktShare) Number of accounts (Accts) Salesperson’s motivation and effort (Rating)

and are listed in Table A1. Their regression model results, including all eight predictor variables, are shown in Fig. A1. This model has a R2 of 92.2% and a standard error of 449, as shown in Table 2. Those territories performing well or poorly were identified by the magnitude of their residual. Those territories whose

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predicted values were greater/less than one standard deviation away are listed in Table 1. Several variables in Cravens et al.’s multiple regression were not significant, so, executing our Step 1 stepwise regression, produced a model with only four predictor variables, as shown in Fig. A2. The stepwise regression produced a model with the following four variables: number of accounts, market potential, market share, and advertising. The R2 decreased a small amount, and the standard error slightly increased in the stepwise regression model, as compared to the eight variable model, as shown in Table 2. Additionally, the stepwise regression found two more territories with residuals more than one standard deviation away, one above and one below, as shown in Table 1. We then performed Step 2 of our methodology, a DEA analysis of the 25 territories using the four significant predictor variables as inputs and sales as the only output variable. The DEA analysis produced:  six efficient territories,  an average DEA efficiency of 87.1%, and  a minimum DEA efficiency of 58.8%. As described in Step 3, we re-executed the stepwise regression including the DEA efficiency score and the other eight predictor variables. This stepwise regression model created a model with the same four variables from Table 1.

Based on regression models good and poor performing territories. Good Performing Territories

Poor Performing Territories

5, 10 5, 8, 10

19, 20, 22 3, 19, 20, 22

Multiple regression (8 predictors) Stepwise regression

Table 2.

Summary of model results.

All eight variables Stepwise – 4 variables Stepwise – 4 variables + DEA efficiency score

R2

Standard Error

92.2 90.0 95.0

449 454 330

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Table 3.

Improvement in the size of the residuals.

Step 1: Stepwise – 4 variables Step 3: Stepwise – 4 variables + DEA efficiency score

Minimum

Maximum


788
437

934 662

Step 1, (# of accounts, market potential, market share, and advertising) plus the DEA efficiency score and is shown in Fig. A3. Step 3’s model has a higher R2 and significantly lower standard error than the initial eight variable model of Cravens et al., as shown in Table 2. Additionally, Table 3 shows the substantial improvement in the residuals from the stepwise regression in Step 1 to the stepwise regression in Step 3. The minimum residual improved (decreased) by about 44% and the maximum residual improved by 29%. Moreover, 15 out of the 25 territories residuals decreased from the stepwise regression to the stepwise regression with the DEA variable (Step 1 to Step 3) with an average decrease of over 50%.

5. CONCLUSIONS In this paper, we will present a new regression forecasting methodology to forecasting comparable units. This approach included in the regression analysis a surrogate measure of the unique weighting of the variables and of performance. This new variable is the relative efficiency of each comparable unit that is generated by DEA. The results of applying this new regression forecasting methodology including a DEA efficiency variable to a data set demonstrated that this may provide a promising rich approach to forecasting comparable units. We plan to perform further testing with other data sets, some with more comparable units and more than one output variables.

REFERENCES Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Cravens, D. W., Woodruff, R. B., & Stamper, J. C. (1972). An analytical approach for evaluating sales territory performance. Journal of Marketing, 36(January), 31–37. Klimberg, R. K. (1998). Model-based health decision support systems: Data envelopment analysis (DEA) models for health systems performance evaluation and benchmarking. In: J. Tan (Ed.), Health decision support systems (pp. 99–126). Aspen.

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Klimberg, R. K., & Kern, D. (1992). Understanding data envelopment analysis (DEA). Boston University School of Management Working Paper, pp. 1–40. Mentzer, J. T., & Bienstock, C. C. (1998). Sales forecasting management. Sage Publications. Rahmlow, H., & Klimberg, R. (2002). Forecasting practices of MBA’s. Advances in business and management forecasting (Vol. 3, pp. 113–123). Amsterdam: Elsevier Science Ltd. Seiford, L. M. (1996). Data envelopment analysis: The evaluation of the state of the art (1978– 1995). The Journal of Productivity Analysis, 9, 99–137. Seiford, L. M., & Thrall, R. M. (1990). Recent developments in DEA: The mathematical programming approach to frontier analysis. Journal of Econometric, 46, 7–38.

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6. APPENDIX Table A1. Territory

Sales

Time

MktPoten

Adver

MktShare

Change

Accts

WkLoad

Rating

3669.88 3473.95 2295.10 4675.56 6125.96 2134.94 5031.66 3367.45 6519.45 4876.37 2468.27 2533.31 2408.11 2337.38 4586.95 2729.24 3289.40 2800.78 3264.20 3453.62

43.1 108.13 13.82 186.18 161.79 8.94 365.04 220.32 127.64 105.69 57.72 23.58 13.82 13.82 86.99 165.85 116.26 42.28 52.84 165.04

74065.11 58117.3 21118.49 68521.27 57805.11 37806.94 50935.26 35602.08 46176.77 42053.24 36829.71 33612.67 21412.79 20416.87 36272 23093.26 26878.59 39571.96 51866.15 58749.82

4582.88 5539.78 2950.38 2243.07 7747.08 402.44 3140.62 2086.16 8846.25 5673.11 2761.76 1991.85 1971.52 1737.38 10694.2 8618.61 7747.89 4565.81 6022.7 3721.1

2.51 5.51 10.91 8.27 9.15 5.51 8.54 7.07 12.54 8.85 5.38 5.43 8.48 7.8 10.34 5.15 6.64 5.45 6.31 6.35

0.34 0.15
0.72 0.17 0.5 0.15 0.55
0.49 1.24 0.31 0.37
0.65 0.64 1.01 0.11 0.04 0.68 0.66
0.1
0.03

74.86 107.32 96.75 195.12 180.44 104.88 256.1 126.83 203.25 119.51 116.26 142.28 89.43 84.55 119.51 80.49 136.58 78.86 136.58 138.21

15.05 19.97 17.34 13.4 17.64 16.22 18.8 19.86 17.42 21.41 16.32 14.51 19.35 20.02 15.26 15.87 7.81 16 17.44 17.98

4.9 5.1 2.9 3.4 4.6 4.5 4.6 2.3 4.9 2.8 3.1 4.2 4.3 4.2 5.5 3.6 3.4 4.2 3.6 3.1

RONALD K. KLIMBERG ET AL.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The 25 Sales Territory Data.

1741.45 2035.75 1578.00 4167.44 2799.97

10.57 13.82 8.13 58.54 21.14

23990.82 25694.86 23736.35 34314.29 22809.53

860.97 3571.51 2845.5 5060.11 3552

7.37 8.39 5.15 12.88 9.14


1.63
0.43 0.04 0.22 0.74

75.61 102.44 76.42 136.58 88.62

20.99 21.66 21.46 24.78 24.96

1.6 3.4 2.7 2.8 3.9

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21 22 23 24 25

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SUMMARY OUTPUT Regression Statistics Multiple R 0.960228698 R Square 0.922039153 Adjusted R Square 0.88305873 Standard Error 449.0256587 Observations 25 ANOVA df 8 16 24

Regression Residual Total

Coefficients -1507.81373 2.009566151 0.03720491 0.150988896 199.0235363 290.855134 5.550960651 19.79389189 8.189283662

Intercept Time MktPoten Adver MktShare Change Accts WkLoad Rating

Fig. A1.

SS 38153564 3225985 41379549 Standard Error 778.6349 1.930654 0.008202 0.047109 67.02792 186.782 4.77555 33.67669 128.5056

MS 4769196 201624

F 23.6539

t Stat -1.93648 1.040873 4.535912 3.20513 2.969263 1.55719 1.162371 0.587762 0.063727

P-value 0.070675 0.313408 0.000338 0.005518 0.009041 0.138983 0.26213 0.564896 0.949977

Significance F 1.81559E-07

Multiple Regression Results with All Eight Predictor Variables

SUMMARY OUTPUT Regression Statistics Multiple R 0.94892 R Square 0.90045 Adjusted R Square 0.88054 Standard Error 453.8362 Observations 25 ANOVA Regression Residual Total

Intercept MktPoten Adver MktShare Accts

df 4 20 24

Coefficients -1441.93 0.038218 0.17499 190.1443 9.213896

Fig. A2.

SS 37260202.46 4119346.464 41379548.93 Standard Error 423.581705 0.007976936 0.036906657 49.74415347 2.865210383

MS 9315051 205967.3

F 45.22587

t Stat -3.40414 4.791005 4.741422 3.822445 3.215783

P-value 0.002814 0.000111 0.000125 0.001065 0.004337

Stepwise Regression Results

Significance F 9.56359E-10

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SUMMARY OUTPUT Regression Statistics Multiple R 0.97466 R Square 0.949961 Adjusted R Square 0.936793 Standard Error 330.1173 Observations 25 ANOVA

Regression Residual Total

Intercept MktPoten Adver MktShare Accts DEA

Fig. A3.

df 5 19 24

SS 39308977 2070572 41379549

Coefficients -3201.59 0.039758 0.179766 138.7031 8.223079 25.13441

Standard Error 509.5426 0.005813 0.026868 38.07893 2.096625 5.796822

MS 7861795 108977.5

F 72.14148

t Stat -6.28326 6.839265 6.69065 3.642515 3.922056 4.335895

P-value 4.95E-06 1.58E-06 2.14E-06 0.001732 0.000916 0.000356

Significance F 1.08742E-11

Stepwise Regression Results with DEA Measure

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DATA MINING RELIABILITY: MODEL-BUILDING WITH MARS AND NEURAL NETWORKS Rod J. Lievano and Eric S. Kyper ABSTRACT Apparent successes have led data-mining proponents to claim that its methods can ‘‘discover unexpected relationships,’’ and ‘‘create new knowledge’’ with minimal guidance. Such claims require proof of the superiority of results as well as of the development of powerful and novel analytical methods. No such proof has been generally forthcoming. But in all of this the point that data mining is a data-analytical exercise seems to have been lost, or at least subjugated. Equally lost is the realization that data-mining techniques are powerful exploratory methods with no formal confirmatory capability. But no matter how powerful the data management and artificial intelligence, importance must be defined; questions must be asked. This study takes a small step toward assessing data mining claims. Specifically, we address the questions: 1. Are new data mining regression techniques reliable model-building methods? 2. Can data analysis methods implemented naively (through default automated routines) yield useful results consistently?

Advances in Business and Management Forecasting, Volume 4, 215–242 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04014-6

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This assessment is done through a 32
2 factorial experiment with replication, with the factors varied being: 1. Regression methods. Classical OLS regression using forward stepwise techniques, feedforward neural networks with sigmoidal activation functions, and multivariate adaptive regression splines (MARS). 2. Type of function being estimated. (a) Linear, (b) nonlinear continuous, and (c) nonlinear discontinuous. 3. Degree of data contamination. (a) Uncontaminated and (b) 5% contamination. Only one level of other potentially important factors such as sparsity (the proportion of significant predictors), dimension (number of variables), magnitude and type of noise (Gaussian vs. not), and multicolinearity are considered in this instance. The analysis protocol in each instance is to use default settings in automated routines: STATISTICA multiple regression forward stepwise and neural networks and Salford Systems MARS software. The assessment criteria are mean square error (normalized for differences in scale across treatments) and two specification measures: underspecification (the omission of significant predictors in a final model) and overspecification (the inclusion of non-significant predictors in a final model).

DATA MINING, RELIABILITY, AND MODEL-BUILDING Many commentators, practitioners, and providers of data mining report upon the ostensible successes of data mining in applications spanning the physical and social sciences, business, and finance. Etheridge, Sriram, and Kathy Hsu (2000) report that financial institutions are using data mining to assist in investment decisions, in fraud detection, in risk assessment, and in improving customer service. Chan and Lewis (2002) give accounts of the health industry’s data-mining genetic and operational data seeking to identify better pharmaceuticals, treatment procedures, and patient care. Two Crows Corporation (1999) describe how natural resource firms and agencies are mining remotely sensed data to analyze weather, pollution, and land-use patterns, and Shaw, Subramaniam, Woo Tan, and Welge (2001) examine the data mining of sales and demographic data to identify market

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segments, to predict sales, to plan and evaluate promotions, and to improve interaction with customers. Of significant concern in these articles is the lack of specificity regarding methods, a dearth of discussion regarding the validity and reliability of the methods applied, and a curious inconsistency regarding the aims and characteristics of data mining. Although most articles dutifully express the exploratory and secondary nature of data mining – as opposed to the primary and confirmatory nature of formal statistics – the language in many articles blurs these distinctions. Hand, Mannila, and Smyth (2001), for example, state that the purpose of data mining is ‘‘ y the analysis of (potentially large) data sets aimed at finding unsuspected relationships which are of interest or value to the database owners y .’’ Fayyad and Stolorz (1997) states that ‘‘Data mining is a process for making discoveries by extracting previously unknown and actionable information from large consolidated databases in support of strategic and tactical decision making. Data mining invokes algorithms that enumerate patterns from, or fit models to, data y ,’’ and Chopoorian, Witherell, Khalil, and Ahmed (2001) contend that ‘‘Therefore, data mining uncovers new knowledge for managers which, in turn, results in more informed decision-making’’ (emphases added). These are strong words. They all imply reliability, validity, and confirmation of the authenticity of a finding. Is such confidence warranted? Much of the proof given is anecdotal and partial. The relatively few systematic evaluations (see, e.g. Banks, Olszewski, & Maxion, 1999; De Veaux et al., 1993; Sephton, 2001; Ziv & Rajagopalan, 2002) are based principally on accuracy of fit (some measure of the degree of determination or of error variance) and have reported mixed results. Another notable aspect of many of these articles is a tone implying extraordinary novelty, as though these types of analyses were unknown before data mining. As Smyth (2001) observes, however, although there is much methodological novelty in data mining, the large majority of applications are those to which established statistical methods have long been applied. If data mining is indeed novel and has value beyond that of traditional data analysis, it must lie primarily in speed. Clearly, the ability to perform rapid analysis of large data sets is of great potential value. But the speed can only be gained through automation of both the mechanical and conceptual aspects of data analysis, and the value can only be realized if the methods indeed consistently discover real information. In the general context of numerical prediction, this information falls into two related categories: (1) predictions of measures of interest and (2)

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characteristics of the correlational structure of measures of interest. For the first, accuracy of fit – usually estimated as a function of prediction errors – is the critical criterion. For the second, accuracy of fit is also important (and ultimately also determinative), but the accuracy of specification – the functional form and the relevant determining factors or variables – is more so. Thus, a good method should be capable of both accuracy of prediction and of the detection of relationship patterns. This paper discusses an experiment intended to evaluate the validity and reliability of two popular non-parametric data mining techniques used for model-building and prediction. Both techniques evaluated – neural networks (NNW) and multivariate adaptive regression splines (MARS) – are essentially multivariate nonlinear regression techniques with novel numeric algorithms, which allow the fitting of curves of high dimensionality and of virtually any shape. Models are also developed using ordinary least squares forward stepwise regression (FSWR) for comparison.

Methods Used in this Study This study focuses on model-building methods. The methods chosen thus had to be capable of (a) estimating parameters, which specify the relationships between dependent and independent variables, and (b) identifying an appropriate subset of predictor variables (specification). Furthermore, the method implementation had to be capable of proceeding to completion with minimal guidance. On these criteria, three methods were chosen (1) FSWR, (2) NNW, and (3) MARS. Forward Stepwise Regression Stepwise regression is a method for estimating the parameters of f(X) in fitting Y ¼ f(X)+e, which minimizes a function of the error e and selects a subset of potential predictors which meets certain criteria such as simplicity, completeness, and lack of redundancy. The basic stepwise procedure involves (1) identifying an initial model, (2) using the ‘‘stepping’’ criteria to add or remove a predictor variable, and (3) continuing until no additional variables meet the stepping criteria or when a specified maximum number of steps has been reached (see Hocking, 1996). The FSWR method employs a combination of forward selection and backward removal of predictors. An eligible predictor variable is added to the model if its marginal contribution to the model’s overall F value exceeds a specified threshold; an eligible predictor variable is removed from the

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model if its marginal contribution to the model’s overall F value is below a specified threshold. The process continues until there are no more eligible predictors or the specified maximum number of steps has been performed. This method has proven effective in guarding against underspecification (not including a significant predictor), but less so in guarding against overspecification (including spurious predictors). In this study, FSWR was implemented in STATISTICA (Statsoft, 2003) with default values for entry (F ¼ 1), removal (F ¼ 0), and number of steps (S ¼ number of independent variables in the data). Box–Cox transformations were performed to account for apparent nonlinearity. Neural Networks Like traditional linear regression methods, NNW attempts to find a specification for the functional form f(X), which will best fit a set of data observations Y, where ‘‘best’’ usually means satisfying an accuracy of fit criterion based on an error function of the type e ¼ Yf(X). Unlike traditional linear regression methods, however, NNW is a non-parametric, data-driven method, which thoroughly explores a functional neighborhood for a solution, and can represent both linear and nonlinear effects. This power comes at the cost of less formal confirmation and thus of the ability to generalize results. An NNW is a model of a biological neural system. The model includes models of individual neurons, models for the propagation and integration of signals, and models for the form of the network, as well as methods for arriving at a suitable solution. The fundamental basis of NNW is a neuron. The model of a neuron:  receives a number of inputs (either from original data, or from the output of other neurons in the network) through a connection, which has a strength (or weight) corresponding to the efficiency of a biological neuron;  has a single input threshold value. The weighted sum of the inputs is formed, and the threshold subtracted, to compose the activation of the neuron (also known as the post-synaptic potential, or PSP, of the neuron); and  passes an activation signal through an activation function (also known as a transfer function) to produce the output of the neuron. The output of a neuron is modeled by choosing a type of activation or transfer function. Common types are step (0–1 binary), linear, radial, or – frequently – the sigmoidal (logistic) function. Network Architecture. The network is composed of input, transfer (hidden), and output neurons working through feedback/feedforward structures (see

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Haykin, 1999). A simple network has a feedforward structure. The hidden and output layer neurons are each connected to all of the units in the preceding layer (fully connected network). Signals flow from inputs, forward through the hidden units, and eventually reach the output units. When the network is executed (used), the input variable values are placed in the input units, and then the hidden and output layer units are progressively executed. Each of them calculates its activation value by taking the weighted sum of the outputs of the units in the preceding layer, and subtracting the threshold. The activation value is passed through the activation function to produce the output of the neuron. When the entire network has been executed, the outputs of the output layer act as the output of the entire network. Perhaps the most popular network architecture in use today is multilayered perceptrons (MLP) (see Rumelhart & McClelland, 1986). In MLP, the units each perform a weighted sum of their inputs and pass this activation level through a transfer function to produce their output; the units have a layered feedforward arrangement. The network is thus a form of input–output model, with the weights and thresholds being the free parameters of the model. Such networks can model functions of great complexity, with the number of layers and the number of units in each layer determining the degree of complexity. Solving the Network: ‘‘Training’’ Multi-Layer Perceptrons. In traditional linear model fitting, it is possible to determine the model configuration, which absolutely minimizes an error function (usually the sum of squared errors). In NNW, the network can be adjusted to lower its error, but finding the minimum point cannot be guaranteed. An error surface can be created as the N+1th dimension of a surface composed of the values of the N weights and thresholds of the network (i.e. the free parameters of the model). For any possible configuration of weights, the error can be plotted in the N+1th dimension, forming an error surface. The objective of network training is to find the lowest point on this surface. The global minimum of this error surface cannot, in general, be found analytically; so neural network training is essentially a search of the error surface for minima. From an initially random configuration of weights and thresholds (i.e. a random point on the error surface), the training algorithm incrementally seeks the global minimum. Typically, this is done by calculating the gradient (slope) of the error surface at the current point, and then using that information to make a downhill move. Eventually, the algorithm stops at a low point, which may be a local minimum, or, hopefully, a global one.

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One of the most used search algorithms is back propagation (BP) (see Haykin, 1999; Fausett, 1994), which uses the data to adjust the network’s weights and thresholds so as to minimize the error in its predictions on the training set. In BP, the gradient vector of the error surface is calculated. This vector points along the line of steepest descent from the current point, so moving along it incrementally will decrease the error. The algorithm therefore progresses iteratively through a number of passes through the data. On each pass, the training cases are each submitted in turn to the network, and target and actual outputs compared and the error calculated. This error, together with the error surface gradient, is used to adjust the weights, and then the process repeats. The initial network configuration is random, and training stops when a given number of passes elapses, or when the error reaches an acceptable level, or when the error stops improving. If the network is properly trained, it has then learned to model the (unknown) function, which relates the input variables to the output variables, and can subsequently be used to make predictions where the output is not known. NNW Implementation in this Study. The default settings of the ‘‘Intelligent Problem Solver’’ in STATISTICA Neural Networks were used in this study. This includes linear, radial, and sigmoidal activation functions, a three-layer MLP architecture, and BP. Multivariate Adaptive Regression Splines Like NNW, MARS is a non-parametric technique, which can represent a large variety of linear and nonlinear relationships. Instead of relying on a dense representation of the error function and massive computation, however, MARS relies on a clever method of representing the response functions of the predictor variables. MARS (Friedman, 1991) builds models by fitting piecewise linear regressions. Each piece (spline) is allowed to vary, permitting the representation of practically any shape. Each spline begins and ends at a ‘‘knot.’’ Which variables to represent in this manner and where to set the knots are determined by an intensive search procedure. These splines are combined through devices called ‘‘basis functions,’’ which are similar to principal components. These basis functions continue to be added until no more can be formed profitably, or until some pre-defined maximum number has been reached. In the second stage of MARS modeling, basis functions are deleted based on their contribution to a linear regression fit until the best model is found.

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The MARS model may be represented as: Yi ¼

K X

bj W j ðX i Þ þ
i

j¼1

where Wj(Xi) is the jth basis function of Xi. Note that Y is linear in the parameters, whereas the basis functions can be of practically any shape. Estimates of the parameters bj are obtained through linear regression. MARS Implementation in this Study. The default settings in the user interface in Salford Systems MARS for Windows (Salford Systems, 2001) were utilized. The most important default setting is a maximum of 15 basis functions. Hypotheses The hypotheses to be tested against their negations are:  H1. The methods consistently achieve an adequate degree of accuracy of fit.  H2. The methods consistently select valid predictors. H2a. The methods do not consistently omit valid predictors (underfit). H2b. The methods do not consistently include invalid predictors (overfit).  H3. One of the methods is dominant at each major factor combination.

BACKGROUND Previous Studies Friedman (1991) reports simulation studies of MARS alone, and related work is described by Barron and Xiao (1991), Breiman (1991), and Gu (1991), among others. Friedman examines several criteria; the main ones are scaled versions of mean integrated squared error (MISE), predictive squared error (PSE), and a criterion based on the ratio of a generalized crossvalidation (GCV) error estimate to PSE. Friedman finds that when the data are pure noise in 5 and 10 dimensions, for sample sizes of 50, 100, and 200, MARS produces estimates of adequate accuracy and is unlikely to find spurious structure. When the data are generated from the additive function

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of five variables with both linear and nonlinear components with five additional noise variables and sample sizes of 50, 100, and 200, MARS had a slight but clear tendency to overfit, especially at the smallest sample sizes. Friedman did not compare MARS against other techniques. Breiman (1991a) notes that Friedman’s examples (except for the pure noise case) have high signal-to-noise ratios (SNRs). De Veaux, Psichogios, and Ungar (1993) compared MARS and NNW. They tested the techniques under a variety of circumstances looking to compare speed and accuracy. The authors evaluated the accuracy of the techniques by comparing mean squared prediction errors (SPE). In running the techniques all parameters were left at their default values, since a major attraction of both MARS and NNW is not to have to worry about finetuning. Their findings indicate that NNW tend to overfit data, especially on smaller data sets. MARS has the ability to ‘‘prune’’ the model in order to minimize redundancy and maximize parsimony. MARS was also found to perform with greater speed on serial computers compared to NNW. MARS creates models that are easier to interpret than NNW. This is stated to be important so as to enable the user to interpret the underlying function, which is the first step in discovering the structure of the system. NNW are not able to provide this function. MARS was found to be not as robust as NNW. Tests removing a single data point caused MARS to generate considerably different final models; this was not the case with NNW. They also found that with correlated and noisy data MARS and NNW perform equally well. However, for low-order interactions MARS outperforms NNW. Banks et al. (1999) compared the performance of many different regression techniques including MARS, NNW, stepwise linear regression, and additive models. They created many datasets each having a different embedded structure; the accuracy of each technique was determined by its ability to correctly identify the structure of each dataset, averaged over all datasets, measured by the MISE. In relation to this paper, it is only important to discuss the differences found between MARS and NNW. MARS outperformed NNW in a variety of tests including linear functions, sets where all variables were spurious, Gaussian functions, small dimensions with correlated Gaussian functions, mixture functions, and product functions. As a result they concluded that NNW are unreliable because, although capable of doing well, they usually have a very large MISE compared to other techniques. MARS is less capable in higher dimensions, but overall performs admirably. MARS rarely has a large MISE compared to other techniques, but also rarely performs the best of any technique.

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Kolarik and Rudorfer (1994) evaluated an NNW approach to the prediction of exchange rates, with mixed results, which did not improve upon straightforward ARIMA models. Zhang, Patuwo, and Hu (2001) evaluated the ability of neural network models in time-series applications using pseudo-data, and found NNW superior to traditional time-series techniques, although difficult to calibrate. In an article of April 2001, Sephton (2001) considered two questions. First, how well does MARS fit historical data, that is how well can MARS predict a recession at time t using information available at time (tk)? Second, how well can MARS predict future recessions? The traditional way to predict recessions is using a probit model. Sephton found that MARS probability estimates are superior to the probit estimates with a root-mean-squared-error of 16.7% for MARS and 28.9% for probit. Recessions were predicted at three-, six-, nine-, and twelve-month horizons. MARS had its lowest root-mean-squared-error at the three-month horizon, and it is highest at the twelve-month horizon at about 24%. At all horizons MARS was superior to the probit model. Sephton argues that this is not to be unexpected since nonlinear non-parametric models excel at explaining relationships in sample. The real question is whether or not MARS can excel at explaining relationships out-of-sample. In this arena Sephton found that MARS specification does not perform as well as the probit model, with root-mean-squared-errors around 30%. Although we should note that the probit model did not vastly outperform MARS using out-of-sample relationships, suggesting there is value in using MARS in place of or in conjunction with the traditional probit model.

ANALYSIS Study Protocol This study consisted of comparing the modeling capabilities of three methods. The evaluation is done through a 32
2 factorial experiment with replication, and a protocol intended to remove as much subjectivity as possible from the model-building:  Ten samples of size n ¼ 1,000 are generated using functions F(x) ¼ G(x)+e. The functions vary in the form of G(x). e is a noise term and is Gaussian in all cases. G(x) is of dimension 5; the generated data set also includes three nuisance variables. Functional forms are: (i) linear in all

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dimensions, (ii) convex in one or more dimensions and concave in one or more dimensions, and (iii) stepped in one or more dimensions. In addition, each data set is modeled without and with 5% contamination (to represent errors of measurement and transcription) in one or more variables. An optimal Box–Cox transformation is performed for estimation by FSWR in nonlinear continuous cases. The estimation is performed in automated mode with software default options using nk ¼ 750 data values (training set). The estimated function is examined, and the degree (number of parameters) of under- and overfitting is recorded. The estimated function is used to predict the remaining k ¼ 250 data values (verification set) The prediction MSE and the inclusion or exclusion of valid and nuisance variables are noted and recorded. A factorial analysis of variance is performed to test for overall accuracy of fit, for accuracy of specification, to rank the techniques, and to assess the degree of interactions between method and data characteristics.

To test the stated hypotheses, we conducted a 32
2 factorial experiment, the factors of which were method, function, and degree of contamination. The levels of the factors and their descriptions are: Methods. The methods compared were FSWR, NNW, and MARS as described previously. Function. Three types of functions were modeled: linear and two types of nonlinear.  Linear: Y i ¼ F ðX Þ ¼ bo þ

K X

bj ðX j Þ þ
i

j¼1

 Nonlinear continuous: Yi ¼ F ðX Þ ¼ 0:1Expð0:5X 1Þ þ 1=½Expð0:2X 2Þ  3X 3 þ 5X 4 þ
This is of the type suggested by Friedman and Silverman (1989) for testing recursive partitioning techniques such as MARS.

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 Nonlinear discontinuous (step): Linear functions with discrete steps at several intervals for two significant predictors. Y i ¼ F ðX Þ

¼

bo þ

K P

bj ðX j Þ þ
i

if X u  S 1 and X v  S 3

j¼1

Y 0i ¼ F ðX Þ Y 00i ¼ F ðX Þ

¼ ¼

Y i þ bu ðX u þ @1 Þ Y i þ bu ðX u þ @1 þ @2 Þ

if X u 4S1 and X v  S3 if X u 4S2 and X v  S3

Y i ¼ F ðX Þ ¼

Y i þ bu ðX u þ @1 þ @2 Þ þbv ðX v þ @2 Þ

if X u 4S2 and X v 4S 3

000

where K ¼ 5 in all cases. Contamination. Uniformly distributed 0–10 random noise was included in 5% of the cases. Contamination obscures correlations.

Data Development The primary (input) data were developed as follows: 1. 2. 3. 4.

Specify the parameters of each function Generate values of the relevant prediction (independent) variables Generate values of the noise factor xN(0,s) with s ¼ 100 Generate values of non-relevant (nuisance) independent variables (three in all cases) 5. Compute values of the response (dependent) variable 6. Repeat r ¼ 10 times for each combination.

The 10 samples of each function were generated with and without contamination using the random number and variable modification functions in STATISTICA, resulting in 3
2 ¼ 6 sets of data with 10 variables (five relevant predictors, three nuisance variables, one noise variable, and one dependent variable) and 1,000 records each. To develop the data for the comparisons, the parameters of the generated functions were estimated with each of the three methods, resulting in 3
60 ¼ 180 estimation results, from which the following were extracted:  The variance of the error (lack of fit) term: MSE in FSWR, the ‘‘Verification’’ MSE in NNW, and the GCV for MARS. To remove the effects of scale and relative variation (SNR), these values were normalized by dividing by the input noise factor. This results in a set of values, which

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measure the proportion of the mean irreducible error resulting from an estimate (PMSE – see below).  The error variances, along with the appropriate degrees of freedom, were also used to calculate an adjusted R2.  The degree of underfit: the number of relevant predictors not included in the final estimate.  The degree of overfit: the number of non-relevant (nuisance) variables included in the final estimate. The tables in the appendix summarize the resulting data set with 180 records, along with the relevant input and response variances, SNRs, and the proportions of the variance of the response variable Y accounted for by the signal (the input function F(X).

Results Accuracy Assessment of the accuracy requires a perspective regarding the variability characteristics of the data sets being modeled. The salient characteristics are the SNR and the proportion of variability associated with the signal. The SNR is the ratio of the variability due to the structure of the relationship between input (independent) and response (dependent) variables inherent in the function F(X) and the variability of the input noise variable e. The proportion of variability associated with the signal (in effect a maximum R2) is the ratio of the variability of the signal relative to the total variability of the response. Thus: Let Y ¼ F(X)+e ¼ response (independent) variable; S ¼ F(X) ¼ signal (stimulus, or function of independent variables); N ¼ e ¼ noise (random disturbances). Total variance: s2y ¼ s2S þ s2N : Variance due to structure of relationship (signal): S ¼ s2y  s2N : SNR: S=N ¼ ðs2y  s2N Þ=s2N ¼ ðs2y =s2N Þ  1: Further, let R2m ¼ proportion of total variance due to signal (thus maximum R2). R2m ¼ s2S =s2y ¼ ðs2y  s2N Þ=s2y ¼ 1  ðs2N =s2y Þ For the data generated in this study: Linear: SNR 2.92. R2m 77.50%. This is a low SNR, thus a low R2m. A ‘‘perfect’’ estimate would have an adjusted R2 of 0.775. Thus, while the

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structure is simple, the data is relatively noisy, partially obscuring the signal. It challenges a technique to extract a relatively weak signal. Nonlinear continuous: SNR 20.97. R2m 95.45%. In this case, the structure is complex, but the noise is relatively low. It challenges a technique to extract a complex (mixed nonlinear, multivariate, additive) structure. Nonlinear discontinuous: SNR 7.77. R2m 88.60%. A medium level of both structure complexity (with the added feature of discontinuity) and noise level. It challenges a technique to extract a discrete (stepped) structure within a moderate amount of noise. The accuracy of fit is evaluated on the basis of two related measures:  R2 adjusted for degrees of freedom in the customary manner.  PMSE (relative or proportional MSE) indicates the multiple of the irreducible (input) error associated with an estimate. A PMSE value of 1 thus indicates a ‘‘perfect’’ estimate. The reciprocal of PMSE measures the proportion of the signal accounted for by the estimate. The most prominent fact in Table 1 for R2 is that even the smallest value (0.535 for the FSWR–NLINA estimate) is well beyond the magnitude required for statistical significance at any level (degrees of freedom for the associated F test range between [4,995] and [8,991]). Furthermore, that smallest value indicates that the estimated equation accounts for more than 50% of the variance of the observed response. Also, the smallest R2 value for the techniques of interest (0.744 for MARS–LIN) indicates an estimate, which accounts for almost 3/4th of the observed response variance. This is all the more remarkable relative to the SNR. In all cases, at least one of the methods achieves a near-perfect fit, with an R2 within two percentage points of the maximum possible. Overall, the techniques achieve linear fits less than 3 percentage points, nonlinear continuous fits less than 16 percentage points, and nonlinear discontinuous fits less than 4 percentage points from the maximum. That means that for each function the techniques extracted 97%, 83%, and 95% of the available information relative to variability, respectively. There are, of course, differences between methods and across factors (function type and contamination) – an aspect discussed below – but overall, hypothesis H1 is amply supported. Of additional interest is the degree of consistency across samples indicated by the standard deviations of the PMSE values in Table 2. Overall, the coefficient of variation of the PMSE estimates ranges between 4% and 18%

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and averages about 10%, with the most consistent being FSWR. This is of particular interest because of the reported susceptibility of both MARS and NNW to respond to random anomalies, thus getting ‘‘trapped’’ in local minima (see Friedman, 1991; De Veaux et al., 1993). That susceptibility is evident in the differences in PMSE for both MARS and NNW between contaminated and uncontaminated nonlinear continuous estimates. For such mild contamination (uniform 0–10 random values in 5% of the cases) to result in over 100% differences in PMSE is very surprising. In contrast, the FSWR PMSEs for the same cases differ by less than 9%. Underfitting The degree to which the techniques fail to detect valid predictors is small overall, with the important exception of NNW. Whereas neither MARS nor FSWR failed to detect a valid predictor even once for any of the cases, NNW missed an average of 0.5 valid predictors overall, and did so quite inconsistently (coefficient of variation 1.3), ranging from 0 in linear fits to 2 in fitting the NLINA C data. Clearly, hypothesis H2a is amply supported for MARS and FSWR and rejected for NNW. Overfitting The results for overfitting shown below are less clear. The number of spurious variables included by NNW averages over 100% greater overall than that of either of the other techniques, but no technique fails to exclude invalid predictors on the average. Furthermore, and perhaps of more concern, all of the techniques show inconsistency, with coefficients of variation averaging 1.33 and ranging up to as high as 2.1. The fact that MARS and FSWR perform well (no spurious inclusions) in several cases mitigates but does not invalidate the general conclusion that hypothesis H2b is not satisfactorily supported. This is consistent with previous findings regarding both MARS and NNW (see Banks et al., 1999; Barron, & Xiao 1991; Breiman, 1991; Friedman, 1991; De Veaux et al., 1993). Factors of Influence The substantial differences in the relevant measures across cases are evident in Tables 1–4. To evaluate the significance of these differences and to identify relevant subsets and dominant techniques, a factorial analysis of variance was performed for each indicator of fit. Overall. For accuracy of fit (Table 5), the type of function modeled has the largest effect, but all main effects and interactions are highly significant. For

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Table 1.

R2 Means.

Method/Data LIN C LIN U NLINA C NLINA U NLINB C NLINB U All Data MARS NNW FSWR All methods

0.744 0.760 0.749 0.751

0.744 0.760 0.750 0.751

0.884 0.873 0.535 0.764

0.951 0.946 0.555 0.818

0.843 0.808 0.877 0.843

0.850 0.809 0.877 0.845

0.836 0.826 0.734 0.795

Abbrev.: MARS, multivariate regression splines estimates; NNW, neural network estimates; FSWR, forward stepwise regression (OLS); LIN, linear function; NLINA, nonlinear continuous function; NLINB, nonlinear discontinuous function; C/U, contaminated/uncontaminated data.

Table 2.

PMSE Means/(SE).

Method/Data LIN C LIN U NLINA C NLINA U NLINB C NLINB U All Data MARS NNW FSWR All methods

1.02 (0.02) 1.01 (0.02) 1.00 (0.01) 1.01 (0.02)

1.02 (0.01) 1.00 (0.02) 1.00 (0.01) 1.01 (0.02)

2.53 (0.14) 2.77 (0.15) 3.44 (0.14) 2.91 (0.12)

1.06 (0.01) 1.17 (0.04) 3.78 (0.07) 2.00 (0.03)

1.32 (0.01) 1.62 (0.06) 1.04 (0.01) 1.33 (0.04)

1.26 (0.01) 1.61 (0.05) 1.04 (0.01) 1.30 (0.03)

1.37 (0.07) 1.53 (0.07) 1.88 (0.16) 1.59 (0.11)

See abbrev. in Table 1.

Table 3.

Underfitting Means/(SD).

Method/Data LIN C LIN U NLINA C NLINA U NLINB C NLINB U All Data MARS NNW FSWR All methods

0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

See abbrev. in Table 1.

0.00 (0.00) 0.10 (0.32) 0.00 (0.00) 0.03 (0.18)

0.00 (0.00) 2.00 (1.15) 0.00 (0.00) 0.67 (0.67)

0.00 (0.00) 0.60 (0.97) 0.00 (0.00) 0.20 (0.56)

0.00 (0.00) 0.10 (0.32) 0.00 (0.00) 0.03 (0.18)

0.00 (0.00) 0.20 (0.42) 0.00 (0.00) 0.07 (0.24)

0.00 (0.00) 0.50 (0.66) 0.00 (0.00) 0.17 (0.38)

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Overfitting Means/(SD).

Table 4. Method

LIN C

LIN U

NLINA C

NLINA U

NLINB C

NLINB U

All Data

MARS

0.60 (0.97) 1.40 (0.70) 0.40 (0.70) 0.80 (0.80)

0.70 (1.06) 0.70 (0.67) 1.30 (0.95) 0.90 (0.91)

0.20 (0.42) 0.60 (0.97) 0.20 (0.42) 0.33 (0.66)

0.00 (0.00) 1.30 (1.25) 0.50 (0.53) 0.60 (0.78)

0.00 (0.00) 0.70 (0.82) 0.00 (0.00) 0.23 (0.48)

0.00 (0.00) 1.10 (1.10) 0.00 (0.00) 0.37 (0.64)

0.25 (0.61) 0.97 (0.94) 0.40 (0.55) 0.54 (0.72)

NNW FSWR All methods

See abbrev. in Table 1.

Table 5. Effect Method Function Contamination Method–function Method–contamination Function–contamination All Error

ANOVA of Accuracy of Fit (PMSE). SS

df Effect

MS Effect

F

p

8.67 75.57 4.46 35.74 4.08 8.27 7.84 5.13

2 2 1 4 2 2 4 162

4.34 37.78 4.46 8.94 2.04 4.13 1.96 0.03

37.05 1194.13 140.87 282.41 64.43 130.67 61.98

0.000 0.000 0.000 0.000 0.000 0.000 0.000

Significant effects (a ¼ 0.05). Levene’s test statistic for homogeneity of variance: 0.116, F ¼ 12.18, p ¼ 0.00.

our immediate purposes, the findings of particular interest are that the methods differ significantly in accuracy overall, and that the effect is magnified by the type of function being estimated and the presence of contamination. It is evident from Table 2 that Levene’s test for homogeneity of variance– covariance fails due to the large differences in variability of the nonlinear continuous contaminated cases for MARS and NNW. This of course distorts the level of significance of the overall findings somewhat, but the disparate cases are only 2 out of 12, so the distortion of the aggregate variance–covariance structure should be small, and thus the F values of the above effects should not be affected to a great extent. Underfit. Table 6 shows that the results for the number of significant predictors omitted parallel those for PMSE. All of the main effects and interactions are highly significant, with the size of effect in the same order as

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Table 6. Effect Method Function Contamination Meth–function Method–contamination Function–contamination All Error

ANOVA of Underfit.

SS

df Effect

MS Effect

F

p

10.00 6.43 0.80 12.87 1.60 2.50 5.00 23.80

2 2 1 4 2 2 4 162

5.00 3.22 0.80 3.22 0.80 1.25 1.25 0.15

34.03 21.90 5.45 21.89 5.45 8.51 8.51

0.000 0.000 0.021 0.000 0.005 0.000 0.000

Significant effects (a ¼ 0.05). Levene’s test statistic for homogeneity of variance: 0.819, F ¼ 13.07, p ¼ 0.00.

Table 7. Effect Method Function Error

ANOVA of Overfit.

SS

df Effect

MS Effect

F

p

17.14 9.54 84.70

2 2 162

8.57 4.77 0.52

16.40 9.13

0.000 0.000

Significant effects (a ¼ 0.05). Levene’s test statistic for homogeneity of variance: 0.996, F ¼ 5.57, p ¼ 0.00.

before, led by the type of function modeled. Again, the relevant finding for our immediate purposes is that the methods differ significantly in the degree of underfit (as can be seen from Table 3, all of the difference is contributed by NNW), and that the effect is magnified by the function being estimated and the presence of contamination. Levene’s test for homogeneity of variance (value is given below Table 6) is not quite appropriate here, since all of the deviations in underfit are for NNW. The question is whether the results for NNW differ significantly from the others in these dimensions. Since the others have means and variances of zero in this measure in all dimensions, the measure is a count and thus non-negative, and the measure for NNW has positive values in all but one case, it is clear from inspection that the mean value for NNW is not zero, that it is affected by both function and contamination, and it is thus significantly different from MARS and FSWR in the degree of underfit. Overfit. The results for overfit (Table 7) are markedly different. Only two effects are significant, and there are no significant interactions. However, the

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two significant effects are of particular interest for our immediate purposes. So the degree of overfit differs significantly by method, and that effect is magnified by the type of function being estimated. An examination of Table 4 reveals that the extreme outliers which contribute heavily to Levene’s statistic are values of zero for MARS and FSWR in the nonlinear discontinuous cases. Again, this lack of homogeneity distorts the levels of significance, but the existence of patterns in this measure is not of great importance for our purposes, since the conclusion that all the methods are unreliable in terms of overfitting has already been made. Methods Comparison: Contrasts The results in this section compare the methods one-to-one on the same terms as the overall analysis. These results are exploratory, since the primary intention was to test the main hypotheses stated previously. Nevertheless, the results can be used to suggest the type and degree of superiority of particular techniques with respect to the three measures of interest. PMSE. MARS outperforms both NNW and FSWR in overall accuracy of fit. In linear fits, NNW dominates, but that is offset by large and highly significant differences in performance for nonlinear continuous fits and with contamination overall. Surprisingly, FSWR dominates the nonlinear discontinuous fits (Tables 8–11). The results in Tables 12 and 13 indicate that data contamination affects MARS estimates less than those of the other techniques. Underfit. MARS and FSWR dominate in the degree of underfit overall and in all cases, with no occurrences of underfitting (Tables 14–19). Again, large and highly significant differences in the relative degree of underfit of MARS and FSWR over NNW, particularly in the nonlinear continuous and contaminated cases, contribute strongly to the highly significant overall difference. Table 8. Contrast FSWR–MARS NNW–MARS See abbrev. in Table 1.

PMSE Contrasts – All Factors.

Difference

Std. Error

t

p

0.521 0.147

0.032 0.032

16.053 4.520

0.000 0.000

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Table 9. PMSE Contrasts – Linear Fits. Contrast FSWR–MARS NNW–MARS

Difference

Std. Error

t

p

0.051 0.118

0.343 0.343

0.149 0.344

0.882 0.732

See abbrev. in Table 1.

PMSE Contrasts – Nonlinear Continuous Fits.

Table 10. Contrast FSWR–MARS NNW–MARS

Difference

Std. Error

t

p

1.838 0.179

0.091 0.091

20.268 1.979

0.000 0.053

See abbrev. in Table 1.

Table 11.

PMSE Contrasts – Nonlinear Discontinuous Fits.

Contrast FSWR–MARS NNW–MARS

Difference

Std. Error

0.255 0.320

0.032 0.032

t

p

7.900 9.900

0.000 0.000

See abbrev. in Table 1.

Table 12. Contrast FSWR–MARS NNW–MARS

PMSE Contrasts – Uncontaminated Data. Difference

Std. Error

t

p

0.834 0.134

0.023 0.023

36.735 5.919

0.000 0.000

See abbrev. in Table 1.

Table 13. Contrast FSWR–MARS NNW–MARS See abbrev. in Table 1.

PMSE Contrasts – Contaminated Data. Difference

Std. Error

t

p

0.208 0.159

0.061 0.061

3.425 2.616

0.000 0.011

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Table 14. Contrast FSWR–MARS NNW–MARS

Underfit Contrasts – All Factors.

Difference

Std. Error

t

p

0.000 0.500

0.070 0.070

0.000 7.145

1.000 0.000

See abbrev. in Table 1.

Table 15. Contrast FSWR–MARS NNW–MARS

Underfit Contrasts – Linear Fits.

Difference

Std. Error

t

p

0.000 0.100

0.272 0.272

0.000 0.368

1.000 0.713

See abbrev. in Table 1.

Underfit Contrasts – Nonlinear Continuous Fits.

Table 16. Contrast FSWR–MARS NNW–MARS

Difference

Std. Error

t

p

0.000 1.300

0.194 0.194

0.000 6.688

1.000 0.000

See abbrev. in Table 1.

Table 17.

Underfit Contrasts – Nonlinear Discontinuous Fits.

Contrast FSWR–MARS NNW–MARS

Difference

Std. Error

t

p

0.000 0.150

0.068 0.068

0.000 2.205

1.000 0.032

See abbrev. in Table 1.

Table 18. Contrast FSWR–MARS NNW–MARS See abbrev. in Table 1.

Underfit Contrasts – Uncontaminated Data. Difference

Std. Error

t

p

0.000 0.300

0.095 0.095

0.000 3.167

1.000 0.002

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Table 19. Contrast FSWR–MARS NNW–MARS

Underfit Contrasts – Contaminated Data. Difference

Std. Error

t

p

0.000 0.700

0.103 0.103

0.000 6.793

1.000 0.000

See abbrev. in Table 1.

Table 20. Contrast FSWR–MARS NNW–MARS

Overfit Contrasts – All Factors.

Difference

Std. Error

t

p

0.150 0.717

0.132 0.132

1.136 5.429

0.258 0.000

See abbrev. in Table 1.

Table 21. Contrast FSWR–MARS NNW–MARS

Overfit Contrasts – Linear Fits.

Difference

Std. Error

t

p

0.200 0.400

0.270 0.270

0.739 1.479

0.462 0.145

See abbrev. in Table 1.

Table 22. Contrast FSWR–MARS NNW–MARS

Overfit Contrasts – Nonlinear Continuous Fits. Difference

Std. Error

t

p

0.250 0.850

0.229 0.229

1.094 3.372

0.279 0.000

See abbrev. in Table 1.

Table 23. Overfit Contrasts – Nonlinear Discontinuous Fits. Contrast FSWR–MARS NNW–MARS See abbrev. in Table 1.

Difference

Std. Error

t

p

0.000 0.900

0.177 0.177

0.000 5.072

1.000 0.000

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Overfit Contrasts – Uncontaminated Data.

Table 24. Contrast

Difference

Std. Error

t

p

0.367 0.800

0.202 0.202

1.811 3.952

0.074 0.000

FSWR–MARS NNW–MARS See abbrev. in Table 1.

Overfit Contrasts – Contaminated Data.

Table 25. Contrast

Difference

Std. Error

t

p

0.067 0.633

0.169 0.169

0.393 3.737

0.695 0.000

FSWR–MARS NNW–MARS See abbrev. in Table 1.

Table 26. Measure

Methods Comparison Summary Best Performance.

Overall

Linear

NLINA

NLINB

C

U

C

U

C

U

C

U

PMSE

MARS

MARS

MARS

FSWR

FSWR

MARS FSWR MARS FSWR

MARS FSWR MARS FSWR

NNW FSWR MARS FSWR None

MARS

Underfit

NNW FSWR MARS FSWR None

MARS FSWR MARS FSWR

MARS FSWR MARS FSWR

MARS FSWR MARS FSWR

MARS FSWR MARS FSWR

Overfit

See abbrev. in Table 1.

Overfit. Much as in overfitting, MARS and FSWR dominate overall, with particular dominance in nonlinear fits (Tables 20–26). No technique dominates overfitting in the linear cases, and MARS and FSWR are essentially equivalent in terms of overfitting with both contaminated and uncontaminated data.

CONCLUSIONS The results generally agree only in part with the results of other studies. MARS outperformed the others in accuracy of fit, and did as well as any in

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under- and overfitting. Surprisingly, NNW did not lead any category but linear fits in performance, tied with FSWR. Also surprisingly, FWSR performed consistently well overall, and tied with MARS for specification honors. In summary, it can be said that:  All the methods used can generate estimates with adequate to high degrees of accuracy of fit across a variety of conditions. The results for MARS and NNW indicate that these techniques have a remarkable capability for highaccuracy curve fitting, even for complex, mixed linear/nonlinear curves.  Even mild data contamination can have significant effects on all aspects of the quality of estimates.  NNW has a marked tendency to omit significant predictors, even in lowdimensional situations such as those in this study (dimensionality of 9, with one response variable, five valid predictors, and three nuisance variables). MARS and FSWR can be relied upon to detect significant predictors in a variety of situations.  None of the techniques can be relied upon to exclude spurious variables, even in simple, low-dimensionality situations. The implications from a modeling standpoint are that all of the techniques can be relied upon for adequate predictions, but none of them can be consistently relied upon for accurate specification (both inclusion and exclusion). Thus, if identification of the structure of relationships is of particular importance, none of the techniques can be applied with confidence. MARS and FSWR are reliable significance detectors, however.  In addition to its undistinguished performance in this study, the ‘‘black box’’ nature of NNW would also be an impediment to good model development. Although included predictors are identified and the network structure is provided, no coefficients are directly given, and of course no means of confirmation of significance are provided.  MARS would be the preferred technique for this type of application. It is an accurate, flexible, and reliable modeling tool (with the safeguards mentioned above), it can be automated, and it provides readily understandable and useful output. Although the overall structure of the resulting model cannot be confirmed, the constructed basis functions are testable statistical constructs akin to principal components, and significance statistics are provided for these basis functions. Another aspect of interest, although not formally studied nor discussed, is that NNW computation takes a considerably longer time (10 or more times as long on a 1.8 GHz machine) than either of the others, even in these lowdimension cases.

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Although this study investigates the behavior of these techniques under variations in the fundamental and important aspects of functional form and data contamination, clearly the study is far from definitive. Other functional forms, smaller SNRs, and greater contamination need to be considered. Furthermore, other potentially important factors such as sparsity (the proportion of significant predictors), dimensionality (number of variables), type of variable measure (nominal, ordinal, ratio), type of noise (Gaussian vs. not), multicolinearity, serial correlation (time-series and cross-sectional time-series data), and sample size should also be evaluated.

REFERENCES Banks, D. L., Olszewski, R. T., & Maxion, R. A. (1999). Comparing methods for multivariate nonparametric regression. CMU-CS-99-102 School of Computer Science. Pittsburgh, PA: Carnegie Mellon University. Barron, A. R., & Xiao, X. (1991). Discussion of multivariate adaptive regression splines, by J.H. Friedman. Annals of Statistics, 19, 67–81. Breiman, L. (1991). Discussion of multivariate adaptive regression splines, by J.H. Friedman. Annals of Statistics, 19, 82–90. Chan, C., & Lewis, B. (2002). A basic primer on data mining. Information Systems Management, 19(4), 56–60. Chopoorian, J. A., Witherell, R., Khalil, O. E. M., & Ahmed, M. (2001). Mind your business by mining your data. S.A.M. Advanced Management Journal, 66(2), 45–51. De Veaux, R. D., Psichogios, D. C., & Ungar, L. H. (1993). A comparison of two nonparametric estimation schemes: MARS and neural networks. Computers and Chemical Engineering, 17(8), 819–837. Etheridge, H. L., Sriram, R. S., & Kathy Hsu, H. Y. (2000). A comparison of selected neural networks that help auditors evaluate client financial viability. Decision Sciences, 11(2), 531–549. Fausett, L. (1994). Fundamentals of neural networks. New York: Prentice-Hall. Fayyad, U. M., & Stolorz, P. (1997). Data mining and KDD: Promise and challenges. Future Generation Computer Systems, 13, 99–115. Friedman, J. H. (1991). Multivariate adaptive regression splines. Annals of Statistics, 19, 1–66. Friedman, J. H., & Silverman, B. W. (1989). Flexible parsimonious modeling. Technometrics, 31, 3–39. Gu, C. (1991). Discussion of the P method for estimating multivariate functions from noisy data, by L. Breiman. Technometrics, 33, 149–154. Hand, D., Mannila, H., & Smyth, P. (2001). Principles of data mining. Cambridge, MA: Bradford Books, MIT Press. Haykin, S. (1999). Neural networks (2nd ed.). New York: Prentice-Hall. Hocking, R. R. (1996). Methods and applications of linear models: Regression and analysis of variance. New York: Wiley.

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Kolarik, T., & Rudorfer, G. (1994). Time series forecasting using neural networks. Proceedings of the international conference on APL 1994, pp. 86–94. Rumelhart, D. E., & McClelland, J. (Eds) (1986). Parallel distributed processing, Vol. 1. Cambridge, MA: MIT Press. Salford Systems (MARS 2001) http://www/salford-systems.com. Sephton, P. (2001). Forecasting recessions: Can we do better on MARS? Review – Federal Reserve Bank of St. Louis, 83, 39–49. Shaw, M. J., Subramaniam, C., Woo Tan, G., & Welge, M. (2001). Knowledge management and data mining for marketing. Decision Support Systems, 31, 127–137. Smyth, P. (2001). Data mining at the interface of computer science and statistics. In: R. L. Grossman, K. Chandrika, K. Philip, K. Vipin & R. N. Raju (Eds), Data mining for scientific and engineering applications. Dordrecht: Kluwer Academic Publishers (Chapter 1). Statsoft, Inc. (STATISTICA 2003). http://www/statsoft.com. Two Crows Corporation. (1999). Introduction to data mining and knowledge discovery (3rd ed.). Potomac, MD: Two Crows Publisher, Two Crows, Inc Zhang, G., Patuwo, E., & Hu, M. (2001). A simulation study of artificial neural networks for nonlinear time series forecasting. Computer and OR, 28, 381–396. Ziv, B.-Y., & Rajagopalan, S. (2002). Template detection via data mining and its applications. Proceedings of WWW 2000, 580–591, Honolulu, HI: ACM Press.

APPENDIX: DATA DESCRIPTION Linear Equation: Y ¼ 230+3.4*X1+8*X212*X3+5*X42.2*X5+e eN(0,100) Sample size 1,000. Descriptive statistics: Variable X1a X2a X3a X4a X5a X6 X7 X8 Y a

Valid N

Mean

Minimum

Maximum

Std. Dev.

1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000

9.979 15.054 20.405 24.888 30.157 12.337 14.868 17.521 195.840

0.021 0.020 0.034 0.024 0.018 0.001 0.006 0.026 –311.329

19.966 29.972 39.971 49.976 59.998 24.989 30.000 35.000 803.343

5.869 8.790 11.824 14.607 17.489 7.315 8.737 10.286 198.045

X1–X5 are valid predictors; X6–X8 are nuisance variables.

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SNR 2.92. Signal accounts for 74.5% of the variance of response Y. Correlations: Variable

X1

X2

X3

X4

X5

X6

X7

X1 X2 X3 X4 X5 X6 X7 X8

1.00 0.03 0.01 0.04 0.00 0.05 0.02 1.00 0.02 0.05 0.03 0.03 0.01 1.00 0.01 0.02 0.01 0.05 1.00 0.02 0.00 0.01 1.00 0.02 0.02 1.00 0.02 1.00

X8

Y

0.01 0.08 0.02 0.31 0.03 0.69 0.02 0.34 0.04 0.17 0.03 0.03 0.03 0.03 1.00 0.01

 Significant at 0.05 level.

Non-linear continuous Equation: Y ¼ 0.1Exp(0.5X1)+1/[Exp(0.2X2)]–3X3+5X4+e eN(0,100). Sample size 1,000. Descriptive statistics: Valid N X1 X2 X3 X4 X5 X6 X7 X8 Y

1000 1000 1000 1000 1000 1000 1000 1000 1000

Mean

Minimum

Maximum

Std. Dev.

10.104 15.225 19.039 24.291 29.489 12.662 15.237 17.513 303.002

0.037 0.007 0.009 0.150 0.037 0.024 0.115 0.006 361.173

19.976 29.975 39.938 49.989 59.910 24.956 29.981 34.890 2194.868

5.899 8.649 11.401 14.136 17.151 7.094 8.809 10.114 468.714

SNR 20.97. Signal accounts for 95.45% of the variance of response Y. Correlations as before. Non-linear discontinuous Equation: Y ¼ 230 þ 6  ðX 1 þ X 1g6 þ X 1g15Þ þ 8  X 2  12  ðX 3 þ X 3jpÞ þ5  X 4  2:2  X 5 þ

 Nð0; 100Þ.

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where X1g6, X1g15, are discrete steps of size 20 at values of X1 greater than 6 and greater than 15, respectively, and X3jp is a discrete step of size 20 at values of X3 between 20 and 30. Variance of Y 83,736. SNR 7.77. Signal accounts for 88.6% of the variance of response Y. The values and correlations of X1–X8 and Y are as above. Contamination Contamination is a uniformly distributed 0–10 random value added to one variable in 5% of the cases to represent mechanical transcription errors. Contamination affects correlations but not the variance of Y and thus not the SNR.

SELECTING FORECASTING INTERVALS TO INCREASE USEFULNESS AND ACCURACY Michael D. Geurts ABSTRACT This paper examines how the changing of the forecasting time frame can improve the accuracy and usefulness of a forecast.

INTRODUCTION The general practice in forecasting has been to forecast commonly used time frames such as month or year forecasts. These time frames are artificial confines (boundaries) that do not necessarily correspond to the time frame that the business is dealing with. For example, Christmas sales encompass part of November, December, and part of January with after Christmas sales and returns. It may be much more useful to forecast ‘‘sales event’’ sales rather than weekly or monthly sales. Buyers for department stores are more interested in knowing what the sales volume will be for the Easter season than they are for March and April sales. By forecasting event sales, the forecaster is often providing more useful information. Also, in many cases the forecast will be more accurate.

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ARTIFICIAL BOUNDARIES In recent years forecasters have concluded that national borders are not a good way to look at markets. The Mexican market does not stop at the USA border, but the southwestern part of the United States is a very important part of the Mexican market. Similarly, it is important for forecasters not to limit their forecasts to traditional time frames or product categories. In some situations a forecast of total company sales may be more accurate than the sum of the product line forecasts. The corporate sales forecast broken down by the percent of total sales that each product line contributes to total sales may be a more accurate way of forecasting product line sales than forecasting the product lines themselves. In a supply chain forecasting situation it becomes important to know the purchase cycle. Some big customers may not be on a monthly cycle. They may be on a 16-day cycle for most of the year and then go to an 8-day purchase cycle for the Christmas season. For production planning it becomes important to have the forecast correspond to the customer purchase cycle.

COMMODITY PURCHASES Many companies are interested in being able to minimize the cost of commodities they buy. For example, a food manufacturer may want to forecast the quantity of potatoes it needs but also a forecast of the price cycle. This will help them minimize cost.

VARIANCE AND FORECASTING ACCURACY Large variances in sales generally make the forecasting of sales less accurate. If the forecaster can reduce the variance of a time series he can generally increase the accuracy. If he can forecast Christmas sales rather than forecasting November, December, and January sales as separate months, then the accuracy can be increased. November sales can be dramatically affected by when Thanksgiving occurs in the United States because the day after Thanksgiving is the start of the Christmas sales seasons and is often the biggest sales day of the entire Christmas season. Because Thanksgiving is always the fourth Thursday in November, it occurs on different dates each

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year and the Christmas season sales can start as early as the 22nd of November or as late as the 28th of November. For products that have a large part of their sales during the Christmas season, November sales are dramatically impacted by whether Thanksgiving is the 22nd instead of the 28th. By forecasting the Christmas sales cycle instead of specific months the forecast may be more accurate because the variance caused by the varying date of Thanksgiving is reduced. Also, Christmas sales can be shifted because of weather. Good late November weather can shift sales from December to November. Shoppers ‘‘take advantage’’ of the good weather to do their shopping hoping to avoid the stormy weather that may occur in December. Generally, Christmas sales are not increased because of an earlier start to the Christmas season. The result is a more accurate forecast when the event, ‘‘Christmas’’ sales, is forecasted rather than monthly sales. Also, Christmas sales may be much more useful as a marketing management tool than monthly sales. Easter and Lent seasons are like Christmas because the event starts at different times and cause a variance from year to year in monthly sales. Athletic events such as The Staley Cup, the NBA playoffs, the Super Bowl, and the World Series occur in different cities from year to year and have a dramatic impact on the sales in the visitor industry for the city in which they occur. Often these events cause forecasters to generate inaccurate forecasts which ignore the impact of the event on sales. A problem encountered in forecasting tourists going to Hawaii is that for some periods (time frames) the occupancy rates at hotels are nearly full and other times there are low occupancy rates. During the low occupancy times, hotel occupancy measures demand for hotel rooms. During high occupancy times, hotel occupancy measures the supply of hotel rooms because some people who want hotel rooms are not able to obtain them. High occupancy occurs during the last two weeks of December and the first week of January. It may be more useful for visitor industry companies to have a forecast for periods of high occupancy and periods of low occupancy.

ACCURACY DETERMINED BY SELECTED INTERVAL When a forecaster is asked to forecast the next year’s sales, he could choose to use past months sales and forecast the next 12 months. Or he could forecast next year using past yearly sales. Generally, the more accurate approach is to use recent months rather than prior years. If he forecasts the next year’s sales by using past months data and then forecasting the next

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12 months sales and totaling them, the forecasting accuracy is aided by two factors. First, the monthly data will be more recent data. Next month is more likely to be like this month than the same month 35 years ago. Second, by forecasting monthly data and adding the months, some months will be over forecasted and some months will be under forecasted. When they are added up there is some canceling out of high and low forecasted months and a more accurate forecast is obtained. When asked to forecast product sales, it is sometimes more accurate to forecast total sales and then let managers decide what percent of total sales each product captures. Generally, total sales are forecasted with more accuracy than individual products. This is partly from cannibalization of sales from an old product’s sales by a new product. When asked to forecast total sales, a forecaster needs to look at whether aggregating sales forecasts to get to the total is more accurate than forecasting the total using past total sales data set. When asked to forecast individual product sales a forecaster needs to evaluate whether forecasting the total of all products sales, and then applying an appropriate percent of the total for each product will be more accurate than forecasting the individual products. When each of the individual products is forecasted best by a different model, the most accurate approach may be to forecast each product with the model that best forecast the individual product and then totaling the individual forecasts. For example, the best way to forecast tourists to Hawaii is to forecast tourist coming from the west and east separately and then combining these two forecasts rather than forecasting the total number of tourists.

CONCLUSION Often more accurate forecasting can be obtained by choosing a time frame to forecast that is different than a calendar period such as months. It is often useful to forecast events like Christmas or World Series sales rather than monthly sales. Also, it is necessary for a forecaster to experiment when forecasting individual units whether it is best to forecast total sales and apply a percent of total for each product or to forecast the individual products.

FORECASTING SIMULTANEOUS BRAND LIFE CYCLE TRAJECTORIES Frenck Waage ABSTRACT This paper develops a tractable method for forecasting competing brands’ market share trajectories over time. Each trajectory is also identically a life cycle time path. The paper thus shows how to predict the time paths of each life cycle complete with their turning points simultaneously for all the brands that compete in a given market. The model is tractable and can be used with ease in business. The leadership of a company will find it a powerful tool with which to manage the life cycles of their brands. The model was developed to meet practical and specific industry planning and forecasting needs. First the paper develops how to estimate consistent historical market share trajectories from historical data for all the brands that concurrently compete in a given market. Thereafter, the paper develops how market share trajectories can be reliably forecasted given the consistent histories. An integral capability of the method is to predict the turning point on brand life cycle curves. An application is described.

Advances in Business and Management Forecasting, Volume 4, 247–263 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04016-X

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1. CONSISTENT MEASUREMENTS OF MARKET SHARE TRAJECTORIES The method shows how to estimate the market shares sequentially, month after month, as competition creates changes. The result is consistent market share histories for the competing brands. The consistent histories are essential for the forecasting method presented in the second part of the paper from Section 10 onward (Armstrong, 1985; Harrison & Stevens, 1971; Maddala, 1977; Wright, Lawrence, & Collopy, 1996.

2. REAL MARKETS AND AN EQUIVALENT URN MODEL In a real market there are k+1 different brands competing for the consumers’ dollars. The total sales of all the k+1 brands in month t are Nt units. Total sales of units of a specific brand j ¼ 1, 2, y , k+1 is Njt units in month t. Skj¼1 þ N jt ¼ N t : Brand j has a market share pjt at time t and pjt ¼ Njt/Nt where 0 r pj,t r 1.00 and Skþ1 j¼1 pj;t ¼ 1:00: Over time all of Nt, Njt and pjt may vary for all j. Abstractly equivalent to the real market is an urn which contains a total number of Nt balls in month t representing total sales. The Nt balls are grouped into k+1 mutually exclusive categories. There is one category for each competing brand. Within a category each ball is identical as are the units of a given brand. Balls in one category are different from the balls in another category. The urn contains a total number Njt balls in category j in month t representing total sales of brand j at time t. The proportion of the balls in category j to the total number of balls in the urn is pjt ¼ Njt/Nt. pjt corresponds to market shares in real markets. In the real markets, the problem is to reliably estimate all the market shares dynamically over time as new successful brands enter the market creating new categories, and as failing brands are withdrawn deleting categories. These changes cause the number of categories to vary over time. Total sales may be changing in the real markets as would be sales of any brand j. In the urn, these changes are represented by changing the quantity Nt of balls in the urn, or by changing the quantity Njt of balls of type j over time. As Nt and Njt vary over time, the proportion pjt will vary over time in the urn as they do in the real markets. The urn is at any time t, a multinomial urn, for any given pjt. However, pjt varies over time and must be considered a random variable.

Forecasting Simultaneous Brand Life Cycle Trajectories

249

3. THE PROBABILITY DENSITY GOVERNING SIMULTANEOUS MARKET SHARES We shall assume that p1t, p2t, y , pkt, pk+1t in the urn, and in the real markets, are random variables distributed by the multi-variable Dirichlet probability density function (1). This assumption offers no significant loss of generality because Dirichlet’s density can assume almost any empirically observed locus for the shares. Derivations of formula and proofs of correctness will not be presented in this paper. The reader is referred to the literature (Wilks, 1950; Gelb, 1974; Lewis, 1986). The means and variances of the shares p1t, p2t, y , pkt, pk+1t in (1), are calculated from (2), (3), (4) and (5) if we know the values of the placement parameters m1t
1, m2t
1, y , mkt
1, mk +1,t
1. If we do not know m1t
1, m2t
1, y , mkt
1, mk +1,t
1, but have observations on the market share means uj(t|t
1) and the market share variances Vj(t|t
1), then mjt
1 is calculated from (6) and (7). Eqs. (6) and (7) are created by solving Eqs. (2) and (3) simultaneously for mj,t
1 and Rt
1.
 Dðpt jmt
1 Þ  D p1t ; p2t ; . . . ; pkþ1;t jm1t
1 ; m2t
1 ; . . . ; mkt
1 ; mkþ1;t
1 " # þ1 Gðm1;t
1 þ m2;t
1 þ    þ mkþ1;t
1 Þ kY mj;t
1
1 ¼ p ð1Þ Gðm1;t
1 ÞGðm2;t
1 Þ . . . Gðmkþ1;t
1 Þ j¼1 j;t mj ðtjt
1Þ ¼

mj;t
1 ¼ mj ðt
1jt
1Þ Rt
1

V j ðtjt
1Þ ¼


 mj;t
1 Rt
1
mjt
1 R2t
1 ðRt
1 þ 1Þ

V j ðtjt
1Þ ¼


mit
1 mjt
1 2 Rt
1 ðRt
1 þ 1Þ

m1;t
1 þ m2;t
1 þ    þ mkþ1;t
1 ¼ Rt
1

(3)

(4) (5)

"

mj;t
1

#
 mj ðtjt
1Þ 1
mj ðtjt
1Þ ¼ mj ðtjt
1Þ
1:00 V jt
1

(2)

"

Rt
1

#
 mj ðtjt
1Þ 1
mj ðtjt
1Þ ¼
1:00 V jt
1

(6)

(7)

250

FRENCK WAAGE

4. USING CURRENT MARKET OUTPUT SIGNALS TO UPDATE THE PRIOR SHARE ESTIMATES Prior market shares pj,t have been estimated using the last month’s information mj,t
1. As soon as updated information becomes available, we update this prior estimate by calculating its posterior estimate. Recording actual sales results in the real markets corresponds to the following urn experiment. Drawing at random a sample of St balls from the multinomial urn at the end of month t, corresponds to sampling St sales in the real markets at the end of month t. The sample will show the new information that z1,t units of brand 1 were sold during month t, z2,t units of brand 2, y , and zk+1,t units of brand k+1 such that St ¼ z1,t+z2,t+ y +zk+1,t. The probability of receiving the outcome z1t, z2t, y , zk+1t from a random sample of St balls drawn from a multinomial urn at the end of month t is governed by the multinomial density function (8).
 Mðzt jpt ; St Þ  M z1t ; z2t ; . . . ; zkþ1;t jp1t ; p2t ; . . . ; pkþ1;t ; S t k þ1h i Y St ! z ¼ pi;ti;t ð8Þ z1;t !z2;t ! . . . zkþ1;t ! i¼1 The means and variances of zi,t and the co-variances of zi,t and zj,t in (8) are measured by Eqs. (9)–(11). mðzj;t Þ ¼ S t pijt

(9)

 
 V zj;t ¼ St pj;t 1
pj;t

(10)


 V zi;t ; zj;t ¼ St pi;t pj;t

(11)

We use this new sampling information to improve the prior estimate of the market share. Achieve the improvement by revising the prior estimate D(pt|mt
1) from (1) to the posterior estimate D(pt|mt
1+zt) calculated by (12). Eq. (12) is Bayes’ theorem (Harrison & Stevens, 1971; Maybeck, 1979; Raiffa & Schlaifer, 1961). The calculation of (12) is carried out by substituting (8) and (1) into (12) and by completing the calculations. The resulting (12) is a Dirichlet density of the same form as the prior density (1). The means and the variances of (12) can therefore immediately be calculated

Forecasting Simultaneous Brand Life Cycle Trajectories

251

from the formula (2) and (3) here updated to (15) and (16) but using the revised parameter values from the (12).


 M zt jpt ; S t D pt jmt
1 
 D pt jmt
1 ; zt ; S t ¼ R
M zt jpt ; S t D pt jmt
1 dp " # þ1 Gðm1;t þ m2;t þ    þ mkþ1;t Þ kY mj;t
1 
 ¼
p ð12Þ G m1;t G m2;t . . . Gðmkþ1;t Þ j¼1 j;t mj;t ¼ mj;t
1 þ zjt

Rt ¼ m1;t þ m2;t þ    þ mkþ1;t ¼

kþ1 X j¼1

mj ðtjtÞ ¼

mj;t
1 þ

(13) kþ1 X

zj;t
1 ¼ Rt
1 þ St

(14)

j¼1

mj;t mj;t
1 zjt ¼ Rt Rt
1 þ S t

(15)

ðmj;t ÞðRt
mj;t Þ ðRt Þ2 ðRt þ 1Þ

(16)

V j ðtjtÞ ¼

Eq. (12) is the posterior density at the end of month t, and it becomes the prior density at the beginning of the next month t+1. The prior-posterior updatings are calculated sequentially as the time index increases to t+2, t+3 and so on. Bayes filter (15) and (16) react very slowly to changes caused by new information, however, because the parameters mj,t ¼ mj,t
1+zjt are cumulative. mjt will become large relative to zjt quickly. The fast growing mjt will come to dominate the market share estimate, and the new information zjt will have a decreasing influence.

5. INCREASING THE RELATIVE WEIGHT OF NEW INFORMATION To increase the responsiveness of Bayes filter (15) and (16), we introduce the weight aj (0ZajZ1) in (13) and replace mj,t ¼ mj,t
1+zjt from (13) with mj,t ¼ aj mj,t
1+zjt defined in (17). If aj ¼ 0 then history mj,t
1 receives no

252

FRENCK WAAGE

weight and the current information zjt is the only information that matters. If aj ¼ 1 history receives full weight and (17) becomes the original (13) mj,t ¼ mj,t
1+zjt. (17) will replace (13) hereafter, and (18) will replace (14). mj;t ¼ aj mj;t
1 þ zjt

Rt ¼ m1;t þ m2;t þ    þ mkþ1;t ¼

kþ1 X j¼1

aj mj;t
1 þ

(17) kþ1 X

zj;t ¼ Rt
1 þ St

(18)

j¼1

An efficient algorithm is needed for sequentially executing the recursive updating as the time index increases efficiently. In the algorithm we shall need Eqs. (19)–(21). Create Eq. (19) by solving (2) and (15) simultaneously after having used (17) and (18) in (15). Substituting (19) into (14) and (16) creates (20) and (21).

6. AN EFFICIENT ALGORITHM FOR REVISING CONDITIONAL PROBABILITIES SEQUENTIALLY To start, set t ¼ 1; and initialize with the prior means mj(0|0), variances Vj(0|0) and co-variances. Calculate from (6) the initial values m10, m20, y , mk+1,0 and from (5) or (7) R0 ¼ m1,0+m2,0 + y +mk+1,0. Next: assign the posterior means mj(t
1|t
1) and variances Vj(t
1|t
1) from the end of month t
1 to become the priors for the next month t, such that: mj ðtjt
1Þ ¼ mj ðt
1jt
1Þ V j ðtjt
1Þ ¼ V j ðt
1jt
1Þ Sample St sales at random at the end of month t. Count the observed sales of every competing brand z1t, z2t, y , zk+1,t in that sample. Calculate mjt for all j and Rt from (17) and (18). Calculate the posterior means mj(t|t), the variances Vj(t|t) and the co-variances from the recursive equations (19), (20) and (21). (19) is created by solving (2) and (15) simultaneously after having used (17) and (18) in (15). Substituting (19) into (14) and (16) creates (20) and (21).     aj Rt
1 zjt mðtjtÞ ¼ mj ðtjt
1Þ þ (19) Rt Rt

Forecasting Simultaneous Brand Life Cycle Trajectories

"
# mj ðtjtÞ 1
mj ðtjtÞ V j ðtjtÞ ¼ Rt þ 1  V i;j ðtjtÞ ¼

 mi ðtjtÞ  mj ðtjtÞ Rt þ 1

253

(20)

(21)

Advance the time index to t+1. Go to next in run the algorithm when t has been replaced by t+1 at the end of month t, which is identically the beginning of month t+1. End of the Recursive Updating Algorithm (Astrom, 1970; Brown & Hwang, 1992; Bozig, 1994; Mehra, 1979; Kalman & Buchy, 1961).

7. AN APPLICATION: MEASURING THE MARKET SHARES OF FIVE COMPETING BRANDS Our company produced and marketed an electronic communications brand which we shall label brand 1. Three brands numbered 3, 4 and 5 competed with it over months t ¼ 1, 2, 3, 4. A new brand numbered 2 entered the market in month 5, which did not exist in months 1, 2, 3 and 4. The forecast method was initialized with the vector u(0|0) ¼ (u1(0|0) ¼ 0.15, u3(0|0) ¼ 0.22, u4(0|0) ¼ 0.25, u5(0|0) ¼ 0.38). The algorithm next generated the priors u(1|0) ¼ u(0|0) at the beginning of month 1. In particular, u(1|0) ¼ (u1(1|0) ¼ 0.15, u3(1|0) ¼ 0.22, u4(1|0) ¼ 0.25, u5(1|0) ¼ 0.38). A random sample of S(1) sales from all the sales that had taken place during month 1 was obtained at the end of month 1. From this sample we obtained the vector of sales by brand: z(1) ¼ (z1(1) ¼ 15, z3(1) ¼ 22, z4(1) ¼ 25, z5(1) ¼ 38) also recorded in Table 1 row 1. Executing (19) we obtained the posterior share estimates u(1|1) ¼ (u1(1|1) ¼ 0.15, u3(1|1) ¼ 0.22, u4(1|1) ¼ 0.25, u5(1|1) ¼ 0.38). These are shown in Table 1, row 1. At the beginning of month t ¼ 2, u(2|1) was generated from u(1|1). At the end of month 2 a random sample of S(2) sales were drawn with the results z(2) shown in row 2 of Table 1. u(2|2) was calculated. Deployment of the algorithm continued. Table 1 records the sampling results z(t), the posteriors u(t|t) and their variances V(t|t) as t ¼ 1, 2, 3, y , 10. Fig. 1 graphs the market share histories x(t|t) from Table 1. The vertical dotted line at month 10 in Fig. 1 marks the end of history. This procedure gave us consistent histories for all of the competing brands simultaneously at the end of each month. The procedure for generating consistent reliable forecasts will be discussed next. An application follows (Singer & Behnke, 1971).

254

Table 1. Month t Initializ

z1,t

z2,t

z3,t

z4,t

z5,t

Consistent Histories for the Competing Brands. u1,t

0 1 2 3 4 5 6 7 8 9 10

15 18 20 18 24 23 29 25 29 27

25 25 27 26 28 25 27 28 27 29

38 34 35 38 32 35 27 27 25 22

0.150 0.172 0.192 0.183 0.223 0.228 0.271 0.256 0.280 0.273

u3,t

u4,t

u5,t

0.000

0.220

0.250

0.380

0.007 0.009 0.031 0.051 0.050 0.099

0.220 0.227 0.194 0.184 0.160 0.160 0.139 0.140 0.140 0.112

0.250 0.250 0.264 0.261 0.274 0.257 0.266 0.276 0.272 0.285

0.380 0.351 0.350 0.371 0.335 0.346 0.293 0.277 0.258 0.231

V1,t

0.00126 0.00084 0.00079 0.00080 0.00077 0.00084 0.00086 0.00095 0.00094 0.00100 0.00101

V2,t

V3,t

V4,t

V5,t

0.00003 0.00004 0.00012 0.00020 0.00022 0.00038

0.00170 0.00114 0.00099 0.00085 0.00079 0.00071 0.00069 0.00063 0.00061 0.00049 0.00045

0.00186 0.00124 0.00107 0.00102 0.00099 0.00100 0.00096 0.00097 0.00099 0.00098 0.00100

0.00233 0.00156 0.00130 0.00121 0.00119 0.00114 0.00114 0.00106 0.00102 0.00103 0.00094

FRENCK WAAGE

Note: a ¼ 0:30

1 1 4 6 5 12

22 23 18 18 15 16 13 14 14 10

u2,t

Forecasting Simultaneous Brand Life Cycle Trajectories

255

0.400 0.350

Fractions of 1.00

0.300 0.250 0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

8

9

10

11

Months

Fig. 1. History of Mean Shares in Months 1 through 10.

8. CONSISTENT, SIMULTANEOUS SHARE AND LIFE CYCLE FORECASTS We now describe a method for forecasting all the future market shares simultaneously when those forecasts are anchored on the history developed above. At the end of month 10, we know the posterior market share estimates for all the preceding months 0, 1, 2, y , 10 from history. Table 1 has recorded them. We shall next discuss how to forecast the future market shares for the future 10 months; month 11 through month 20. The share forecasts will be measured by u(11), u(12), u(13), u(14), u(15), u(16), u(17), u(18), u(19), and u(20). 1. Establish two mutually exclusive market share categories. Category 1 contains brands whose market shares will be declining over the entire future. Category 2 contains brands with non-declining market shares over the entire future. We shall also observe the following convention. When an existing brand is re-designed, and then re-introduced into the market, even under the same brand name, the re-designed brand is treated as a new entering brand. The ‘‘old’’ jettisoned design is treated as a brand that has been withdrawn from the market.

256

FRENCK WAAGE

2. Form a panel of knowledgeable persons.A panel of experts was formed to serve in a Delphi experiment (Dalkey & Helmer, 1963; Dietz, 1987; Helmer, 1975; Jolson & Rassow, 1978; Linstone & Turoff, 1975; Riggs, 1983; Spinelli, 1983). On the panel were individuals each with deep knowledge of own brands, of competitors’ brands and of markets. The experts were selected from the departments of brand management, research and brand development, sales and marketing. The panel’s responsibility coincided with the forecast’s objective, namely to develop the most likely locus of the trajectories that the market shares would follow into the future.

9. FORECAST THE SHARES OF THE BRANDS IN CATEGORY 1 FOR DECLINING BRANDS Relying on the history shown in Table 1 for months 1 through 10, and plotted in Fig. 1, brands 3 and 5 were judged to belong in Category 1. We validated that verdict by verifying that the customers placed brands 3 and 5 lowest on their relative preference orderings and that they were not inclined to change that opinion. Brands 1, 2, 4 were placed in Category 2. The Delphi Experiment was led by a manager. The manager’s job was to develop initial share trajectories for each of brands 3 and 5. To develop this initial forecast view, he used his own insights, experience and knowledge of patterns he knew from past equivalent brands in equivalent descending circumstances. His initial view of the descending trajectories were then graphed. The specific supporting assumptions were typed on the graph. The graph was then distributed to each expert panel member. Each expert was instructed to do one of the two things. Either, amend the locus of the trajectory if that were his best judgment and to write down the explicit assumptions supporting that judgment. Or, leave the initial trajectory unchanged, if that were his best judgment, stating the assumptions behind this judgment. The amended forecast trajectories were returned to the forecast manager with the new and added information from round 1. The forecast manager’s job now was to develop a revised most likely locus for each declining Category 1 brand on the basis of the new information from ‘‘round 1’’. The new most likely view of the forecast trajectory was developed and graphed. The enriched information was posted with the graph. The graph with the information was sent to the experts and ‘‘round 2’’ started. After ‘‘round 2’’ there was a ‘‘round 3’’ and a sequence of rounds followed. The sequence ended when round t generated the same result as round t
1. Convergence on the most likely trajectories had been obtained. Fig. 2

Forecasting Simultaneous Brand Life Cycle Trajectories

257

Market share History Tracking from Month 1 to 10. Market share Forecast for Months 11 to 20 0.400 0.350

Fractions of 1.00

0.300 0.250 0.200 0.150 0.100 0.050 0.000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Months

Fig. 2.

Market share History Tracking from Month 1 to 10. Market share Forecast for Months 11–20.

shows the final view of the two declining trajectories, m3t and m5t, for brands 3 and 5 in Category 1. This became the official forecast for brands 3 and 5. To the left of the vertical dotted line at month 10 in Fig. 1 is history. To the right are the forecast trajectories.

10. FORECAST THE SHARES OF THE BRANDS IN CATEGORY 2 OF NON-DECLINING BRANDS We now knew the market share trajectories m3t and m5t of the decreasing brands in Category 1. We also knew that the sum of market shares m1t+m2t+m4t would follow the specific trajectory (1.00 – m3t – 5t ). Brands 1, 2 and 3 were in Category 2. Their future market share trajectories now had to be forecasted.

11. IDENTIFY THE DOMINANT GROWTH BRAND IN CATEGORY 2 Fig. 3 depicts a simple technique for identifying the dominant growth brand. Plot any two of the market shares in Category 2 against the third. Using the

258

FRENCK WAAGE

Searching for the Dominant Product 0. 300

Market Shares of Products 1 and 4

0. 280 0. 260 0. 240 0. 220 0. 200 0. 180 0. 160 0. 140 0. 120 0. 100 0.000

0.020

0.040

0.060

0.080

0.100

0.120

Market Shares of Product 2

Fig. 3.

Searching for the Dominant Product.

history for month 0 through month 10 from Table 1, the shares m1t and m4t were plotted along the ordinate axis and m2t along the abscissa. The result is shown in Fig. 3. As the shares of brand 2 increase the shares of brands 1 and 4 are growing at decreasing rates. This implied that brand 2 acquired customers from brands 1 and 4, and also from the declining brands 3 and 5. It also implied that brand 2 managed to retain most of its customers. Brand 2 was identified as the dominant growth brand.

12. FORECAST THE MARKET SHARE OF THE DOMINANT GROWTH BRAND The Delphi Experiment manager developed and graphed an initial estimate of brand 2’s share trajectory. The specific supporting assumptions were typed on the graph. The graph with the initial trajectory, and the supporting assumptions, were then distributed to each expert panel member starting round 1. Each expert either amended the locus of the trajectory also writing down the explicit assumptions that supported that judgment, or he left the initial trajectory unchanged stating the assumptions behind this judgment.

Forecasting Simultaneous Brand Life Cycle Trajectories

259

Share Forecast for Product 2 in Solid Line. Plot of difference (1.00 - u2t - u3t - u5t) in dashed line 0. 800

Fractions of 1.00

0. 700

Trajectory u2t

0. 600 0. 500 0. 400 0. 300

Trajectory (100 u2t + u 3t + u 5t )

0. 200 0. 100 0. 000 0

Fig. 4.

1

2

3

4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Months. History 0 to 10. Forecast 11 to 20

Share Forecast for Product 2 in Solid Line. Plot of Difference (1.00
u2t
u3t
u5t) in Dashed Line.

The amended forecast trajectories were returned to the forecast manager with the new and added information from round 1. The forecast manager now developed a revised new most likely locus for brand 2. The new most likely view of the forecast trajectory was graphed. The enriched information was posted with the graph. The graph with the information was sent to the experts and ‘‘round 2’’ started. The sequence of rounds ended when round t generated the same result as round t
1. Fig. 4 shows the final view of the growth trajectory m2t. This became the official forecast for brands 2. To the left of the vertical dotted line at month 10 in Fig. 4 is history. To the right are the forecast trajectories. Next, we had to forecast the share trajectories of the remaining brands, 1 and 4, from Category 2. We knew that the sum of market shares m1t+m4t would follow the specific trajectory (1.00–m2t–m3t–m5t ), also plotted in Fig. 4, because we knew the trajectories m2t, m3t and m5t.

13. FORECAST THE SHARES OF THE REMAINING BRANDS IN CATEGORY 2 Because the trajectory (1.00–m2t
m3t
m5t) first ascends, then peaks, and thereafter descends, the forecast trajectory of the sum m1t+m4t will identically first ascend, then peak and thereafter descend. To obtain the market share forecasts for brands 1 and 4 separately, the Delphi Experiment

260

FRENCK WAAGE 0.800 0.700

Fractions of 1.00

0.600 0.500 0.400 0.300 0.200 0.100 0.000 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 Months

Fig. 5.

Market share History Months 0 through 10. Market share Forecasts Months 11 through 20.

was repeated using the same panel members. The experts’ final trajectories for m1t and m 4t became the official forecasts. Fig. 5 shows the final view of all the forecasted trajectories. Our method had forecasted the turning point on a brand’s life cycle.

14. FORECAST COMPARED WITH ACTUAL In this specific application we forecasted at the beginning of month 10 that the life cycle peak of brands 1 and 4 would take place in month 14. It occurred in month 15. The reason for this error was the forecast of brand 2’s growth. It actually followed a trajectory with a slightly smaller slope than forecasted. The timed demises of brands 3 and 5 took place two and one months sooner than had been forecasted, respectively. Table 1 records the history from month 0 through month 10. Table 2 records the forecasts from month 11 through month 20. The tracking history of posteriors u(t|t) for t ¼ 1, 2, 3, y , 19, 20 is also recorded in Table 2.

Consistent Forecasts for the Competing Brands.

Month t

z1,t

z2,t

z3,t

z4,t

z5,t

u1,t

u2,t

u3,t

u4,t

u5,t

V1,t

V2,t

V3,t

V4,t

V5,t

11 12 13 14 15 16 17 18 19 20

30 36 35 37 26 21 21 18 14 10

13 16 17 18 27 36.5 43 53.5 61 75

10 7 6 6 5 3 4 2 1 2

26 22 24 22 26 25 19 15 14 5

20 19 18 17 16 14.5 13 11.5 10 8

0.294 0.340 0.347 0.363 0.291 0.234 0.217 0.191 0.155 0.117

0.122 0.149 0.164 0.175 0.242 0.328 0.399 0.494 0.575 0.698

0.104 0.080 0.066 0.062 0.054 0.037 0.039 0.026 0.015 0.018

0.269 0.235 0.238 0.226 0.250 0.250 0.208 0.167 0.148 0.079

0.211 0.196 0.185 0.174 0.164 0.151 0.136 0.121 0.106 0.088

0.0015 0.0016 0.0016 0.0016 0.0014 0.0012 0.0012 0.0011 0.0009 0.0007

0.0007 0.0009 0.0010 0.0010 0.0013 0.0015 0.0017 0.0017 0.0017 0.0015

0.0007 0.0005 0.0004 0.0004 0.0004 0.0002 0.0003 0.0002 0.0001 0.0001

0.0014 0.0013 0.0013 0.0012 0.0013 0.0013 0.0011 0.0010 0.0009 0.0005

0.0012 0.0011 0.0010 0.0010 0.0010 0.0009 0.0008 0.0007 0.0007 0.0006

Forecasting Simultaneous Brand Life Cycle Trajectories

Table 2.

Note: a ¼ 0:30

261

262

FRENCK WAAGE Probability Density for Forecast of Months 15 (dashed) and 20 (solid)

Probability of a Market Share

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.00

Fig. 6.

0.05

0.10

0.15

0.20 0.25 0.30 Possible Market Shares

0.35

0.40

0.45

0.50

Probability Density for Forecast of Months 15 (dashed) and 20 (solid).

The actual probability densities that governs the forecasts for any two months, say month 15 and month 20, are given in Fig. 6. These densities are the two-variable reduction of the Dirichlet density known as a Beta density. They were obtained by substitution into (17) and (18).

15. SUMMARY AND CONCLUSIONS This paper first established how we develop measurements for a consistent simultaneous market share history for all the competing brands. The paper secondly developed a method for forecasting the market shares and the brand life cycles of all the competing brands simultaneously given the consistent history. The new posterior u(t|t) was generated, and the forecast for the next 10 months from month 11 to month 20 was completed anchored on u(t|t). Table 1 records the history of sampling results z(t), the posteriors u(t|t) and their variances V(t|t). Fig. 1 graphs the market share histories from Table 1. Thus, the paper presents a method for solving the frequently occurring, and nearly always demanding problem of simultaneously forecasting market shares of all those brands that compete in a given market.

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REFERENCES Armstrong, J. S. (1985). Long range forecasting: From crystal ball to computer. New York: Wiley. Astrom, K. J. (1970). Introduction to stochastic control theory. New York: Academic Press. Brown, R. G., & Hwang, P. Y. C. (1992). Introduction to random signals and applied Kalman filtering (2nd ed). New York: Wiley. Bozig, S. M. (1994). Digital and Kalman filtering (2nd ed.). Edward Arnold of the Hodder Headline Group. Dalkey, N. C., & Helmer, O. (1963). An experimental application of the Delphi method to the use of experts. Management Science, 9, 458–467. Dietz, T. (1987). Methods for analyzing data from Delphi panels: Some evidence from a forecasting study. Technological Forecasting and Social Change, 31, 79–85. Gelb, A. (1974). Applied optimal estimation. Cambridge, MA: Massachusetts Institute of Technology Press. Harrison, P. J., & Stevens, C. F. (1971). Bayesian Forecasting. Journal of the Royal Statistical Society, 3B(Series B), 205–247. Helmer, O. (1975). Foreward. In: H. Linstone & M. Turoff (Eds), The Delphi method: Techniques and applications. London: Addison-Wesley Publishers. Kalman, R. E., & Buchy, R. S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83, 95–108. Jolson, M. A., & Rossow, G. (1978). The Delphi process in marketing decision making. Journal of Marketing Research, 8, 443–448. Lewis, R. (1986). Optimal estimation with an introduction to stochastic control theory. New York: Wiley. Linstone, H. A., & Turoff, M. (1975). The Delphi method: Techniques and applications. London: Addison-Wesley Publishers. Maddala, G. S. (1977). Econometrics. New York: McGraw-Hill Book Company, Chapter 17. Maybeck, P. S. (1979). Stochastic models, estimation and control. New York: Academic Press, Inc. Mehra, R. K. (1979). Kalman filters and their applications to forecasting. TIMS studies in the management sciences (Vol. 12, pp. 75–94). The Hague, Netherlands: North-Holland Publishing Co. Raiffa, H., & Schlaifer, R. (1961). Applied statistical decision theory. Boston, MA: Harvard Business School Press. Riggs, W. E. (1983). The Delphi method: An experimental evaluation. Technological Forecasting and Social Change, 23, 89–94. Singer, R., & Behnke, K. (1971). Real time tracking filter evaluation and selection for tactical evaluations. IEEE Trans. Aerospace Electron Systems, AES-7, January, 100–110. Spinelli, T. (1983). The Delphi decision-making process. Journal of Psychology, 113, 73–81. Wilks, S. S. (1950). Mathematical statistics. New Jersey: Princeton University Press. Wright, G., Lawrence, M. J., & Collopy, F. (1996). The role and validity of judgment in forecasting. International Journal of Forecasting, 12, 1–8.

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A TYPOLOGY OF PSYCHOLOGICAL BIASES IN FORECASTING ANALYSIS Paul Dishman ABSTRACT Forecasters, just as any business analysts, are subject to a variety of biases that affect the forecaster’s ability to provide a prediction in which error is minimized. This article proposes that biases are derived from processes related to cognition, philosophical orientation, organizational influences, and operational influence. A proposed typology is suggested, which identifies various types of bias and their accompanying characteristics.

INTRODUCTION Numerous studies in forecasting show that there are imbedded biases in the forecast process including optimism (Dugar & Nathan, 1995), data collection inefficiency (Abarbanell, 1991), and irrationality (Brown et al., 1993; Trueman, 1994; Ehrbeck & Waldmann, 1996; Keane & Runkle, 1998). Lo¨ffler (1998) argues that much of Ehrbeck & Waldmann’s findings may not be due to strategic biases, but cognitive-based biases. Geurts (2002a) also warns forecasters to be aware of human-based bias. It is the nature of these Advances in Business and Management Forecasting, Volume 4, 265–275 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04017-1

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cognitive-based biases that deserve more attention by researchers as they provide many significant factors that provide unintentional bias in forecast analysis.

THE COGNITIVE PROCESS The forecasting analysis process is not as simplistic as data manipulation. Other factors influence this process in subtle but sublime ways. As much as a forecasting analyst might desire to purify their process by quantifying everything that they do, their own psychological thought process cannot be ignored especially when it may create bias in analysis (Tversky & Kahneman, 1974; Kahneman & Tversky, 1982). A forecasters’ thought process consist of internal factors, such as the complex stimulus combined though receptor organs, the sensory processing system, and both types of memory. These components are filled with stimuli from personal experience, educational training, professional experience, personal cultural values, and ego structure (Johnson, 1992). The combination of these two sets of factors is further impacted by the organizational culture and the ways in which the forecaster must operate and perform within that organization. Thus, the psychological process is defined by all of the above and serves to formulate biases that affect forecasting analysis. Considering how omnipresent these factors are, one cannot ignore their impact on an individual forecaster’s abilities, much less their impact on a team or forecasting department.

TYPES OF PSYCHOLOGICAL BIASES As with any human endeavor, the psychological process of forecasting has its inherent strengths and weaknesses. These are constituted in the cognitive process, the philosophical orientation of the forecaster, the organization in which he or she must function, and the operational environment in which the forecaster must perform. Due to all of these factors, the forecaster (and the forecasting process) cannot be with biases, either intentional or unintentional. Biases in and of themselves are not good or bad. However, forecasters are primarily concerned with bias affects when affects prevent proper interpretation of data and contribute to inaccurate forecasts.

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Cognitive The primary bias in the cognitive genera is expectancy bias, which is the tendency to predict what one expected to predict (Jervis, 1976; Betts, 1978). This is where the findings are already aligned with a preconceived notion that may be based on the familiarity of the forecaster to the problem. The bias might also manifest itself in the continued use of an analytical technique that may not necessarily provide the required forecasting accuracy due to changes in the data from previous applications. Contributing to this is the phenomenon that, as an analyst learns from each forecast that is performed, the expectation set of anticipated outcomes increase. Thus, the more experienced the analyst, the more they might have a tendency to exploit unconsciously their expanded repertoire of expectations. It is believed by some researchers that there is no real propensity for analysts to perform using or understanding those reference items that may be outside the expectations set (Heuer, 1999). As stated, experiential knowledge creates a larger expectations set. Thus, the larger the set of anticipated outcomes (or questioned assumptions), the lower the probability for error in the forecast. On the other hand, extensive expertise tends to reinforce previous created expectations, which limits the perspective of analysis and generated bias. This expertise bias tends to create expectancy sets for the learning process that provides, may not increase problem resolution or forecasting accuracy. Naivety bias is the reverse of expertise bias. This is a forecasting perspective that is void of any professional reference set from which to apply to the problem at hand. This bias may be due to lack or training, lack of experience with the specific forecasting problem, or unawareness of analysis methodology. Although such bias may be disadvantageous to properly formulating a research problem utilizing complicated analytical techniques, it can be beneficial to the forecaster in employing individuals, which might bring a fresh perspective to an old problem or questioning long-held assumptions. Progressive perception bias is a psychological phenomenon that affects accurate forecasts when a single forecaster works on one problem for such a long time that they do not notice tiny, incremental changes in the data that they are analyzing (Jervis, 1976). This might especially be true of corollary data that influences the forecasting adjustment such as subtle changes in the customer base or industry structure. This ‘‘data creep’’ phenomenon sometimes has to produce a large accumulated affect before being noticed and processed by the forecaster. Unfortunately, the accumulated affect might occur over such a long period that the forecaster may not gain any real benefit from the forecast accuracy adjustment.

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The comparative order in which the forecaster receives information also influences the strength that the forecaster attributes to the information. This is not data that will be provided a predetermined weight within the forecasting model, but information that may directly relate to the formulation of assumptions pertaining to the forecasting model such as economic information or sales staff presumptions. A primacy effect exists when the forecaster places more importance on the information that is received first and it is used as the comparator reference for future data. If the forecaster unconsciously places more importance on the most recently acquired information and uses that to compare with all other information, this is bias from recency effect. A forecaster analyst might use a ‘‘natural’’ starting point, perhaps from a previous forecast, as a data anchor. This point is then used as an estimate for assumptions or perhaps even as an approximation of the prediction. When such anchoring occurs, it may create biases by creating intellectual inertia that inhibits full and proper adjustments of estimates (Heuer, 1999). (This is a common issue with forecasts adjustments that have identified ‘‘base-rate’’ problems.) Prior hypothesis or a priori hypotheses bias is where an analyst improperly utilizes a previously proven hypothesis (or set of assumptions) to an inapplicable new forecasting problem (Fleisher & Bensoussan, 2002). This may also include the failure to reject previous held hypotheses (Watson, 1960; Gettys et al., 1980). DeRosia, Christensen, and Whitlark (2004) further state that a priori hypotheses are unable to provide for effectiveness and impartiality in qualitative research methodology. Ambiguity bias arises from data that provide for difficult interpretations, definitions, misleading information, or possesses confounding variables. In some cases, data may not arise from a point of contextual evaluation. Thus, the same information may be viewed differently within different contexts (Katz & Vardi, 1991; Bernhardt, 1993). A set of bias effects may be created where the value of initial (but perhaps erroneous) interpretations linger on. Heuer (1999) states that the quality and the quantity of the data/ information required to overcome an initial interpretation (or hypothesis, or assumption) are much greater than the amount of data/information required to form an immediate impression. Additionally, exposure to ambiguous information negatively interferes with accurate conclusions even after information that is more reliable becomes available later. This is known as lingering ambiguity bias. Exposure effects can also create bias. The longer that a forecaster is exposed to ambiguous data, the stronger the clarity that new information

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has to be to gain recognition as important (longitudinal ambiguity bias). In addition, the more ambiguous the initial information, the stronger its clarity has to be in order for it to be recognized (Jervis, 1976). A forecaster viewing an initial set of facts (which might help formulate assumptions to a model) at the stage when they may be particularly ambiguous, may be at an analytical disadvantage compared to other forecasters that enter the project at a later stage when the assumptions have already been formulated (Heuer, 1999). Although Geurts and Whitlark (2002) propose that there is a sale forecasting accuracy response curve that states that there is a positive relationship between dollars spent on data and the forecasting accuracy, other analytical processes show that increased dollars spent produce diminishing returns with regard to improving analytical or prediction ability. This is especially true when the data collected is ambiguous and contradictory (Heuer, 1999). Several experiments have also shown that increasing the amount of information does not necessarily improve the accuracy of the prediction (Goldberg, 1968). However, additional data did tend to increase the analysts’ confidence in their predictive ability (Table 1). Philosophical Orientation Since people tend to be partial to processing different kinds of data and information, there are also biases related to these processing preferences. Table 1.

A Typology of Psychological Biases in Forecasting. Psychological Biases

Cognitive Expectancy Expertise Naivety Progressive perception Comparative order Data anchoring A priori hypothesis Ambiguity Exposure

Phylosophical orientation Data type Prima facie Technique/model

Organizational Ideological Bureaucratic/ political Premature closure Collaborative Forecaster tenure Extraorganizational Cognitive grex Strategic

Operational Systemic and systematic Information limitation Censored data

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Primarily, these types of biases fall along the lines of preferences toward processing quantitative or qualitative data. This preference can cause conflict between the forecasting analysts and the data providers. This frequently occurs in the case of sale personnel required to convert verbal (or experiential) customer information to numeric data in order for it to fit into the forecaster’s qualitative model. On the other hand, mathematic myopia is where other facts and knowledge items from the firm (key forecasting intelligence) that could improve the accuracy of forecasts, such as personal experience, sales lead, and customer-specific history are not tracked or taken into consideration by the forecaster (Kahn & Adams, 2000). Related to this type of bias, is prima facie bias, where the forecaster does not examine the collected data without questioning its source, creditability, or reliability. In the same way that forecasters (and data providers) can prefer a particular type of data, forecasters can have a preference for a particular modeling technique, which, if used inappropriately may lead to biased results.

Organizational In organizations, where there is a culture that is permeated with a particular political or philosophical belief can contribute to ideological bias. If these beliefs are ubiquitous in the organization, they may form into dogmatic tenets that influence all aspects of the business including forecasting. Such belief structures may include a belief of competitive superiority, customer service importance, or a besieged mentality. Related to this are biases from bureaucratic or political factors where the forecaster is affected by various pressures and constraints including how they must work, budget, type of forecasting product, or how they report their findings. Some forecasters relate that their predictions are not without internal influences from their supervisors. In such an environment, a forecaster might learn to adjust their predictions to make them more palatable to management or clients. Such political influences might create the inclination to over forecast with ‘‘puffing’’ from intrinsic optimism or fear from upper management or other departments (Reese, 2000; Geurts, 2002a). Conversely, some organizations would pressure for under forecasts when forecasts tend to become operating targets or sales quotas. Premature closure bias problems are sometimes manageable by the forecaster. These analytical limitations placed on the forecaster are due to time constraints, changing deadlines, or external competitive pressures that require an accelerated report. Forecasts must be provided to management by

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specified deadlines; however, it may be that not all of the underlying data have been accounting for by the time necessary to process the forecast. This can be related to the accuracy adjustment problem in that, if early information leads to one forecast prediction, but subsequent information points to another prediction, forecasters may have an inherent interest in sustaining the original prediction. Collaborative forecasting processes are created in order to overcome limited perspectives, limited experience, data type bias, and increase executive involvement in the forecasting process (Diehn, 2000; Reese, 2000; Wilson, 2001). Although there is much support in the literature for improving the forecasting process using collaborative efforts, care should be taken so that the collaboration in itself does not create biases. These biases could be any of the organizational or operational biases. However, in the collaboration process, there is evidence that biases may actually compound themselves to create complex biases that may actually increase error arithmetically (Heuer, 1999). Forecasting tenure can create biases from due to the length of time that a forecaster has spent within the same organization, the profession and, perhaps, on a specific task. Tenure can provide positive and negative contributions to forecast improvement. It is believed that forecasting accuracy is improved by the experiential expertise that has been provided by insights gained over the years. It may also be that, due to ‘‘negation evaluation,’’ burnout or job dissatisfaction may occur (Geurts, 2002b). Similar sorts of affects can occur when a forecaster has spent a great deal of time in the same industry or analyzing the same forecasting problem, period after period giving way to progressive perception bias, among others. There are also extra-organizational biases, caused by factors extrinsic to the organization. This can occur when a forecaster is heavily influenced by predictions made by other forecasters in the same field. Commonly known on Wall Street as herding behavior, it can occur as cognitare grex among analysts as well and traders. This bias is only possible when one forecaster has access to the findings of other forecasters working on the same general problem. A derivative of this bias is a priori cognitare grex when a biased forecast is produced in anticipation of what other (industry) forecasters will predict (Fleisher & Bensoussan, 2002). Lo¨ffler (1998) restates that there are motivating reasons for a forecaster to intentionally bias his/her forecasts either for the purpose of institutional benefit or for the purpose of client benefit. He labeled these intentional biases ‘‘strategic biases’’ as they do so in an attempt to gain some economic benefit. Although these are conscious biases on the part of the forecaster,

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they are included here as they are intended to evoke a psychological response from the anticipated recipient. Operational As one examines the entire forecasting process, two related types of biases might be uncovered – systematic and systemic. Systematic bias is caused by an inherently flawed process or procedure that leads to error in the forecast. This might occur when a forecaster uses a model where the mathematical assumptions do not reflect reality to the degree that is required. Systemic bias is caused when the entire forecasting method is faulty, and (perhaps, unknowingly) the fault is perpetuated into subsequent forecasts. In information limitation bias, not all the requisite data is available to the forecaster or it cannot be collected in a timely fashion. Thus, out of necessity, a subset of the data must be used. Forecasters sometimes witness forecast contribution reluctance – reticence on the part of, perhaps, a sales person to provide meaningful data into the forecasting process. This bias might also occur when all of the data is collected from a single source such as prior sales, but the forecasting model does not include future new product introductions, price changes, or competitive actions. A subsidiary of this bias is the censored data problem, were one cannot collect data that would reflect true demand (Zeni, 2002).

CATEGORIES OF BIASES To aid in understanding biases and their impact on forecasting, it might serve to examine them along three criteria – where they might occur in the forecasting process domain, their instigation locus, and their state of intent. Biases in forecasting can transpire in the collection stage, the analysis stage, or across the process domain. They can originate intrinsically from within the psychological processes of the forecaster or they can be thrust upon the forecaster from external forces. Most biases are unconscious to the forecaster and, therefore, their effects are an unintentional consequence. However, some biases are intentional, such as strategic bias, as are consciously part of the forecasting procedure and report. Intentional biases are rooted in the execution of analysis in which the findings or methodology are manipulated for advantage of one or more entities. These entities may include the forecaster, the forecaster’s organization, or the subject firm or industry (Table 2).

A Typology of Psychological Biases in Forecasting Analysis

Table 2. Bias Cognitive Expectancy Expertise Naivety Progressive perception Comparative order Ambiguity Exposure Philosophical orientation Data type Technique/model Organizational Ideological Bureaucratic/political Premature closure Collaborative Forecasting tenure Extra-organizational Strategic Operational Systemic and systematic Information limitation Censored data

273

Categorizations of Biases.

Domain

Locus

Intent

Analysis Analysis Analysis Analysis/collection Analysis Analysis Analysis

Intrinsic Intrinsic Intrinsic Intrinsic Intrinsic Intrinsic Intrinsic

Unintentional Unintentional Unintentional Unintentional Unintentional Unintentional Unintentional

Analysis Analysis

Intrinsic Intrinsic

Unintentional Unintentional

Process Process Process Process Process Process Process

Extrinsic Extrinsic Extrinsic Extrinsic Intrinsic/extrinsic Extrinsic Extrinsic

Unintentional Unintentional/intentional Unintentional Unintentional Unintentional Unintentional Intentional

Process Collection Collection

Extrinsic Extrinsic Extrinsic

Unintentional Unintentional Unintentional

SUMMARY It is interesting to note that systematic errors in forecasts conform to patterns discussed in the psychological literature. Additionally, cognitive biases tend to linger and persist even given the disciplinary forces of the market (Lo¨ffler, 1998). It is hoped that through the identification of bias within the forecasting process, that their effects can be minimized and that mathematic, psychological, and organizational model be put in place, which can alleviate most of the effects of such bias. It is recommended that further research be conducted in the areas of the measurement of error from bias. These should include studies that examine the psychological processes that are unique to forecasting and their creation and perpetuation of cognitive processes that create unnecessary bias in the forecasting process.

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REFERENCES Abarbanell, J. (1991). Do analysts’ earnings forecasts incorporate information in prior stock exchange prices? Journal of Accounting and Economics, 14, 147–165. Bernhardt, D. (1993). Perfectly legal competitor intelligence: How to get it, use it and profit from it. London, UK: Financial Times/Pitman Publishing. Betts, R. (1978). Analysis, war and decision: Why intelligence failures are inevitable. World Politics, 31(October), 84–85. Brown, L., Lawrence, D., O’Hanlon, J., Thomas, J. K., Brown, P., & Zmijewski, M. E. (1993). Earnings forecasting research: Its implications for capital markets research. International Journal of Forecasting, 9(3), 295–320. DeRosia, E., Christensen, G., & Whitlark, D. (2004). Improving sales forecasts by testing underlying hypotheses about consumer behavior: A proposed qualitative method. Advances in business and management forecasting (Vol. 4, pp. 183–197). Kidlington, UK: JAI, Elsevier Science. Diehn, D. (2000). Seven steps to build a successful collaborative forecasting process. The Journal of Business Forecasting, (Winter 2000–2001), 23–29. Dugar, A., & Nathan, S. (1995). The effect of investment banking relationships on financial analyst’ earnings forecasts and investment recommendations. Contemporary Accounting Research, 12, 131–160. Ehrbeck, T., & Waldmann, R. (1996). Why are professional forecasters biased? Agency versus behavioral explanations. Quarterly Journal of Economics, 111, 21–40. Fleisher, C., & Bensoussan, B. (2002). Strategic and competitive analysis: Methods and techniques for analyzing business competition. Upper Saddle River, NJ: Prentice-Hall. Gettys, C. F., Manning, C., Mehle, T., & Fisher, S. (1980). Hypothesis generation: A final report on three years of research. Technical report, May 10, 1980. University of Oklahoma, Decision Processes Laboratory. Norman, OK. Geurts, M. (2002a). What to do when sales forecasts are not accurate enough. Advances in business and management forecasting (Vol. 3, pp. 73–84). Kidlington, UK: JAI, Elsevier Science. Geurts, M. (2002b). Managing to retain sales forecasters. Advances in business and management forecasting (Vol. 3, pp. 155–160). Kidlington, UK: JAI, Elsevier Science. Geurts, M., & Whitlark, D. (2002). The relationship between accuracy and expenditures in forecasting. Advances in business and management forecasting (Vol. 3, pp. 85–91). Kidlington, UK: JAI, Elsevier Science. Goldberg, L. (1968). Simple models or simple processes? Some research on clinical judgments. American Psychologist, 23, 261–265. Heuer, R. (1999). Psychology of intelligence analysis. Washington, DC: Center for the Study of Intelligence. Central Intelligence Agency. Jervis, R. (1976). Perceptions and misperception in international politics (pp. 195–197). Princeton, NJ: Princeton University Press. Johnson, G. (1992). In the palaces of memory: How we build the worlds inside our heads. New York, NY: Vantage Books. Kahn, K.B., & Adams, M.E. (2000). Sales forecasting as a knowledge management process. The Journal of Business Forecasting, (Winter 2000–2001), 19–22. Kahneman, D., & Tversky, A. (1982). Intuitive prediction: Biases and corrective procedures. In: D. Kahneman, P. Slovic & A. Tversky (Eds), Judgment under uncertainty: Heuristics and biases (pp. 414–421). Cambridge, UK: Cambridge University Press.

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Katz, Y., & Vardi, Y. (1991). Strategies for data gathering and evaluation in the intelligence community. International Journal of Intelligence and Counterintelligence, 5(3), 313–328. Keane, M., & Runkle, D. (1998). Are financial analysts’ forecasts of corporate profits rational? Journal of Political Economy, 106(4), 768–805. Lo¨ffler, G. (1998). Biases in analyst forecasts: Cognitive, strategic or second-best? International Journal of Forecasting, 14, 261–275. Reese, S. (2000). The human aspects of collaborative forecasting. The Journal of Business Forecasting, (Winter 2000–2001), 3–9. Trueman, B. (1994). Analyst forecasts and herding behavior. Review of Financial Studies, 7, 97– 124. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(27th September), 1124–1131. Watson, P. C. (1960). On the failure to eliminate hypotheses in a conceptual task. The Quarterly Journal of Experimental Psychology, 12(3), 129–140. Wilson, N. (2001). Game plan for successful collaborative forecasting process. The Journal of Business Forecasting, (Spring), 3–6. Zeni, R. H. (2002). Estimating airline demand from censored data. Advances in business and management forecasting (Vol. 3, pp. 21–36). Kidlington, UK: JAI, Elsevier Science.

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A FORECAST COMBINATION METHODOLOGY FOR DEMAND FORECASTING J. Gaylord May and Joanne M. Sulek ABSTRACT A number of previous studies have shown that a combination of forecasts typically outperforms any component forecast. Managers may wish to use forecast combination to improve forecast accuracy in predicting retail sales. In this study, revenue data from an actual service company is used to generate and test a least absolute value (LAV) regression model for forecast combination. The LAV forecast, developed by the authors, is determined by minimizing weighted deviations from the component forecasts. The accuracy of this approach is compared to the accuracy of a traditional method.

INTRODUCTION For many businesses, accurate forecasting represents a greater challenge than ever before. As customers have become less satisfied with waiting for product orders or service delivery, the need for more accurate demand forecasts has grown. Using modern computer technology, managers can now easily apply a variety of quantitative forecasting models (e.g., Advances in Business and Management Forecasting, Volume 4, 277–287 Copyright r 2006 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 1477-4070/doi:10.1016/S1477-4070(05)04018-3

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exponential smoothing, trend analysis) to demand data; they can also tap expert opinion to develop purely judgmental (or qualitative) forecasts. Consequently, managers frequently have access to several forecasts of future demand. Given multiple forecasts for a single product or service, they face the task of effectively combining all the forecast information they have at their disposal into one forecast. For at least 30 years, researchers have argued that a combination of forecasts tends to be more accurate than any of the individual forecasts comprising the combination (see, for instance, de Menzes, Bunn, & Taylor, 2000). Forecast accuracy tends to improve even if only two forecasts are combined (Russell & Adam, 1987; Fildes, Hibon, Madridakis, & Mead, 1998). This study utilized revenue data from an actual service company to generate and test a forecast combination model based on least absolute value (LAV) regression. The accuracy of this approach was then compared to the accuracy of a standard forecast combination method. As discussed in the next section, a variety of quantitative models for forecast combination have been presented in the forecasting literature during the past three decades.

FORECAST COMBINATION Forecast combination is not a new idea. During the past 30 years, a number of researchers have proposed a wide variety of forecast combination techniques, including both judgmental and quantitative techniques. Among the quantitative approaches, four techniques for forecast combination have received a good deal of attention. These include: (1) the simple average approach, (2) the out-performance technique devised by Bunn (1975), (3) the optimal method proposed by Bates and Granger (1969) and (4) ordinary least squares (OLS) regression. Each of these traditional methods exhibits both strengths and weaknesses which will be briefly summarized here. Bunn’s (1996) review paper contains a good survey of non-traditional approaches to forecast combination. Perhaps the most basic forecast combination methodology is the simple average approach. In this method, a combined forecast for period t (denoted by Ft) is generated by taking the arithmetic mean of two or more component forecasts for period t. Thus, Ft ¼

Xf

i

n

(1)

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where fi is an individual forecast and n the number of individual forecasts to be combined. The simple average approach is easy to use and has performed well empirically. Several studies have demonstrated instances where the simple average outperformed more complex models (Clemen, 1989). The accuracy of the combined forecast produced by the arithmetic mean reflects the accuracy of the component forecasts (Gupta & Wilton, 1987). Thus, a disadvantage of this approach is that inaccurate individual forecasts reduce the accuracy of the combined forecast (Ashton & Ashton, 1985). Since the simple average is actually a weighted average in which all the weights are the same, the simple average ‘‘treats the forecasts as though they are interchangeable: i.e., indistinguishable from one another’’ (Gupta & Wilton, 1987, p. 357). In practice, the component forecast models may differ from one another in terms of model structure and individual forecast accuracy. If it is known that some component models in a combined forecast are more accurate than other component models, the weights in the combined forecast should somehow reflect this fact. Obviously, a disadvantage of the simple average approach is that it provides no information about the relative performance of the component models. Like the simple average, the ‘‘outperformance’’ technique proposed by Bunn (1975) is easy to use; it also avoids the ‘‘exchangeability’’ assumption inherent in an equal weighting approach. The ‘‘outperformance’’ method is based on a weighted average in which each weight is the proportion of time the corresponding forecast model performed the best in the past. Thus, an advantage of the outperformance method is that it utilizes individual forecast weights, which have intuitive meaning for the decision-maker. An additional advantage of this method is that it performs well when data are sparse. Finally, this method is well suited to situations in which expert judgment influences the choice of the weights (Bunn, 1985; De Menzes, Bunn, & Taylor, 2000). Despite its advantages, the outperformance model does not utilize all available information about the component forecasts. Specifically, ‘‘information regarding the relative performances within the set of outperformed models is ignored’’ (Gupta & Wilton, 1987, p. 359). The optimal method for forecast combination proposed by Bates and Granger (1969) represents another common weighted average approach. In this model, linear weights are constructed to minimize the error variance of the forecast combination, assuming that individual forecasts are unbiased. Granger and Ramanthan (1984) have demonstrated that the optimal method is equivalent to a least squares regression model, which omits the intercept and constrains the weights to sum to one. They have also noted that

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the optimal method may fail to produce an unbiased forecast when component forecasts are biased. In addition, De Menzes et al. (2000) have observed that this method requires the covariance matrix S of forecast errors to be properly estimated. They argued that ‘‘in practice, S is often not stationary, in which case it is estimated on the basis of a short history of forecasts and thus becomes an adaptive approach to combining forecasts’’ (p. 192). The OLS regression models constitute another frequently used methodology for forecast combination. In this approach, component forecasts serve as the independent variables, while the observed value for the forecasted variable is the dependent variable (Coulson & Robins, 1993). Thus, if two component forecasts are used in the combination, the model has the form: yt ¼ b0 þ b1 f 1t þ b2 f 2t þ et

(2)

where yt is the actual value of the forecasted variable for period t; fit the forecast for period t generated by component forecast model i; b0 the constant term; bi the regression coefficient for component forecast i; et the error term for period t. De Menzes et al. (2000, p. 192) report that this regression approach is superior to the optimal model because ‘‘an unbiased combined forecast is produced regardless of whether the constituent forecasts are biased.’’ However, Narula and Korhonen (1994, p. 71) have argued that, despite its popularity, the OLS model may not be the best regression approach in some instances. They have observed that the OLS model ‘‘implicitly assumes that the loss function is proportional to the square of the errors’’ and that it ‘‘is known that in many situations the quadratic loss function is inappropriate.’’ Furthermore, the results of OLS forecasts are frequently reported in terms of relative percentage error, which is based on the absolute value of the ratio of the error term to the observed value. Given this practice, Narula and Korhonen (1994) conclude that ‘‘it is more appropriate to consider a loss function proportional to the absolute value of the errors rather than the square of the errors’’ (p. 71). They recommend that LAV regression be considered as an alternative to OLS regression since LAV regression finds the coefficients and constant term which minimize the sum of the absolute values of the error terms. The LAV model has long been viewed as an alternative to OLS regression; however, in contrast with OLS regression, there are no formulas for estimating the slope and intercept of the LAV regression line. Several algorithms exist for calculating these estimates (Birkes & Dodge, 1993); in particular, the linear (goal) programming approach has received much

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attention (Hanna, 1992). Regression lines estimated by OLS are ‘‘more severely affected by outliers or extreme data points’’ than those estimated with the LAV model; furthermore, the LAV method may provide better estimates of regression coefficients than the OLS model when normality assumptions are not met (Dielman & Pfaffenberger, 1998, p. 734). A final approach to forecast combination relies on the use of expert judgment. In their study, Flores and White (1989) found that the accuracy of subjective forecast combinations equaled and at times surpassed the accuracy of such traditional mathematical methods as the simple average and the optimal method. They also suggested that at most four component forecasts be included in the combination process.

CASE STUDY Competition within the cellular phone industry has increased dramatically during the past several years. Cell phone customers find it easier than ever to move from one service provider to another, alter their service usage patterns or substitute an alternative service product for the current one (Fildes & Kumar, 2002). Low customer loyalty and increasing phone replacement rates have resulted in an average churn rate of 30% in recent years (Kumar, Nagpal, & Venkatsen, 2002). This has forced providers of cellular phone services to intensify their efforts in attracting new customers while maintaining current customers. The cellular phone company which served as the research context for this case study, faced such a challenge. Located in the southeast United States, this company once had only one major competitor in its market and had characterized demand for its services as ‘‘limitless.’’ This is no longer the case. Now the company is uncertain about how demand will grow in the future. In response to this uncertainty, the company is conducting customer surveys and focus groups to a greater extent than in the past. In addition, improved forecasting methodologies are needed. The company is particularly interested in improving its forecasts of total revenue, which is the sum of phone access and phone usage revenues. Working with recent company data on total revenue, the authors developed combined forecast models for this situation and then examined the accuracy of these combination models. Two years of monthly total revenue figures comprised the data set. The research effort proceeded in the following stages:

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Stage 1: Data Preparation The raw data was smoothed to adjust for the fact that the number of days per month varied. An index was developed for each month to ‘‘deseasonalize’’ the data. Plots of the deseasonalized data indicated the presence of a trend in total revenue. The adjusted revenue data is denoted by r(t).

Stage 2: Generation of the Component Forecast Models The first 12 monthly total revenue values (i.e., Year 1 data) were used to develop two individual forecast models. The first model, REG(t), was a simple linear regression model, which estimated a trend line for the 12 values. The second model, SEST(t), was an exponentially smoothed forecast with trend adjustment. In the development of this model, a search was conducted to determine the values of the smoothing constants (a and b) which ensured the greatest accuracy.

Stage 3: Generation of Alternative Combined Forecasts Each model developed in Stage 2 was used to generate a series of monthly revenue forecasts for Year 1. These forecasts were then used as input to two alternative models for forecast combination. The first combined model was the frequently used multiple regression model described in (2). The second combined model was an LAV regression model, denoted by weighted least absolute value (WLAV), which was developed by the authors. For month t ¼ 2; 3; . . . ; 12 in Year 1, the model formulates a weighted absolute deviation from the WLAV forecast to each of the pair of values obtained from the two component forecasts. Linear (goal) programming approach was used to find the slope and intercept for the regression equation, which minimized the sum of the weighted absolute deviations. The model formulation is given in Exhibit 1. For each month t ¼ 2; 3; . . . ; 12; weights w1(t) and w2(t) were computed for the deviations from the WLAV regression line to each of the two component forecasts: f1(t) ¼ SEST(t) and f2(t) ¼ REG(t). Determination of these weights reflects two basic ideas. (1) The component forecast, which is closest to the actual revenue, should have the largest deviation weight. Hence, the weights reflect the accuracy of the component forecasts. (2) In determining the WLAV regression equation for the current time period, the

A Forecast Combination Methodology for Demand Forecasting

Exhibit 1.

283

WLAV Regression Model.

Let WLAV(t) denote the weighted LAV forecast for month(t). The form of the WLAV regression equation is given as WLAV(t) ¼ m  t+b, where m and b are the slopes and constant terms of the regression line. The linear programming formulation is given: Let o(t) and u(t) denote the deviations (over or under) of WLAV(t) from SEST(t) in month(t), t ¼ 2, y ,12. Let a(t) and b(t) denote the deviations (over or under) of WLAV(t) from REG(t) in month(t), t ¼ 2, y ,12. Let w1(t) and w2(t) denote the weights assigned to the absolute deviations of WLAV(t) from SEST(t) and REG(t), respectively, t ¼ 2, y , 12. Objective X X minimize w1ðtÞ  ðoðtÞ þ uðtÞÞ þ w2ðtÞ  ðaðtÞ þ bðtÞÞ; t ¼ 2; . . . ; 12. Constraints ðm  t þ bÞ þ uðtÞ
oðtÞ ¼ SESTðtÞ;

t ¼ 2; . . . ; 12

ðm  t þ bÞ þ bðtÞ
aðtÞ ¼ REGðtÞ;

t ¼ 2; . . . ; 12

u(t), o(t), b(t), a(t)Z0, all t; m, b unrestricted in value.

weights associated with earlier time periods should be reduced. Hence, deviations from component forecasts for earlier time periods will be less important when establishing the regression line. The following formulas were used in computing weights wi(t). For each month t ¼ 2; 3; . . . 12 and each component forecast fi(t), compute: 1. Fraction absolute error: eiðtÞ ¼

jrðtÞ
fiðtÞj rðtÞ

(3)

2. Fraction accurate: aiðtÞ ¼ 1
eiðtÞ

(4)

3. Time period weight: wðtÞ ¼ b12
t ;

0ob  1

(5)

(In our example b ¼ 0:9) 4. Absolute deviation weight: wiðtÞ ¼ wðtÞ  aiðtÞ

(6)

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J. GAYLORD MAY AND JOANNE M. SULEK

The model uses time (months) as the independent variable. In particular, the WLAV model developed in this study differs from traditional applications of LAV regression to forecast combination, in that the individual forecasts were not used as predictor variables.

Stage 4: Model Comparisons in Year 1 Year 1 reseasonalized monthly forecasts generated by the OLS and WLAV methods are given in Table 1 and are illustrated in Fig. 1. Year 1 results revealed that the OLS combination forecast produced a smaller MAD (582562.4803) than the WLAV method MAD (892207.9364). Year 1 results also showed that the MAPE for the WLAV model (0.1076) exceeded the MAPE of the OLS model (0.0763).

Stage 5: Generation of Forecasts for Year 2 The OLS and WLAV models found in Stage 3 were used to predict monthly total revenue over a 12-month forecast horizon (Year 2). The 12-month forecasts for Year 2 were then compared to the actual revenues that ultimately materialized over Year 2 (see Table 1 and Fig. 1). The WLAV method clearly outperformed the OLS approach with the MAD for the OLS model (3149967.079) nearly three times the MAD for the WLAV model (1063622.78). In addition, the MAPE for the WLAV model (0.05187) was less than one-third the MAPE for the OLS model (0.16862).

DISCUSSION Results indicate that while the OLS method appeared more accurate than the WLAV approach initially, the WLAV method actually provided much better predictions of total revenue for the forecast horizon (Year 2). Both the MAD and the MAPE for the WLAV model were smaller in Year 2 than the corresponding values for the OLS model. Additional research is needed to determine if the WLAV approach will continue to outperform the OLS method for other variables of interest to this company. Such variables include peak minutes used, minutes per customer and number of customers. Additional research is also needed to ascertain how well the WLAV approach compares to other common methods for forecast combination.

Multiple Regression and WLAV Regression Forecasts.

Month

Actual Revenue

Multiple Regression Forecasts

WLAV Regression Forecasts

Multiple Regression MAD (Months 2– 12)

2 3 4 5 6 7 8 9 10 11 12

13166957 12912701 13858640 14167864 14871489 14245411 14751254 14881695 15140529 16497933 15330133

12925582.44 13452705.28 14122918.88 15129021.03 14064428.26 13614506.28 14424827.56 14081742.96 16006107.51 16210616.39 16014266.25

12262015 13176757.13 13478398.5 15092424.63 13411125 15068242.75 15384248 14272957.5 16744270.25 14847512.5 15891529.5

241374.2363 540004.5151 264278.5456 961156.9389 807061.1815 630904.6019 326426.2268 799952.2903 865578.8517 287316.684 684133.2114

13 14 15 16 17 18 19 20 21 22 23 24

15532047 15949844 15496587 18787710 19328223 19356764 18677290 18877698 18789560 18794540 21149140 20763433

17460140.25 16459665.14 18023267.1 20592627.99 21359612.12 21120889.65 22893630.43 22585179.85 23317348.84 26339895.78 22421482.95 26728701.23

16193170.88 14923877.75 15996622.5 17912290 18228295.25 17701377.75 18860305.75 18304660.5 18606301.88 20708699.75 17380100.38 20440332

MAD ¼ 82562.4803

Multiple Regression MAD (Months 13– 24)

WLAV Regression MAD (Months 2– 12)

WLAV Regression MAD (Months 13– 24)

904941.68 264056.365 380241.83 924560.535 1460364.44 822831.87 632994.21 608737.75 1603741.59 1650420.57 561396.46 1928093.235 509820.8721 2526680.414 1804917.5 2031388.854 1764125.7 4216340.503 3707481.384 4527789.211 7545356.209 1272343.146 5965267.918

892207.9364

661123.855 1025966.52 500035.81 875420.49 1099928.02 1655386.2 183015.82 573037.97 183257.755 1914160.18 3769039.425 323101.31

A Forecast Combination Methodology for Demand Forecasting

Table 1.

MAD ¼ 3149967.079 MAD ¼ 1063622.78

285

286

J. GAYLORD MAY AND JOANNE M. SULEK 30000000 25000000

Revenue

20000000 15000000 10000000 5000000 0

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Month Actual Revenue

Fig. 1.

Mult. Reg. Forecasts

WLAV Reg. Forecasts

Multiple Regression and WLAV Forecasts vs. Actual Revenue.

Since this study represents only an initial investigation of the use of combination methods for services forecasting, it is impossible to conclude that the WLAV approach is a superior methodology. However, it should be noted that the WLAV model possesses some advantages that the OLS model does not. Unlike the OLS model, the WLAV model can be written as a linear program and sensitivity analysis can be used to determine the effect of an outlier or misspecified point on model parameters. Sensitivity analysis can also indicate whether a data point can be dropped from the WLAV analysis entirely without affecting the WLAV line. This may prove useful when the forecaster must consider how to deal with a misspecified point. In summary, this study offers only preliminary results on the performance of WLAV regression and OLS regression as combination methodologies in service forecasting. The initial findings presented in this paper do suggest, however, that the WLAV model developed in this paper has potential as a forecasting tool.

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