Handbook of Financial Time Series

Handbook of Financial Time Series

Torben G. Andersen • Richard A. Davis Jens-Peter Kreiß • Thomas Mikosch Editors

Handbook of Financial Time Series

Torben G. Andersen Department of Finance Kellogg School of Management Northwestern University 2001 Sheridan Road Evanston, IL 60208 U.S.A. [email protected]

Richard A. Davis Department of Statistics Columbia University 1255 Amsterdam Avenue New York, NY 10027 U.S.A. [email protected]

Jens-Peter Kreiß Institut für Mathematische Stochastik Technische Universität Braunschweig Pockelsstrasse 14 38106 Braunschweig Germany [email protected]

Thomas Mikosch Department Mathematics University of Copenhagen Universitetsparken 5 2100 Copenhagen Denmark [email protected]

ISBN 978-3-540-71296-1

e-ISBN 978-3-540-71297-8

Library of Congress Control Number: 2008943984 © 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Foreword

The Handbook of Financial Time Series, edited by Andersen, Davis, Kreiss and Mikosch, is an impressive collection of survey articles by many of the leading contributors to the field. These articles are mostly very clearly written and present a sweep of the literature in a coherent pedagogical manner. The level of most of the contributions is mathematically sophisticated, and I imagine many of these chapters will find their way onto graduate reading lists in courses in financial economics and financial econometrics. In reading through these papers, I found many new insights and presentations even in areas that I know well. The book is divided into five broad sections: GARCH-Modeling, Stochastic Volatility Modeling, Continuous Time Processes, Cointegration and Unit Roots, and Special Topics. These correspond generally to classes of stochastic processes that are applied in various finance contexts. However, there are other themes that cut across these classes. There are several papers that carefully articulate the probabilistic structure of these classes, while others are more focused on estimation. Still others derive properties of extremes for each class of processes, and evaluate persistence and the extent of long memory. Papers in many cases examine the stability of the process with tools to check for breaks and jumps. Finally there are applications to options, term structure, credit derivatives, risk management, microstructure models and other forecasting settings. The GARCH family of models is nicely introduced by Teräsvirta and then the mathematical underpinning is elegantly and readably presented by Lindner with theorems on stationarity, ergodicity and tail behavior. In the same vein, Giraitis, Leipus and Surgailis examine the long memory properties of infinite order ARCH models with memory decay slower than GARCH, and Davis and Mikosch derive tail properties of GARCH models showing that they satisfy a power law and are in the maximum domain of attraction of the Fréchet distribution. The multivariate GARCH family is well surveyed by Silvennoinen and Teräsvirta. Linton and Čížek and Spokoiny, respectively, specify models which are non- or semi-parametric or which are only constant over intervals of homogeneity.

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Foreword

The section on Stochastic Volatility Modelling (SV) brings us up to date on the development of alternatives to GARCH style models. Davis and Mikosch in two chapters develop the somewhat easier underlying mathematical theory and tail properties of SV. They derive an important difference from GARCH models. While both stochastic volatility and GARCH processes exhibit volatility clustering, only the GARCH has clustering of extremes. Long memory is conveniently described by SV models in Hurvich and Soulier. Chib, Omori and Asai extend these analyses to multivariate systems although they do not envision very high dimensions. Estimation is covered in several chapters by Renault, Shephard and Andersen, and Jungbacker and Koopman. The continuous time analysis begins with familiar Brownian motion processes and enhances them with jumps, dynamics, time deformation, correlation with returns and Lévy process innovations. Extreme value distributions are developed and estimation algorithms for discretely sampled processes are analyzed. Lindner discusses the idea of continuous time approximations to GARCH and SV models showing that the nature of the approximation must be carefully specified. The continuous time framework is then applied to several finance settings such as interest rate models by Björk, option pricing by Kallsen, and realized volatility by Andersen and Benzoni. The book then returns to analysis of first moments with surveys of discrete time models with unit roots, near unit roots, fractional unit roots and cointegration. Finally, a remaining 13 chapters are collected in a section called Special Topics. These include very interesting chapters on copulas, non-parametric models, resampling methods, Markov switching models, structural break models and model selection. Patton and Sheppard examine univariate and multivariate volatility forecast comparisons. They show the advantages of a GLS correction, discuss multiple comparisons and economic loss functions. Bauwens and Hautsch survey a wide range of models for point processes that have been used in the finance literature to model arrival times of trades and quotes. The survey is well grounded in the statistical literature and the economics literature. Embrechts, Furrer and Kaufmann discuss different types of risk— market, credit, operational and insurance—and some of the leading approaches to estimation. Christoffersen applies the filtered historical simulation or FHS method to univariate and multivariate simulation based calculation of VaR, Expected Shortfall and active portfolio risks. Lando surveys the structural and reduced form approaches to modeling credit spreads. He focuses on CDS spreads and default dependence and gives a nice description of tests between contagion and factor structures in formulating dependence. So make yourself a double cappuccino and relax in a comfortable chair, or adjust your headphones at 30,000 ft. over the Pacific, and dig in. There are treats in lots of different areas just waiting to be discovered.

New York, September 2008

Robert Engle

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torben G. Andersen, Richard A. Davis, Jens-Peter Kreiss and Thomas Mikosch References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

13

Part I Recent Developments in GARCH Modeling An Introduction to Univariate GARCH Models . . . . . . . . . . . . . Timo Teräsvirta 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Generalized ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Why Generalized ARCH? . . . . . . . . . . . . . . . . . . . . . . 3.2 Families of univariate GARCH models . . . . . . . . . . . 3.3 Nonlinear GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Time-varying GARCH . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Markov-switching ARCH and GARCH . . . . . . . . . . . 3.6 Integrated and fractionally integrated GARCH . . . 3.7 Semi- and nonparametric ARCH models . . . . . . . . . 3.8 GARCH-in-mean model . . . . . . . . . . . . . . . . . . . . . . . 3.9 Stylized facts and the first-order GARCH model . . 4 Family of Exponential GARCH Models . . . . . . . . . . . . . . . . . 4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stylized facts and the first-order EGARCH model . 4.3 Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Comparing EGARCH with GARCH . . . . . . . . . . . . . . . . . . . . 6 Final Remarks and Further Reading . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)–Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander M. Lindner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 19 20 23 26 27 28 30 30 31 34 34 35 36 37 38 39 43 43

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2

Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Strict stationarity of ARCH(1) and GARCH(1, 1) . 2.2 Strict stationarity of GARCH(p, q) . . . . . . . . . . . . . . 2.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Weak stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The ARCH(∞) Representation and the Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence of Moments and the Autocovariance Function of the Squared Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Moments of ARCH(1) and GARCH(1, 1) . . . . . . . . . 4.2 Moments of GARCH(p, q) . . . . . . . . . . . . . . . . . . . . . . 4.3 The autocorrelation function of the squares . . . . . . 5 Strong Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Some Distributional Properties . . . . . . . . . . . . . . . . . . . . . . . . . 7 Models Defined on the Non-Negative Integers . . . . . . . . . . . . 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45 49 52 53 54 55 56 57 60 62 64 66 67 67

ARCH(∞) Models and Long Memory Properties . . . . . . . . . . . . Liudas Giraitis, Remigijus Leipus and Donatas Surgailis 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stationary ARCH(∞) Process . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Volterra representations . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dependence structure, association, and central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Infinite variance and integrated ARCH(∞) . . . . . . . 3 Linear ARCH and Bilinear Model . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Tour in the Asymptotic Theory of GARCH Estimation . . . Christian Francq and Jean-Michel Zakoïan 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Least–Squares Estimation of ARCH Models . . . . . . . . . . . . . 3 Quasi–Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . 3.1 Pure GARCH models . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 ARMA–GARCH models . . . . . . . . . . . . . . . . . . . . . . . 4 Efficient Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Alternative Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Self–weighted LSE for the ARMA parameters . . . . 5.2 Self–weighted QMLE . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Lp –estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Least absolute deviations estimators . . . . . . . . . . . . . 5.5 Whittle estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Moment estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Properties of Estimators when some GARCH Coefficients are Equal to Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

71 73 73 75 77 79 82

85 87 89 90 94 95 99 100 100 101 102 103 104 104

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6.1 6.2 6.3

Fitting an ARCH(1) model to a white noise . . . . . . On the need of additional assumptions . . . . . . . . . . . Asymptotic distribution of the QMLE on the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Application to hypothesis testing . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Eric Zivot 1 2 3

4

5

6

7

8

105 106 106 107 109 109

Issues in the Analysis of Univariate GARCH Models 113 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Stylized Facts of Asset Returns . . . . . . . . . . . . . . . . . . . The ARCH and GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conditional mean specification . . . . . . . . . . . . . . . . . . 3.2 Explanatory variables in the conditional variance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The GARCH model and stylized facts of asset returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Temporal aggregation . . . . . . . . . . . . . . . . . . . . . . . . . Testing for ARCH/GARCH Effects . . . . . . . . . . . . . . . . . . . . . 4.1 Testing for ARCH effects in daily and monthly returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Numerical accuracy of GARCH estimates . . . . . . . . 5.2 Quasi-maximum likelihood estimation . . . . . . . . . . . 5.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Evaluation of estimated GARCH models . . . . . . . . . 5.5 Estimation of GARCH models for daily and monthly returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GARCH Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Asymmetric leverage effects and news impact . . . . . 6.2 Non-Gaussian error distributions . . . . . . . . . . . . . . . . Long Memory GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Testing for long memory . . . . . . . . . . . . . . . . . . . . . . . 7.2 Two component GARCH model . . . . . . . . . . . . . . . . . 7.3 Integrated GARCH model . . . . . . . . . . . . . . . . . . . . . 7.4 Long memory GARCH models for daily returns . . . GARCH Model Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 GARCH and forecasts for the conditional mean . . . 8.2 Forecasts from the GARCH(1,1) model . . . . . . . . . . 8.3 Forecasts from asymmetric GARCH(1,1) models . . 8.4 Simulation-based forecasts . . . . . . . . . . . . . . . . . . . . . 8.5 Forecasting the volatility of multiperiod returns . . . 8.6 Evaluating volatility predictions . . . . . . . . . . . . . . . .

113 114 115 118 119 119 121 121 122 123 125 126 126 127 127 131 131 135 137 139 139 140 141 142 142 143 144 145 145 146

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8.7

Forecasting the volatility of Microsoft and the S&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Semiparametric and Nonparametric ARCH Modeling . . . . . . . Oliver B. Linton 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Nonparametric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Error density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Functional form of volatility function . . . . . . . . . . . . 3.3 Relationship between mean and variance . . . . . . . . . 3.4 Long memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Locally stationary processes . . . . . . . . . . . . . . . . . . . . 3.6 Continuous time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Varying Coefficient GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . Pavel Čížek and Vladimir Spokoiny 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conditional Heteroscedasticity Models . . . . . . . . . . . . . . . . . . 2.1 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Test of homogeneity against a change–point alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adaptive Nonparametric Estimation . . . . . . . . . . . . . . . . . . . . 3.1 Adaptive choice of the interval of homogeneity . . . . 3.2 Parameters of the method and the implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Real–Data Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Finite–sample critical values for the test of homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stock index S&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme Value Theory for GARCH Processes . . . . . . . . . . . . . . . Richard A. Davis and Thomas Mikosch 1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Strict Stationarity and Mixing Properties . . . . . . . . . . . . . . . 3 Embedding a GARCH Process in a Stochastic Recurrence Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Tails of a GARCH Process . . . . . . . . . . . . . . . . . . . . . . . . 5 Limit Theory for Extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Convergence of maxima . . . . . . . . . . . . . . . . . . . . . . . .

157 157 157 158 158 159 162 163 164 164 165 165 169 169 171 173 173 175 176 176 179 179 180 183 183 187 187 188 189 190 194 194

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5.2 Convergence of point processes . . . . . . . . . . . . . . . . . 195 5.3 The behavior of the sample autocovariance function 197 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Multivariate GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annastiina Silvennoinen and Timo Teräsvirta 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Models of the conditional covariance matrix . . . . . . 2.2 Factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Models of conditional variances and correlations . . 2.4 Nonparametric and semiparametric approaches . . . 3 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hypothesis Testing in Multivariate GARCH Models . . . . . . 4.1 General misspecification tests . . . . . . . . . . . . . . . . . . . 4.2 Tests for extensions of the CCC–GARCH model . . 5 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 203 204 207 210 215 218 218 219 221 222 224 226

Part II Recent Developments in Stochastic Volatility Modeling Stochastic Volatility: Origins and Overview . . . . . . . . . . . . . . . . . . Neil Shephard and Torben G. Andersen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Origin of SV Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Second Generation Model Building . . . . . . . . . . . . . . . . . . . . . 3.1 Univariate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multivariate models . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Inference Based on Return Data . . . . . . . . . . . . . . . . . . . . . . . 4.1 Moment–based inference . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simulation–based inference . . . . . . . . . . . . . . . . . . . . . 5 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic Properties of Stochastic Volatility Models . . . . . . Richard A. Davis and Thomas Mikosch 1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stationarity, Ergodicity and Strong Mixing . . . . . . . . . . . . . . 2.1 Strict stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ergodicity and strong mixing . . . . . . . . . . . . . . . . . . . 3 The Covariance Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Moments and Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Asymptotic Theory for the Sample ACVF and ACF . . . . . .

233 233 235 240 240 241 242 242 243 246 246 247 250 255 255 256 256 257 258 261 263

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Moment–Based Estimation of Stochastic Volatility Models . . . Eric Renault 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Use of a Regression Model to Analyze Fluctuations in Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The linear regression model for conditional variance 2.2 The SR–SARV(p) model . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Exponential SARV model . . . . . . . . . . . . . . . . . . 2.4 Other parametric SARV models . . . . . . . . . . . . . . . . . 3 Implications of SV Model Specification for Higher Order Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fat tails and variance of the variance . . . . . . . . . . . . 3.2 Skewness, feedback and leverage effects . . . . . . . . . . 4 Continuous Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Measuring volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Moment-based estimation with realized volatility . . 4.3 Reduced form models of volatility . . . . . . . . . . . . . . . 4.4 High frequency data with random times separating successive observations . . . . . . . . . . . . . . . . . . . . . . . . 5 Simulation–Based Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Simulation-based bias correction . . . . . . . . . . . . . . . . 5.2 Simulation-based indirect inference . . . . . . . . . . . . . . 5.3 Simulated method of moments . . . . . . . . . . . . . . . . . . 5.4 Indirect inference in presence of misspecification . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter Estimation and Practical Aspects of Modeling Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Borus Jungbacker and Siem Jan Koopman 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Quasi-Likelihood Analysis Based on Kalman Filter Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kalman filter for prediction and likelihood evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Smoothing methods for the conditional mean, variance and mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Practical considerations for analyzing the linearized SV model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Monte Carlo Likelihood Analysis . . . . . . . . . . . . . . . . . . . . . 3.1 Construction of a proposal density . . . . . . . . . . . . . . 3.2 Sampling from the importance density and Monte Carlo likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Some Generalizations of SV Models . . . . . . . . . . . . . . . . . . . . .

269 270 272 272 274 277 279 281 281 284 286 287 288 292 293 295 296 298 300 304 305 307 313 313 316 319 320 321 322 323 325 327

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4.1 Basic SV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Multiple volatility factors . . . . . . . . . . . . . . . . . . . . . . 4.3 Regression and fixed effects . . . . . . . . . . . . . . . . . . . . 4.4 Heavy-tailed innovations . . . . . . . . . . . . . . . . . . . . . . . 4.5 Additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Leverage effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Stochastic volatility in mean . . . . . . . . . . . . . . . . . . . 5 Empirical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Standard & Poor’s 500 stock index: volatility estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Standard & Poor’s 500 stock index: regression effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Daily changes in exchange rates: dollar–pound and dollar–yen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 328 329 330 331 331 333 333

Stochastic Volatility Models with Long Memory . . . . . . . . . . . . . Clifford M. Hurvich and Philippe Soulier 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Properties of the LMSV Model . . . . . . . . . . . . . . . . . . . 3 Parametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Semiparametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Generalizations of the LMSV Model . . . . . . . . . . . . . . . . . . . . 6 Applications of the LMSV Model . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

334 335 337 340 340 342

345 346 347 349 352 352 353

Extremes of Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . Richard A. Davis and Thomas Mikosch 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Tail Behavior of the Marginal Distribution . . . . . . . . . . 2.1 The light-tailed case . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The heavy-tailed case . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Point Process Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Application to stochastic volatility models . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

355

Multivariate Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siddhartha Chib, Yasuhiro Omori and Manabu Asai 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic MSV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 No-leverage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Leverage effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Heavy-tailed measurement error models . . . . . . . . . .

365

355 356 356 357 358 358 360 364

366 369 369 373 377

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3

Factor MSV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Volatility factor model . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mean factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bayesian analysis of mean factor MSV model . . . . . 4 Dynamic Correlation MSV Model . . . . . . . . . . . . . . . . . . . . . . 4.1 Modeling by reparameterization . . . . . . . . . . . . . . . . 4.2 Matrix exponential transformation . . . . . . . . . . . . . . 4.3 Wishart process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

379 379 382 384 388 388 390 391 396 397

Part III Topics in Continuous Time Processes An Overview of Asset–Price Models . . . . . . . . . . . . . . . . . . . . . . . . . Peter J. Brockwell 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Shortcomings of the BSM Model . . . . . . . . . . . . . . . . . . . . . . . 3 A General Framework for Option Pricing . . . . . . . . . . . . . . . . 4 Some Non-Gaussian Models for Asset Prices . . . . . . . . . . . . . 5 Further Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ornstein–Uhlenbeck Processes and Extensions . . . . . . . . . . . . . . . Ross A. Maller, Gernot Müller and Alex Szimayer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 OU Process Driven by Brownian Motion . . . . . . . . . . . . . . . . 3 Generalised OU Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Background on bivariate Lévy processes . . . . . . . . . 3.2 Lévy OU processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Self-decomposability, self-similarity, class L, Lamperti transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Discretisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Autoregressive representation, and perpetuities . . . 4.2 Statistical issues: Estimation and hypothesis testing 4.3 Discretely sampled process . . . . . . . . . . . . . . . . . . . . . 4.4 Approximating the COGARCH . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jump–Type Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ernst Eberlein 1 Probabilistic Structure of Lévy Processes . . . . . . . . . . . . . . . . 2 Distributional Description of Lévy Processes . . . . . . . . . . . . . 3 Financial Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples of Lévy Processes with Jumps . . . . . . . . . . . . . . . . 4.1 Poisson and compound Poisson processes . . . . . . . .

403 404 409 410 411 415 416 421 422 422 424 424 426 429 430 430 431 431 432 435 435 439 439 443 446 449 449

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4.2 Lévy jump diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hyperbolic Lévy processes . . . . . . . . . . . . . . . . . . . . . 4.4 Generalized hyperbolic Lévy processes . . . . . . . . . . . 4.5 CGMY and variance gamma Lévy processes . . . . . . 4.6 α-Stable Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Meixner Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

450 450 451 452 453 453 454

Lévy–Driven Continuous–Time ARMA Processes . . . . . . . . . . . . Peter J. Brockwell 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Second–Order Lévy–Driven CARMA Processes . . . . . . . . . . 3 Connections with Discrete–Time ARMA Processes . . . . . . . 4 An Application to Stochastic Volatility Modelling . . . . . . . . 5 Continuous–Time GARCH Processes . . . . . . . . . . . . . . . . . . . 6 Inference for CARMA Processes . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

457

Continuous Time Approximations to GARCH and Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander M. Lindner 1 Stochastic Volatility Models and Discrete GARCH . . . . . . . 2 Continuous Time GARCH Approximations . . . . . . . . . . . . . . 2.1 Preserving the random recurrence equation property 2.2 The diffusion limit of Nelson . . . . . . . . . . . . . . . . . . . 2.3 The COGARCH model . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Weak GARCH processes . . . . . . . . . . . . . . . . . . . . . . . 2.5 Stochastic delay equations . . . . . . . . . . . . . . . . . . . . . 2.6 A continuous time GARCH model designed for option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Continuous Time Stochastic Volatility Approximations . . . . 3.1 Sampling a continuous time SV model at equidistant times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Approximating a continuous time SV model . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Likelihood and Gaussian Estimation of Continuous Time Models in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter C. B. Phillips and Jun Yu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exact ML Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 ML based on the transition density . . . . . . . . . . . . . . 2.2 ML based on the continuous record likelihood . . . . 3 Approximate ML Methods Based on Transition Densities . . 3.1 The Euler approximation and refinements . . . . . . . . 3.2 Closed–form approximations . . . . . . . . . . . . . . . . . . . .

458 460 470 474 476 478 479 481 481 482 483 484 486 488 489 490 491 491 493 495 497 498 499 499 502 503 504 509

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3.3 Simulated infill ML methods . . . . . . . . . . . . . . . . . . . 3.4 Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Approximate ML Methods Based on the Continuous Record Likelihood and Realized Volatility . . . . . . . . . . . . . . . 5 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Estimation Bias Reduction Techniques . . . . . . . . . . . . . . . . . . 6.1 Jackknife estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Indirect inference estimation . . . . . . . . . . . . . . . . . . . 7 Multivariate Continuous Time Models . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric Inference for Discretely Sampled Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Sørensen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Asymptotics: Fixed Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 3 Likelihood Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Martingale Estimating Functions . . . . . . . . . . . . . . . . . . . . . . . 5 Explicit Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 High Frequency Asymptotics and Efficient Estimation . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torben G. Andersen and Luca Benzoni 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Measuring Mean Return versus Return Volatility . . . . . . . . . 3 Quadratic Return Variation and Realized Volatility . . . . . . . 4 Conditional Return Variance and Realized Volatility . . . . . . 5 Jumps and Bipower Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Efficient Sampling versus Microstructure Noise . . . . . . . . . . . 7 Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Early work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Volatility forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The distributional implications of the no-arbitrage condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Multivariate quadratic variation measures . . . . . . . . 7.5 Realized volatility, model specification and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Possible Directions for Future Research . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

512 514 516 519 520 521 522 524 527 527 531 531 532 536 538 543 548 551 555 556 557 559 561 563 564 566 566 567 568 568 569 569 570

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Estimating Volatility in the Presence of Market Microstructure Noise: A Review of the Theory and Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yacine Aït-Sahalia and Per A. Mykland 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The parametric volatility case . . . . . . . . . . . . . . . . . . 2.2 The nonparametric stochastic volatility case . . . . . . 3 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multi-scale realized volatility . . . . . . . . . . . . . . . . . . . 3.2 Non-equally spaced observations . . . . . . . . . . . . . . . . 3.3 Serially-correlated noise . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Noise correlated with the price signal . . . . . . . . . . . . 3.5 Small sample edgeworth expansions . . . . . . . . . . . . . 3.6 Robustness to departures from the data generating process assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Computational and Practical Implementation Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Calendar, tick and transaction time sampling . . . . . 4.2 Transactions or quotes . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Selecting the number of subsamples in practice . . . 4.4 High versus low liquidity assets . . . . . . . . . . . . . . . . . 4.5 Robustness to data cleaning procedures . . . . . . . . . . 4.6 Smoothing by averaging . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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577 577 579 579 582 585 585 586 587 589 591 591 592 592 592 593 594 594 595 596 596

Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Kallsen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Arbitrage Theory from a Market Perspective . . . . . . . . . . . . . 3 Martingale Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Arbitrage Theory from an Individual Perspective . . . . . . . . . 5 Quadratic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Utility Indifference Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

599

An Overview of Interest Rate Theory . . . . . . . . . . . . . . . . . . . . . . . Tomas Björk 1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Interest Rates and the Bond Market . . . . . . . . . . . . . . . . . . . . 3 Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Modeling under the Objective Measure P . . . . . . . . . . . . . . . 4.1 The market price of risk . . . . . . . . . . . . . . . . . . . . . . . 5 Martingale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Affine term structures . . . . . . . . . . . . . . . . . . . . . . . . .

615

599 600 603 605 606 607 611

615 618 620 621 622 623 624

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5.2 Short rate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inverting the yield curve . . . . . . . . . . . . . . . . . . . . . . . 6 Forward Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The HJM drift condition . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Musiela parameterization . . . . . . . . . . . . . . . . . . 7 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Forward measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 LIBOR Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Caps: definition and market practice . . . . . . . . . . . . 8.2 The LIBOR market model . . . . . . . . . . . . . . . . . . . . . 8.3 Pricing caps in the LIBOR model . . . . . . . . . . . . . . . 8.4 Terminal measure dynamics and existence . . . . . . . . 9 Potentials and Positive Interest . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Flesaker–Hughston fractional model . . . . . . . . . 9.3 Connections to the Riesz decomposition . . . . . . . . . 9.4 Conditional variance potentials . . . . . . . . . . . . . . . . . 9.5 The Rogers Markov potential approach . . . . . . . . . . 10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

625 627 629 629 631 632 632 635 635 638 638 640 641 641 642 642 644 646 647 648 650 651

Extremes of Continuous–Time Processes . . . . . . . . . . . . . . . . . . . . . Vicky Fasen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Extremes of discrete–time processes . . . . . . . . . . . . . 2.2 Extremes of continuous–time processes . . . . . . . . . . 2.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Generalized Ornstein-Uhlenbeck (GOU)–Model . . . . . . 3.1 The Ornstein–Uhlenbeck process . . . . . . . . . . . . . . . . 3.2 The non–Ornstein–Uhlenbeck process . . . . . . . . . . . . 3.3 Comparison of the models . . . . . . . . . . . . . . . . . . . . . . 4 Tail Behavior of the Sample Maximum . . . . . . . . . . . . . . . . . . 5 Running sample Maxima and Extremal Index Function . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

653 653 654 655 656 656 657 658 659 661 661 663 664 665

Part IV Topics in Cointegration and Unit Roots Cointegration: Overview and Development . . . . . . . . . . . . . . . . . . 671 Søren Johansen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 1.1 Two examples of cointegration . . . . . . . . . . . . . . . . . . 672

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1.2 Three ways of modeling cointegration . . . . . . . . . . . . 1.3 The model analyzed in this article . . . . . . . . . . . . . . 2 Integration, Cointegration and Granger’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of integration and cointegration . . . . . . . 2.2 The Granger Representation Theorem . . . . . . . . . . . 2.3 Interpretation of cointegrating coefficients . . . . . . . . 3 Interpretation of the I(1) Model for Cointegration . . . . . . . . 3.1 The models H(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Normalization of parameters of the I(1) model . . . . 3.3 Hypotheses on long-run coefficients . . . . . . . . . . . . . . 3.4 Hypotheses on adjustment coefficients . . . . . . . . . . . 4 Likelihood Analysis of the I(1) Model . . . . . . . . . . . . . . . . . . . 4.1 Checking the specifications of the model . . . . . . . . . 4.2 Reduced rank regression . . . . . . . . . . . . . . . . . . . . . . . 4.3 Maximum likelihood estimation in the I(1) model and derivation of the rank test . . . . . . . . . . . . . . . . . . 5 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Asymptotic distribution of the rank test . . . . . . . . . 5.2 Asymptotic distribution of the estimators . . . . . . . . 6 Further Topics in the Area of Cointegration . . . . . . . . . . . . . 6.1 Rational expectations . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The I(2) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Series with Roots on or Near the Unit Circle . . . . . . . . . . Ngai Hang Chan 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Unit Root Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 AR(p) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Miscellaneous Developments and Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Willa W. Chen and Clifford M. Hurvich 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Type I and Type II Definitions of I(d) . . . . . . . . . . . . . . . . . . 2.1 Univariate series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multivariate series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Models for Fractional Cointegration . . . . . . . . . . . . . . . . . . . . 3.1 Parametric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Tapering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Semiparametric Estimation of the Cointegrating Vectors . .

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673 674 675 675 677 678 680 680 681 681 682 683 683 683 684 686 686 687 689 689 690 691 692 695 695 696 697 699 702 704 705 709 709 710 710 713 715 716 717 718

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Testing for Cointegration; Determination of Cointegrating Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Part V Special Topics – Risk Different Kinds of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paul Embrechts, Hansjörg Furrer and Roger Kaufmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Risk factor mapping and loss portfolios . . . . . . . . . . 3 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reduced form models . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Credit risk for regulatory reporting . . . . . . . . . . . . . . 4 Market Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Market risk models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conditional versus unconditional modeling . . . . . . . 4.3 Scaling of market risks . . . . . . . . . . . . . . . . . . . . . . . . . 5 Operational Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Insurance Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Life insurance risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Modeling parametric life insurance risk . . . . . . . . . . 6.3 Non-life insurance risk . . . . . . . . . . . . . . . . . . . . . . . . . 7 Aggregation of Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Value–at–Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Christoffersen 1 Introduction and Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . . 2 A Univariate Portfolio Risk Model . . . . . . . . . . . . . . . . . . . . . . 2.1 The dynamic conditional variance model . . . . . . . . . 2.2 Univariate filtered historical simulation . . . . . . . . . . 2.3 Univariate extensions and alternatives . . . . . . . . . . . 3 Multivariate, Base–Asset Return Methods . . . . . . . . . . . . . . . 3.1 The dynamic conditional correlation model . . . . . . . 3.2 Multivariate filtered historical simulation . . . . . . . . . 3.3 Multivariate extensions and alternatives . . . . . . . . . 4 Summary and Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729 729 732 732 735 736 737 737 738 738 739 740 740 742 744 744 745 747 748 749 750 753 753 755 756 757 759 760 761 761 763 764 764

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Copula–Based Models for Financial Time Series . . . . . . . . . . . . . Andrew J. Patton 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Copula–Based Models for Time Series . . . . . . . . . . . . . . . . . . . 2.1 Copula–based models for multivariate time series . 2.2 Copula–based models for univariate time series . . . 2.3 Estimation and evaluation of copula–based models for time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications of Copulas in Finance and Economics . . . . . . . 4 Conclusions and Areas for Future Research . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Credit Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Lando 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling the Probability of Default and Recovery . . . . . . . . 3 Two Modeling Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Credit Default Swap Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Corporate Bond Spreads and Bond Returns . . . . . . . . . . . . . 6 Credit Risk Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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767 767 771 772 773 775 778 780 781 787 787 788 789 792 795 795 797

Part V Special Topics – Time Series Methods Evaluating Volatility and Correlation Forecasts . . . . . . . . . . . . . . Andrew J. Patton and Kevin Sheppard 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Direct Evaluation of Volatility Forecasts . . . . . . . . . . . . . . . . 2.1 Forecast optimality tests for univariate volatility forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 MZ regressions on transformations of σ t2 . . . . . . . . . 2.3 Forecast optimality tests for multivariate volatility forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Improved MZ regressions using generalised least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Direct Comparison of Volatility Forecasts . . . . . . . . . . . . . . . 3.1 Pair–wise comparison of volatility forecasts . . . . . . . 3.2 Comparison of many volatility forecasts . . . . . . . . . . 3.3 ‘Robust’ loss functions for forecast comparison . . . . 3.4 Problems arising from ‘non–robust’ loss functions . 3.5 Choosing a “robust” loss function . . . . . . . . . . . . . . . 3.6 Robust loss functions for multivariate volatility comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

801 801 803 804 805 806 807 808 810 815 816 817 818 819 823 825

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3.7 Direct comparison via encompassing tests . . . . . . . . Indirect Evaluation of Volatility Forecasts . . . . . . . . . . . . . . . 4.1 Portfolio optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Tracking error minimisation . . . . . . . . . . . . . . . . . . . . 4.3 Other methods of indirect evaluation . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

828 830 831 832 833 835 835

Structural Breaks in Financial Time Series . . . . . . . . . . . . . . . . . . Elena Andreou and Eric Ghysels 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Consequences of Structural Breaks in Financial Time Series 3 Methods for Detecting Structural Breaks . . . . . . . . . . . . . . . . 3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Historical and sequential partial–sums change–point statistics . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multiple breaks tests . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Change–Point Tests in Returns and Volatility . . . . . . . . . . . . 4.1 Tests based on empirical volatility processes . . . . . . 4.2 Empirical processes and the SV class of models . . . 4.3 Tests based on parametric volatility models . . . . . . 4.4 Change–point tests in long memory . . . . . . . . . . . . . 4.5 Change–point in the distribution . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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839 840 843 844 845 848 851 851 854 858 861 863 865 866

An Introduction to Regime Switching Time Series Models . . . Theis Lange and Anders Rahbek 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Markov and observation switching . . . . . . . . . . . . . . . 2 Switching ARCH and CVAR . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Switching ARCH and GARCH . . . . . . . . . . . . . . . . . 2.2 Switching CVAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Likelihood–Based Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hannes Leeb and Benedikt M. Pötscher 1 The Model Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A general formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Model selection procedures . . . . . . . . . . . . . . . . . . . . . 2 Properties of Model Selection Procedures and of Post–Model–Selection Estimators . . . . . . . . . . . . . . . . . . . . . . . 2.1 Selection probabilities and consistency . . . . . . . . . . .

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Risk properties of post-model-selection estimators Distributional properties of post-model-selection estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Model Selection in Large- or Infinite–Dimensional Models . 4 Related Procedures Based on Shrinkage and Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonparametric Modeling in Financial Time Series . . . . . . . . . . . Jürgen Franke, Jens-Peter Kreiss and Enno Mammen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nonparametric Smoothing for Time Series . . . . . . . . . . . . . . . 2.1 Density estimation via kernel smoothing . . . . . . . . . 2.2 Kernel smoothing regression . . . . . . . . . . . . . . . . . . . . 2.3 Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonparametric Quantile Estimation . . . . . . . . . . . . . . . . . . . . 5 Advanced Nonparametric Modeling . . . . . . . . . . . . . . . . . . . . . 6 Sieve Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling Financial High Frequency Data Using Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luc Bauwens and Nikolaus Hautsch 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamental Concepts of Point Process Theory . . . . . . . . . . 2.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . . . 2.2 Compensators, intensities, and hazard rates . . . . . . 2.3 Types and representations of point processes . . . . . 2.4 The random time change theorem . . . . . . . . . . . . . . . 3 Dynamic Duration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 ACD models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dynamic Intensity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hawkes processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Autoregressive intensity processes . . . . . . . . . . . . . . . 4.3 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

903 906 908 915 916 916 927 927 929 929 932 935 937 940 942 944 947 953 953 954 955 955 956 959 960 960 963 964 965 967 967 969 973 975 976

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Contents

Part V Special Topics – Simulation Based Methods Resampling and Subsampling for Financial Time Series . . . . . . Efstathios Paparoditis and Dimitris N. Politis 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Resampling the Time Series of Log–Returns . . . . . . . . . . . . . 2.1 Parametric methods based on i.i.d. resampling of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonparametric methods based on i.i.d. resampling of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Markovian bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Resampling Statistics Based on the Time Series of Log–Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Regression bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Wild bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Local bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Subsampling and Self–Normalization . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

983

Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Johannes and Nicholas Polson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview of MCMC Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Clifford–Hammersley theorem . . . . . . . . . . . . . . . . . . 2.2 Constructing Markov chains . . . . . . . . . . . . . . . . . . . . 2.3 Convergence theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Financial Time Series Examples . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Geometric Brownian motion . . . . . . . . . . . . . . . . . . . . 3.2 Time-varying expected returns . . . . . . . . . . . . . . . . . . 3.3 Stochastic volatility models . . . . . . . . . . . . . . . . . . . . 4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1001

Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Johannes and Nicholas Polson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Particle Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Exact particle filtering . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Auxiliary particle filtering algorithms . . . . . . . . . . . . 4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

983 986 986 988 990 992 992 993 994 995 997

1001 1002 1002 1003 1007 1008 1008 1009 1010 1011 1012 1015 1015 1017 1019 1021 1024 1026 1027 1028

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031

List of Contributors

Yacine Aït-Sahalia Princeton University and NBER, Bendheim Center for Finance, Princeton University, U.S.A.. Torben G. Andersen Kellogg School of Management and NBER, Northwestern University, U.S.A. and CREATES, Aarhus, Denmark. Elena Andreou Department of Economics, University of Cyprus, Nicosia, Cyprus. Manabu Asai Faculty of Economics, Soka University, Tokyo, Japan. Luc Bauwens CORE, Université Catholique de Louvain, Belgium. Luca Benzoni Federal Reserve Bank of Chicago, U.S.A.. Thomas Björk Department of Finance, Stockholm School of Economics, Sweden. Peter J. Brockwell Department of Statistics, Colorado State University, Fort Collins, U.S.A.. Ngai Hang Chan Department of Statistics, Chinese University of Hong Kong, Shatin, NT, Hong Kong. Willa W. Chen Department of Statistics, Texas A&M University, College Station, U.S.A..

xxvi

List of Contributors

Siddhartha Chib Olin Business School, Washington University in St. Louis, U.S.A.. Peter Christoffersen Desautels Faculty of Management, McGill University, Quebec, Canada. Pavel Čížek Department of Econometrics & OR, Tilburg University, The Netherlands. Richard A. Davis Department of Statistics, Columbia University, New York, U.S.A.. Ernst Eberlein Department of Mathematical Stochastics, University of Freiburg, Germany. Paul Embrechts Department of Mathematics, ETH Zürich, Switzerland. Vicky Fasen Zentrum Mathematik, Technische Universität München, Germany. Christian Francq University Lille III, EQUIPPE-GREMARS, France. Jürgen Franke Department of Mathematics, Universität Kaiserslautern, Germany. Hansjörg Furrer Swiss Life, Zürich, Switzerland. Eric Ghysels Department of Economics, University of North Carolina at Chapel Hill, U.S.A.. Liudas Giraitis Department of Economics, Queen Mary University of London, United Kingdom. Nikolaus Hautsch Institute for Statistics and Econometrics, Humboldt–Universität zu Berlin, Germany. Clifford M. Hurvich Leonard N. Stern School of Business, New York University, U.S.A.. Michael Johannes Graduate School of Business, Columbia University, New York, U.S.A.. Søren Johansen Department of Applied Mathematics and Statistics, University of Copenhagen, Denmark.

List of Contributors

xxvii

Borus Jungbacker Department of Econometrics, Vrije Universiteit Amsterdam, The Netherlands. Jan Kallsen Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Germany. Roger Kaufmann AXA Winterthur, Winterthur, Switzerland. Siem Jan Koopman Department of Econometrics, Vrije Universiteit Amsterdam, The Netherlands. Jens-Peter Kreiss Institut für Mathematische Stochastik, Technische Universität Braunschweig, Germany. David Lando Copenhagen Business School, Department of Finance, Denmark. Theis Lange Department of Economics, University of Copenhagen, Denmark. Hannes Leeb Department of Statistics, Yale University, U.S.A.. Remigijus Leipus Vilnius University and Institute of Mathematics and Informatics, Vilnius, Lithuania. Alexander M. Lindner Technische Universität Braunschweig, Institut für Mathematische Stochastik, Germany. Oliver B. Linton Department of Economics, London School of Economics and Political Science, United Kingdom. Ross A. Maller School of Finance & Applied Statistics and Centre for Mathematics & its Applications, Australian National University, Canberra, Australia. Enno Mammen Abteilung Volkswirtschaftslehre, Universität Mannheim, Germany. Thomas Mikosch Laboratory of Actuarial Mathematics, University of Copenhagen, Denmark. Gernot Müller Zentrum Mathematik, Technische Universität München, Germany.

xxviii

List of Contributors

Per A. Mykland Department of Statistics, The University of Chicago, U.S.A.. Yasuhiro Omori Faculty of Economics, University of Tokyo, Japan. Efstathios Paparoditis Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus. Andrew J. Patton Department of Economics and Oxford-Man Institute of Quantitative Finance, University of Oxford, United Kingdom. Peter C. B. Phillips Cowles Foundation for Research in Economics, Yale University, U.S.A.; University of Auckland; University of York; and Singapore Management University. Benedikt M. Pötscher Department of Statistics, University of Vienna, Austria. Dimitris N. Politis Department of Mathematics, University of California, San Diego, U.S.A.. Nicholas Polson Graduate School of Business, University of Chicago, U.S.A.. Anders Rahbek Department of Economics, University of Copenhagen, Denmark. Eric Renault Department of Economics, University of North Carolina, Chapel Hill, U.S.A.. Neil Shephard Oxford-Man Institute and Department of Economics, University of Oxford, United Kingdom. Kevin Sheppard Department of Economics and Oxford-Man Institute of Quantitative Finance, University of Oxford, United Kingdom. Annastiina Silvennoinen School of Finance and Economics, University of Technology Sydney, Australia. Michael Sørensen Department of Mathematical Sciences, University of Copenhagen, Denmark. Philippe Soulier Department of Mathematics, University Paris X, France.

List of Contributors

xxix

Vladimir Spokoiny Weierstrass–Institut, Berlin, Germany. Donatas Surgailis Vilnius University and Institute of Mathematics and Informatics, Vilnius, Lithuania. Alex Szimayer Fraunhofer-Institut für Techno-und Wirtschaftsmathematik, Kaiserslautern, Germany. Timo Teräsvirta CREATES, School of Economics and Management, University of Aarhus, Denmark and Department of Economic Statistics, Stockholm School of Economics, Sweden. Jun Yu School of Economics, Singapore Management University, Singapore. Jean-Michel Zakoïan University Lille III, EQUIPPE-GREMARS, and CREST, France. Eric Zivot Department of Economics, University of Washington, Seattle, U.S.A..

An Introduction to Univariate GARCH Models Timo Teräsvirta

Abstract This paper contains a survey of univariate models of conditional heteroskedasticity. The classical ARCH model is mentioned, and various extensions of the standard Generalized ARCH model are highlighted. This includes the Exponential GARCH model. Stochastic volatility models remain outside this review.

1 Introduction Financial economists are concerned with modelling volatility in asset returns. This is important as volatility is considered a measure of risk, and investors want a premium for investing in risky assets. Banks and other financial institutions apply so-called value-at-risk models to assess their risks. Modelling and forecasting volatility or, in other words, the covariance structure of asset returns, is therefore important. The fact that volatility in returns fluctuates over time has been known for a long time. Originally, the emphasis was on another aspect of return series: their marginal distributions were found to be leptokurtic. Returns were modelled as independent and identically distributed over time. In a classic work, Mandelbrot (1963) and Mandelbrot and Taylor (1967) applied so-called stable Paretian distributions to characterize the distribution of returns. Rachev and Mittnik (2000) contains an informative discussion of stable Paretian distributions and their use in finance and econometrics. Observations in return series of financial assets observed at weekly and higher frequencies are in fact not independent. While observations in these Timo Teräsvirta CREATES, School of Economics and Management, University of Aarhus, DK-8000 Aarhus C, and Department of Economic Statistics, Stockholm School of Economics, Box 6501, SE113 83 Stockholm, e-mail: [email protected]

T.G. Anderson et al., Handbook of Financial Time Series, DOI: 10.1007/978-3-540-71297-8_1, © Springer-Verlag Berlin Heidelberg 2009

17

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T. Teräsvirta

series are uncorrelated or nearly uncorrelated, the series contain higherorder dependence. Models of Autoregressive Conditional Heteroskedasticity (ARCH) form the most popular way of parameterizing this dependence. There are several articles in this Handbook devoted to different aspects of ARCH models. This article provides an overview of different parameterizations of these models and thus serves as an introduction to autoregressive conditional heteroskedasticity. The article is organized as follows. Section 2 introduces the classic ARCH model. Its generalization, the Generalized ARCH (GARCH) model is presented in Section 3. This section also describes a number of extensions to the standard GARCH models. Section 4 considers the Exponential GARCH model whose structure is rather different from that of the standard GARCH model, and Section 5 discusses ways of comparing EGARCH models with GARCH ones. Suggestions for further reading can be found at the end.

2 The ARCH Model The autoregressive conditional heteroskedasticity (ARCH) model is the first model of conditional heteroskedasticity. According to Engle (2004), the original idea was to find a model that could assess the validity of the conjecture of Friedman (1977) that the unpredictability of inflation was a primary cause of business cycles. Uncertainty due to this unpredictability would affect the investors’ behaviour. Pursuing this idea required a model in which this uncertainty could change over time. Engle (1982) applied his resulting ARCH model to parameterizing conditional heteroskedasticity in a wage-price equation for the United Kingdom. Let εt be a random variable that has a mean and a variance conditionally on the information set Ft−1 (the σ-field generated by εt−j , j ≥ 1). The ARCH model of εt has the following properties. First, E{εt |Ft−1 } = 0 and, second, the conditional variance ht = E{ε2t |Ft−1 } is a nontrivial positive-valued parametric function of Ft−1 . The sequence {εt } may be observed directly, or it may be an error or innovation sequence of an econometric model. In the latter case, εt = yt − μt (yt )

(1)

where yt is an observable random variable and μt (yt ) = E{yt |Ft−1 }, the conditional mean of yt given Ft−1 . Engle’s (1982) application was of this type. In what follows, the focus will be on parametric forms of ht , and for simplicity it is assumed that μt (yt ) = 0. Engle assumed that εt can be decomposed as follows: 1/2

ε t = z t ht

(2)

An Introduction to Univariate GARCH Models

19

where {zt } is a sequence of independent, identically distributed (iid) random variables with zero mean and unit variance. This implies εt |Ft−1 ∼ D(0, ht ) where D stands for the distribution (typically assumed to be a normal or a leptokurtic one). The following conditional variance defines an ARCH model of order q: q  ht = α0 + αj ε2t−j (3) j=1

where α0 > 0, αj ≥ 0, j = 1, . . . , q−1, and αq > 0. The parameter restrictions in (3) form a necessary and sufficient condition for positivity of the conditional variance. Suppose the unconditional variance Eε2t = σ 2 < ∞. The definition of εt through the decomposition (2) involving zt then guarantees the white noise property of the sequence {εt }, since {zt } is a sequence of iid variables. Although the application in Engle (1982) was not a financial one, Engle and others soon realized the potential of the ARCH model in financial applications that required forecasting volatility. The ARCH model and its generalizations are thus applied to modelling, among other things, interest rates, exchange rates and stock and stock index returns. Bollerslev et al. (1992) already listed a variety of applications in their survey of these models. Forecasting volatility of these series is different from forecasting the conditional mean of a process because volatility, the object to be forecast, is not observed. The question then is how volatility should be measured. Using ε2t is an obvious but not necessarily a very good solution if data of higher frequency are available; see Andersen and Bollerslev (1998) and Andersen and Benzoni (2008) for discussion.

3 The Generalized ARCH Model 3.1 Why Generalized ARCH? In applications, the ARCH model has been replaced by the so-called generalized ARCH (GARCH) model that Bollerslev (1986) and Taylor (1986) proposed independently of each other. In this model, the conditional variance is also a linear function of its own lags and has the form ht = α0 +

q  j=1

αj ε2t−j +

p 

βj ht−j .

(4)

j=1

The conditional variance defined by (4) has the property that the unconditional autocorrelation function of ε2t , if it exists, can decay slowly, albeit still exponentially. For the ARCH family, the decay rate is too rapid compared to what is typically observed in financial time series, unless the maximum

20

T. Teräsvirta

lag q in (3) is long. As (4) is a more parsimonious model of the conditional variance than a high-order ARCH model, most users prefer it to the simpler ARCH alternative. The overwhelmingly most popular GARCH model in applications has been the GARCH(1,1) model, that is, p = q = 1 in (4). A sufficient condition for the conditional variance to be positive is α0 > 0, αj ≥ 0, j = 1, . . . , q; βj ≥ 0, j = 1, . . . , p. The necessary and sufficient conditions for positivity of the conditional variance in higher-order GARCH models are more complicated than the sufficient conditions just mentioned and have been given in Nelson and Cao (1992). The GARCH(2,2) case has been studied in detail by He and Teräsvirta (1999). Note that for the GARCH model to be identified if at least one βj > 0 (the model is a genuine GARCH model) one has to require that also at least one αj > 0. If α1 = . . . = αq = 0, the conditional and unconditional variances of εt are equal and β1 , . . . , βp are unidentified nuisance parameters. p The GARCH(p,q) process is weakly stationary if and q only if j=1 αj + j=1 βj < 1. The stationary GARCH model has been slightly simplified by ’variance targeting’, see Engle and This implies replacing the interq Mezrich(1996). p cept α0 in (4) by (1 − j=1 αj − j=1 βj )σ 2 where σ 2 = Eε2t . The estimate T σ 2 = T −1 t=1 ε2t is substituted for σ 2 before estimating the other parameters. As a result, the conditional variance converges towards the ’long-run’ unconditional variance, and the model contains one parameter less than the standard GARCH(p,q) model. It may be pointed out that the GARCH model is a special case of an infinite-order (ARCH(∞)) model (2) with ht = α0 +

∞ 

αj ε2t−j .

(5)

j=1

The ARCH(∞) representation is useful in considering properties of ARCH and GARCH models such as the existence of moments and long memory; see Giraitis et al. (2000). The moment structure of GARCH models is considered in detail in Lindner (2008).

3.2 Families of univariate GARCH models Since its introduction the GARCH model has been generalized and extended in various directions. This has been done to increase the flexibility of the original model. For example, the original GARCH specification assumes the response of the variance to a shock to be independent of the sign of the shock and just be a function of the size of the shock. Several extensions of the GARCH model aim at accommodating the asymmetry in the response.

An Introduction to Univariate GARCH Models

21

These include the GJR-GARCH model of Glosten et al. (1993), the asymmetric GARCH models of Engle and Ng (1993) and the quadratic GARCH of Sentana (1995). The GJR-GARCH model has the form (2), where ht = α0 +

q 

{αj + δj I(εt−j > 0)}ε2t−j +

j=1

p 

βj ht−j .

(6)

j=1

In 6), I(εt−j > 0) is an indicator function obtaining value one when the argument is true and zero otherwise. In the asymmetric models of both Engle and Ng, and Sentana, the centre of symmetry of the response to shocks is shifted away from zero. For example, ht = α0 + α1 (εt−1 − γ)2 + β1 ht−1

(7)

with γ = 0 in Engle and Ng (1993). The conditional variance in Sentana’s Quadratic ARCH (QARCH) model (the model is presented in the ARCH form) is defined as follows: ht = α0 + α εt−1 + εt−1 Aεt−1

(8)

where εt = (εt , . . . , εt−q+1 ) is a q × 1 vector, α = (α1 , . . . , αq ) is a q × 1 parameter vector and A a q × q symmetric parameter matrix. In (8), not only squares of εt−i but also cross-products εt−i εt−j , i = j, contribute to the conditional variance. When α = 0, the QARCH generates asymmetric responses. The ARCH equivalent of (7) is a special case of Sentana’s model. Constraints on parameters required for positivity of ht in (8) become clear by rewriting (8) as follows:     A α/2  εt−1 . (9) ht = εt−1 1 α /2 α0 1 The conditional variance ht is positive if and only if the matrix in the quadratic form on the right-hand side of (9) is positive definite. Some authors have suggested modelling the conditional standard deviation instead of the conditional variance: see Taylor (1986), Schwert (1990), and for an asymmetric version, Zakoïan (1994). A further generalization of this idea appeared in Ding et al. (1993). These authors proposed a GARCH model for hkt where k > 0 is also a parameter to be estimated. Their GARCH model is (2) with q p   hkt = α0 + αj |εt−j |2k + βj hkt−j , k > 0. (10) j=1

j=1

The authors argued that this extension provides flexibility lacking in the original GARCH specification of Bollerslev (1986) and Taylor (1986). The proliferation of GARCH models has inspired some authors to define families of GARCH models that would accommodate as many individual

22

T. Teräsvirta

models as possible. Hentschel (1995) defined one such family . The first-order GARCH model has the general form λ/2

−1

ht

λ

λ/2

λ/2

= ω + αht−1 f ν (zt−1 ) + β

ht−1 − 1 λ

(11)

where λ > 0 and f ν (zt ) = |zt − b| − c(zt − b). Family (11) contains a large number of well-known GARCH models. The Box1/2 Cox type transformation of the conditional standard deviation ht makes it possible, by allowing λ → 0, to accommodate models in which the logarithm of the conditional variance is parameterized, such as the exponential GARCH model to be considered in Section 4. Parameters b and c in f v (zt ) allow the inclusion of different asymmetric GARCH models such as the GJR-GARCH or threshold GARCH models in (11). Another family of GARCH models that is of interest is the one He and Teräsvirta (1999) defined as follows: hkt =

q  j=1

g(zt−j ) +

p 

cj (zt−j )hkt−j ,

k>0

(12)

j=1

where {g(zt )} and {c(zt )} are sequences of independent and identically distributed random variables. In fact, the family was originally defined for q = 1, but the definition can be generalized to higher-order models. For example, the standard GARCH(p, q) model is obtained by setting g(zt ) = α0 /q and 2 + βj , j = 1, . . . , q, in (12). Many other GARCH modcj (zt−j ) = αj zt−j els such as the GJR-GARCH, the absolute-value GARCH, the Quadratic GARCH and the power GARCH model belong to this family. Note that the power GARCH model itself nests several well-known GARCH models; see Ding et al. (1993) for details. Definition (12) has been used for deriving expressions of fourth moments, kurtosis and the autocorrelation function of ε2t for a number of first-order GARCH models and the standard GARCH(p, q) model. The family of augmented GARCH models, defined by Duan (1997), is a rather general family. The first-order augmented GARCH model is defined as follows. Consider (2) and assume that |λφt − λ − 1| if λ = 0 ht = (13) exp{φt − 1} if λ = 0 where φt = α0 + ζ1,t−1 φt−1 + ζ2,t−1 .

(14)

In (14), (ζ1t , ζ2t ) is a strictly stationary sequence of random vectors with a continuous distribution, measurable with respect to the available information

An Introduction to Univariate GARCH Models

23

until t. Duan defined an augmented GARCH(1,1) process as (2) with (13) and (14), such that ζ1t = α1 + α2 |εt − c|δ + α3 max(0, c − εt )δ |εt − c|δ − 1 max(0, c − εt )δ − 1 + α5 . ζ2t = α4 δ δ This process contains as special cases all the GARCH models previously mentioned, as well as the Exponential GARCH model to be considered in Section 4. Duan (1997) generalized this family to the GARCH(p, q) case and derived sufficient conditions for strict stationarity for this general family as well as conditions for the existence of the unconditional variance of εt . Furthermore, he suggested misspecification tests for the augmented GARCH model.

3.3 Nonlinear GARCH 3.3.1 Smooth transition GARCH As mentioned above, the GARCH model has been extended to characterize asymmetric responses to shocks. The GJR-GARCH model, obtained as setq 2 + βj , j = ting j=1 g(zt−j ) = α0 and cj (zt−j ) = (αj + ωj I(zt−j > 0))zt−j 1, . . . , q, in (12), is an example of that. A nonlinear version of the GJRGARCH model is obtained by making the transition between regimes smooth. Hagerud (1997), Gonzales-Rivera (1998) and Anderson et al. (1999) proposed this extension. A smooth transition GARCH (STGARCH) model may be defined as equation (2) with ht = α10 +

q 

α1j ε2t−j + (α20 +

j=1

q 

α2j ε2t−j )G(γ, c; εt−j ) +

j=1

p 

βj ht−j (15)

j=1

where the transition function G(γ, c; εt−j ) = (1 + exp{−γ

K

(εt−j − ck )})−1 , γ > 0.

(16)

k=1

When K = 1, (16) is a simple logistic function that controls the change of the coefficient of ε2t−j from α1j to α1j + α2j as a function of εt−j , and similarly for the intercept. In that case, as γ → ∞, the transition function becomes a step function and represents an abrupt switch from one regime to the other. Furthermore, at the same time setting c1 = 0 yields the GJRGARCH model because εt and zt have the same sign. When K = 2 and, in addition, c1 = −c2 in (16), the resulting smooth transition GARCH model is still symmetric about zero, but the response of the conditional variance to a

24

T. Teräsvirta

shock is a nonlinear function of lags of ε2t . Smooth transition GARCH models are useful in situations where the assumption of two distinct regimes is not an adequate approximation to the asymmetric behaviour of the conditional variance. Hagerud (1997) also discussed a specification strategy that allows the investigator to choose between K = 1 and K = 2 in (16). Values of K > 2 may also be considered, but they are likely to be less common in applications than the two simplest cases. The smooth transition GARCH model (15) with K = 1 in (16) is designed for modelling asymmetric responses to shocks. On the other hand, the standard GARCH model has the undesirable property that the estimated model often exaggerates the persistence in volatility (the estimated sum of the αand β-coefficients is close to one). This in turn results in poor volatility forecasts. In order to remedy this problem, Lanne and Saikonnen (2005) proposed a smooth transition GARCH model whose first-order version has the form ht = α0 + α1 ε2t−1 + δ1 G1 (θ; ht−1 ) + β1 ht−1 .

(17)

In (17), G1 (θ; ht−1 ) is a continuous bounded function such as (16): Lanne and Saikkonen use the cumulative distribution function of the gammadistribution. A major difference between (15) and (17) is that in the latter model the transition variable is a lagged conditional variance. In empirical examples given in the paper, this parameterization clearly alleviates the problem of exaggerated persistence. The model may also be generalized to include a term of the form G1 (θ; ht−1 )ht−1 , but according to the authors, such an extension appeared unnecessary in practice.

3.3.2 Threshold GARCH and extensions If (15) is defined as a model for the conditional standard deviation such 1/2 1/2 that ht is replaced by ht , ht−j by ht−j , j = 1, . . . , p, and ε2t−j by |εt−j |, j = 1, . . . , q, then choosing K = 1, c1 = 0 and letting γ → ∞ in (16) yields the threshold GARCH (TGARCH) model that Zakoïan (1994) considered. The TGARCH(p, q) model is thus the counterpart of the GJR-GARCH model in the case where the entity to be modelled is the conditional standard deviation instead of the conditional variance. Note that in both of these models, the threshold parameter has a known value (zero). In Zakoïan (1994), the conditional standard deviation is defined as follows: 1/2 ht

= α0 +

q  j=1

+ (α+ j εt−j

−

− α− j εt−j )

+

q 

1/2

βj ht−j

(18)

j=1

− + − where ε+ t−j = max(εt−j , 0), εt−j = min(εt−j , 0), and αj , αj , j = 1, . . . , q, are parameters. Rabemananjara and Zakoïan (1993) introduced an even more 1/2 general model in which ht can obtain negative values, but it has not gained

An Introduction to Univariate GARCH Models

25

wide acceptance. Nevertheless, these authors provide evidence of asymmetry in the French stock market by fitting the TGARCH model (18) to the daily return series of stocks included in the CAC 40 index of the Paris Bourse. The TGARCH model is linear in parameters because the threshold parameter is assumed to equal zero. A genuine nonlinear threshold model is the Double Threshold ARCH (DTARCH) model of Li and Li (1996). It is called a double threshold model because both the autoregressive conditional mean and the conditional variance have a threshold-type structure as defined in Tong (1990). The conditional mean is defined as follows: yt =

K 

(φ0k +

k=1

pk 

(m)

(m)

φjk yt−j )I(ck−1 < yt−b ≤ ck ) + εt

(19)

j=1

and the conditional variance has the form p L   (v) (v) (α0 + αj ε2t−j )I(c−1 < yt−d ≤ c ). ht = =1

(20)

j=1

Furthermore, k = 1, . . . , K,  = 1, . . . , L, and b and d are delay parameters, b, d ≥ 1. The number of regimes in (19) and (20), K and L, respectively, need not be the same, nor do the two threshold variables have to be equal. Other threshold variables than lags of yt are possible. For example, replacing yt−d in (20) by εt−d or ε2t−d may sometimes be an interesting possibility. Another variant of the DTARCH model is the model that Audrino and Bühlmann (2001) who introduced it called the Tree-Structured GARCH model. It has an autoregressive conditional mean: yt = φyt−1 + εt

(21)

where εt is decomposed as in (2), and the first-order conditional variance ht =

K 

2 (α0k + α1k yt−1 + βk ht−1 )I{(yt−1 , ht−1 ) ∈ Rk }.

(22)

k=1

In (22), Rk is a subset in a partition P = {R1 , . . . , RK } of the sample space of (yt−1 , ht−1 ). For example, if K = 2, either R1 = {yt−1 > cy , ht−1 > 0} or R1 = {−∞ < yt−1 < ∞, ht−1 > ch }, ch > 0, and R2 is the complement of R1 . Note that, strictly speaking, equation (22) does not define a GARCH model unless φ = 0 in (21), because the squared variable in the equation 2 , not ε2t−1 . A practical problem is that the tree-growing strategy of is yt−1 Audrino and Bühlmann (2001) does not seem to prevent underidentification: if K is chosen too large, (22) is not identified. A similar problem is present in the DTARCH model as well as in the STGARCH one. Hagerud (1997) and Gonzales-Rivera (1998), however, do provide linearity tests in order to avoid this problem in the STGARCH framework.

26

T. Teräsvirta

3.4 Time-varying GARCH An argument brought forward in the literature, see for instance Mikosch and Stărică (2004), is that in applications the assumption of the GARCH models having constant parameters may not be appropriate when the series to be modelled are long. Parameter constancy is a testable proposition, and if it is rejected, the model can be generalized. One possibility is to assume that the parameters change at specific points of time, divide the series into subseries according to the location of the break-points, and fit separate GARCH models to the subseries. The main statistical problem is then finding the number of break-points and their location because they are normally not known in advance. Chu (1995) has developed tests for this purpose. Another possibility is to modify the smooth transition GARCH model (15) to fit this situation. This is done by defining the transition function (16) as a function of time: G(γ, c; t∗ ) = (1 + exp{−γ

K

(t∗ − ck )})−1 , γ > 0

k=1

where t∗ = t/T , t = 1, . . . , T , and T is the number of observations. Standardizing the time variable between zero and unity makes interpretation of the parameters ck , k = 1, . . . , K, easy as they indicate where in relative terms the changes in the process occur. The resulting time-varying parameter GARCH (TV-GARCH) model has the form ht = α0 (t) +

q  j=1

αj (t)ε2t−j +

p 

βj (t)ht−j

(23)

j=1

where α0 (t) = α01 + α02 G(γ, c; t∗ ), αj (t) = αj1 + αj2 G(γ, c; t∗ ), j = 1, . . . , q, and βj (t) = βj1 + βj2 G(γ, c; t∗ ), j = 1, . . . , p. This is the most flexible parameterization. Some of the time-varying parameters in (23) may be restricted to constants a priori. For example, it may be assumed that only the intercept α0 (t) is time-varying. This implies that the ’baseline volatility’ or unconditional variance is changing over time. If change is allowed in the other GARCH parameters then the model is capable of accommodating systematic changes in the amplitude of the volatility clusters that cannot be explained by a constant-parameter GARCH model. This type of time-varying GARCH is mentioned here because it is a special case of the smooth transition GARCH model. Other time-varying parameter models of conditional heteroskedasticity, such as nonstationary ARCH models with locally changing parameters, are discussed in Čížek and Spokoiny (2008).

An Introduction to Univariate GARCH Models

27

3.5 Markov-switching ARCH and GARCH Markov-switching or hidden Markov models of conditional heteroskedasticity constitute another class of nonlinear models of volatility. These models are an alternative way of modelling volatility processes that contains breaks. Hamilton and Susmel (1994) argued that very large shocks, such as the one affecting the stocks in October 1987, may have consequences for subsequent volatility so different from consequences of small shocks that a standard ARCH or GARCH model cannot describe them properly. Their Markov-switching ARCH model is defined as follows: ht =

k 

(i) (α0

+

i=1

q 

(i)

αj ε2t−j )I(st = i)

(24)

j=1

where st is a discrete unobservable random variable obtaining values from the set S = {1, . . . , k} of regime indicators. It follows a (usually first-order) homogeneous Markov chain: Pr{st = j|st = i} = pij , i, j = 1, . . . , k. (i)

Cai (1994) considered a special case of (24) in which only the intercept α0 is switching, and k = 2. But then, his model also contains a switching conditional mean. Furthermore, Rydén et al. (1998) showed that a simplified (i) version of (24) where αj = 0 for j ≥ 1 and all i, is already capable of generating data that display most of the stylized facts that Granger and Ding (1995) ascribe to high-frequency, daily, say, financial return series. This suggests that a Markov-switching variance alone without any ARCH structure may in many cases explain a large portion of the variation in these series. Nevertheless, it can be argued that shocks drive economic processes, and this motivates the ARCH structure. If the shocks have a persistent effect on volatility, however, a parsimonious GARCH representation may be preferred to (24). Generalizing (24) into a GARCH model involves one major difficulty. A straightforward (first-order) generalization would have the following form: (i)

(i)

(i)

ht = (α0 + α1 ε2t−1 + β1 ht−1 )I(st = i).

(25)

From the autoregressive structure of (25) it follows that ht is completely path-dependent: its value depends on the unobservable st−j , j = 0, 1, 2, . . . t. This makes the model practically impossible to estimate because in order to evaluate the log-likelihood, these unobservables have to be integrated out of this function. Simplifications of the model that circumvent this problem can be found in Gray (1996) and Klaassen (2002). A good discussion about their models can be found in Haas et al. (2004). These authors present another MS-GARCH model whose fourth-moment structure they are able to work out. That does not seem possible for the other models. The MS-GARCH

28

T. Teräsvirta

model of Haas et al. (2004) is defined as follows: εt = z t

k 

1/2

hit I(st = i)

i=1

where st is defined as in (24). Furthermore, ht = α0 + α1 ε2t−1 + Bht−1 where ht = (h1t , . . . , h1k ) , αi = (αi1 , . . . , αik ) , i = 0, 1, and B = diag(β11 , . . . , β1k ) . Thus, each volatility regime has its own GARCH equation. The conditional variance in a given regime is only a function of the lagged conditional variance in the same regime, which is not the case in the other models. The identification problem mentioned in Section 3.3.2 is present here as well. If the true model has fewer regimes than the specified one, the latter contains unidentified nuisance parameters. More information about Markov-switching ARCH and GARCH models can be found in Lange and Rahbek (2008).

3.6 Integrated and fractionally integrated GARCH In applications it often occurs that the estimated sum of the parameters α1 and β1 in the standard first-order GARCH model (4) with p = q = 1 is close to unity. Engle and Bollerslev (1986), who first paid attention to this phenomenon, suggested imposing the restriction α1 + β1 = 1 and called the ensuing model an integrated GARCH (IGARCH) model. The IGARCH process is not weakly stationary as Eε2t is not finite. Nevertheless, the term ”integrated GARCH” may be somewhat misleading as the IGARCH process is strongly stationary. Nelson (1991) showed that under mild conditions for {zt } and assuming α0 > 0, the GARCH(1,1) process is strongly stationary if E ln(α1 + β1 zt2 ) < 0

(26)

(recall that Ezt2 = 1). The IGARCH process satisfies (26). The analogy with integrated processes, that is, ones with a unit root, is therefore not as straightforward as one might think. For a general discussion of stationarity conditions in GARCH models, see Lindner (2008). Nelson (1991) also showed that when an IGARCH process is started at some finite time point, its behaviour depends on the intercept α0 . On the one hand, if the intercept is positive then the unconditional variance of the process grows linearly with time. In practice this means that the amplitude of the clusters of volatility to be parameterized by the model on the average increases over time. The rate of increase need not, however, be particularly

An Introduction to Univariate GARCH Models

29

rapid. One may thus think that in applications with, say, a few thousand observations, IGARCH processes nevertheless provide a reasonable approximation to the true data-generating volatility process. On the other hand, if α0 = 0 in the IGARCH model, the realizations from the process collapse to zero almost surely. How rapidly this happens, depends on the parameter β1 . Although the investigator may be prepared to accept an IGARCH model as an approximation, a potentially disturbing fact is that this means assuming that the unconditional variance of the process to be modelled does not exist. It is not clear that this is what one always wants to do. There exist other explanations to the fact that the sum α1 + β1 estimates to one or very close to one. First Diebold (1986) and later Lamoureux and Lastrapes (1990) suggested that this often happens if there is a switch in the intercept of a GARCH model during the estimation period. This may not be surprising as such a switch means that the underlying GARCH process is not stationary. Another, perhaps more puzzling, observation is related to exponential GARCH models to be considered in Section 4. Malmsten (2004) noticed that if a GARCH(1, 1) model is fitted to a time series generated by a stationary first-order exponential GARCH model (see Section 4), the probability of the estimated sum α1 + β1 exceeding unity can sometimes be rather large. In short, if the estimated sum of these two parameters in a standard GARCH(1,1) model is close to unity, imposing the restriction α1 + β1 = 1 without further investigation may not necessarily be the most reasonable action to take. Assuming p = q = 1, the GARCH(p, q) equation (4) can also be written in the ”ARMA(1,1) form” by adding ε2t to both sides and moving ht to the right-hand side: ε2t = α0 + (α1 + β1 )ε2t−1 + νt − β1 νt−1

(27)

where {νt } = {ε2t − ht } is a martingale difference sequence with respect to ht . For the IGARCH process, (27) has the ”ARIMA(0,1,1) form” (1 − L)ε2t = α0 + νt − β1 νt−1 .

(28)

Equation (28) has served as a starting-point for the fractionally integrated GARCH (FIGARCH) model. The FIGARCH(1,d,0) model is obtained from (28) by replacing the difference operator by a fractional difference operator: (1 − L)d ε2t = α0 + νt − β1 νt−1 .

(29)

The FIGARCH equation (29) can be written as an infinite-order ARCH model by applying the definition νt = ε2t − ht to it. This yields ht = α0 (1 − β1 )−1 + λ(L)ε2t

30

T. Teräsvirta

 j 2 where λ(L) = {1 − (1 − L)d (1 − β1 L)−1 }ε2t = ∞ j=1 λj L εt , and λj ≥ 0 for all j. Expanding the fractional difference operator into an infinite sum yields the result that for long lags j, λj = {(1 − β1 )Γ (d)−1 }j −(1−d) = cj −(1−d) , c > 0

(30)

where d ∈ (0, 1) and Γ (d) is the gamma function. From (30) it is seen that the effect of the lagged ε2t on the conditional variance decays hyperbolically as a function of the lag length. This is the reason why Ballie et al. (1996) introduced the FIGARCH model, as it would conveniently explain the apparent slow decay in autocorrelation functions of squared observations of many daily return series. The FIGARCH model thus offers a competing view to the one according to which changes in parameters in a GARCH model are the main cause of the slow decay in the autocorrelations. The first-order FIGARCH model (29) can of course be generalized into a FIGARCH(p, d, q) model. The probabilistic properties of FIGARCH processes such as stationarity, still an open question, are quite complex, see, for example, Davidson (2004) and Giraitis et al. (2008) for discussion. The hyperbolic GARCH model introduced in the first-mentioned paper contains the standard GARCH and the FIGARCH models as two extreme special cases; for details see Davidson (2004).

3.7 Semi- and nonparametric ARCH models The ARCH decomposition of returns (2) has also been used in a semi- or nonparametric approach. The semiparametric approach is typically employed in situations where the distribution of zt is left unspecified and is estimated nonparametrically. In nonparametric models, the issue is the estimation of the functional form of the relationship between ε2t and ε2t−1 , . . . , ε2t−q . Semiand nonparametric ARCH models are considered in detail in Linton (2008).

3.8 GARCH-in-mean model GARCH models are often used for predicting the risk of a portfolio at a given point of time. From this it follows that the GARCH type conditional variance could be useful as a representation of the time-varying risk premium in explaining excess returns, that is, returns compared to the return of a riskless asset. An excess return would be a combination of the unforecastable difference εt between the ex ante and ex post rates of return and a function of the conditional variance of the portfolio. Thus, if yt is the excess return at time t,

An Introduction to Univariate GARCH Models

yt = εt + β + g(ht ) − Eg(ht )

31

(31)

where ht is defined as a GARCH process (4) and g(ht ) is a positive-valued 1/2 function. Engle et al. (1987) originally defined g(ht ) = δht , δ > 0, which corresponds to the assumption that changes in the conditional standard deviation appear less than proportionally in the mean. The alternative g(ht ) = δht has also appeared in the literature. Equations (31) and (4) form the GARCHin-mean or GARCH-M model. It has been quite frequently applied in the applied econometrics and finance literature. Glosten et al. (1993) developed their asymmetric GARCH model as a generalization of the GARCH-M model. The GARCH-M process has an interesting moment structure. Assume that Ezt3 = 0 and Eε4t < ∞. From (31) it follows that the kth order autocovariance E(yt − Eyt )(yt−k − Eyt ) = Eεt−k g(ht ) + cov(g(ht ), g(ht−k )) = 0. This means that there is forecastable structure in yt , which may contradict some economic theory if yt is a return series. Hong (1991) showed this in a special case where g(ht ) = δht , Eε4t < ∞, and ht follows a GARCH(p,q) model. In that situation, all autocorrelations of yt are nonzero. Furthermore, E(yt − Eyt )3 = 3Eht {g(ht ) − Eg(ht )} + E{g(ht ) − Eg(ht )}3 = 0.

(32)

It follows from (32) that a GARCH-M model implies postulating a skewed marginal distribution for yt unless g(ht ) ≡ constant. For example, if g(ht ) = 1/2 δht , δ < 0, this marginal distribution is negatively skewed. If the model builder is not prepared to make this assumption or the one of forecastable structure in yt , the GARCH-M model, despite its theoretical motivation, does not seem an appropriate alternative to use. For more discussion of this situation, see He et al. (2006).

3.9 Stylized facts and the first-order GARCH model As already mentioned, financial time series such as high-frequency return series constitute the most common field of applications for GARCH models. These series typically display rather high kurtosis. At the same time, the autocorrelations of the absolute values or squares of the observations are low and decay slowly. These two features are sometimes called stylized facts of financial time series. Granger and Ding (1995) listed a few more such features. Among them is the empirical observation that in a remarkable number of financial series, the autocorrelations of the powers of observations, |εt |k , peak around k = 1. Granger and Ding called this stylized fact the Taylor effect as Taylor (1986) was the first to draw attention to it (by comparing the autocorrelations of ε2t and |εt |).

32

T. Teräsvirta

One way of evaluating the adequacy of GARCH models is to ask how well they can be expected to capture the features or stylized facts present in the series to be modelled. The expressions for kurtosis and the autocorrelation function of absolute values and squared observations are available for the purpose. They allow one to find out, for example, whether or not a GARCH(1,1) model is capable of producing realizations with high kurtosis and low, slowly decaying autocorrelations. The results of Malmsten and Teräsvirta (2004) who have used these expressions, illustrate the well known fact, see, for example, Bollerslev et al. (1994), that a GARCH model with normally distributed errors does not seem to be a sufficiently flexible model for explaining these two features in financial return series. This is shown in Figure 1. The panels contain a number of isoquants for which the sum

Fig. 1 Kurtosis/first-order autocorrelation isoquants for the GARCH(1,1) model, from highest to lowest: α + β = 0.9, 0.95, 0.99, 0.999, and corresponding combinations estimated from data: Upper left panel: Daily returns of the 27 most actively traded stocks at the Stockholm Stock Exchange; Upper right panel: Returns of five major daily exchange rates, divided into 34 subseries; Lower left panel: Daily returns of the S&P 500 index from 3 January 1928 to 19 September 2001, divided into 20 equally long subseries; Lower right panel: All observations

An Introduction to Univariate GARCH Models

33

α1 + β1 remains constant as a function of the kurtosis and the first-order autocorrelation of squared observations. Note that α1 + β1 is the exponential decay rate of the autocorrelation function, that is, the jth autocorrelation ρj = (α1 + β1 )j−1 ρ1 for j ≥ 1. They also contain combinations of the kurtosis and the first-order autocorrelation estimated directly from time series. It is seen that very often the kurtosis/autocorrelation combinations do not tend to lie in the vicinity of these isoquants even when α1 + β1 is very close to one. The isoquants are od course only defined for combinations of α1 and β1 for which Eε4t < ∞. Malmsten and Teräsvirta (2004) also demonstrated how the situation can be improved, as is customary in practice, by replacing the normal error distribution by a more fat-tailed one. In Figure 2 it is seen how increasing the ”baseline kurtosis”, that is, the kurtosis of the distribution of zt , the error, helps the GARCH(1,1) model to capture the stylized fact of high kurtosis/low autocorrelation. The isoquants are moved to the right because the baseline

Fig. 2 Isoquants of pairs of kurtosis and first-order autocorrelation of squared observations in the GARCH(1,1) model with t(7)-distributed (left-hand panel) and t(5)-distributed errors (right-hand panel), for (from above) α + β = 0.90, 0.95, 0.99 and 0.999, and corresponding observations (the same ones as in the lower right panel of Figure 1).

kurtosis increases. At the same time it seems that the level of the autocorrelations decreases. But then, this does not simultaneously affect the decay rate α1 + β1 of the autocorrelations. Recently, Kim and White (2004) suggested that the standard estimator of kurtosis exaggerates the true kurtosis and that robust measures yield more reliable results. It follows that high kurtosis values estimated from return series are a result of a limited number of outliers. If this is the case, then the use of a non-normal (heavy-tailed) error distribution may not necessarily be an optimal extension to the standard normal-error GARCH model. However, Teräsvirta and Zhao (2006) recently studied 160 daily return series and, following Kim and White (2004), used robust kurtosis and autocorrelation

34

T. Teräsvirta

estimates instead of standard ones. Their results indicate that leptokurtic distributions for zt are needed in capturing the kurtosis-autocorrelation stylized fact even when the influence of extreme observations is dampened by the use of robust estimates. As to the Taylor effect, He and Teräsvirta (1999) defined a corresponding theoretical property, the Taylor property, as follows. Let ρ(|εt |k , |εt−j |k ) be the jth order autocorrelation of {|εt |k }. The stochastic process has the Taylor property when ρ(|εt |k , |εt−j |k ) is maximized for k = 1 for j = 1, 2, . . .. In practice, He and Teräsvirta (1999) were able to find analytical results for the AVGARCH(1,1) model, but they were restricted to comparing the first-order autocorrelations for k = 1 and k = 2. For this model, ρ(|εt |, |εt−1 |) > ρ(ε2t , ε2t−1 ) when the kurtosis of the process is sufficiently high. The corresponding results for the standard GARCH(1,1) model (4) with p = q = 1 and normal errors are not available as the autocorrelation function of {|εt |} cannot be derived analytically. Simulations conducted by He and Teräsvirta (1999) showed that the GARCH(1,1) model probably does not possess the Taylor property, which may seem disappointing. But then, the results of Teräsvirta and Zhao (2006) show that if the standard kurtosis and autocorrelation estimates are replaced by robust ones, the evidence of the Taylor effect completely disappears. This stylized fact may thus be a consequence of just a small number of extreme observations in the series.

4 Family of Exponential GARCH Models 4.1 Definition and properties The Exponential GARCH (EGARCH) model is another popular GARCH model. Nelson (1991) who introduced it had three criticisms of the standard GARCH model in mind. First, parameter restrictions are required to ensure positivity of the conditional variance at every point of time. Second, the standard GARCH model does not allow an asymmetric response to shocks. Third, if the model is an IGARCH one, measuring the persistence is difficult since this model is strongly but not weakly stationary. Shocks may be viewed persistent as the IGARCH process looks like a random walk. However, the IGARCH model with α0 > 0 is strictly stationary and ergodic, and when α0 = 0, the realizations collapse into zero almost surely, as already indicated in Section 3.6. The second drawback has since been removed as asymmetric GARCH models such as GJR-GARCH (Glosten et al. (1993)) or smooth transition GARCH have become available. A family of EGARCH(p, q) models may be defined as in (2) with

An Introduction to Univariate GARCH Models

ln ht = α0 +

q  j=1

gj (zt−j ) +

35 p 

βj ln ht−j .

(33)

j=1

When gj (zt−j ) = αj zt−j + ψj (|zt−j | − E|zt−j |), j = 1, . . . , q, (33) becomes the EGARCH model of Nelson (1991). It is seen from (33) that no parameter restrictions are necessary to ensure positivity of ht . Parameters αj , j = 1, . . . , q, make an asymmetric response to shocks possible. 2 When gj (zt−j ) = αj ln zt−j , j = 1, . . . , q, (2) and (33) form the logarithmic GARCH (LGARCH) model that Geweke (1986) and Pantula (1986) proposed. The LGARCH model has not become popular among practitioners. A principal reason for this may be that for parameter values encountered in practice, the theoretical values of the first few autocorrelations of {ε2t } at short lags tend to be so high that such autocorrelations can hardly be found in financial series such as return series. This being the case, the LGARCH model cannot be expected to provide an acceptable fit when applied to financial series. Another reason are the occasional small values of ln ε2t that complicate the estimation of parameters. As in the standard GARCH case, the first-order model is the most popular EGARCH model in practice. Nelson (1991) derived existence conditions for moments of the general infinite-order Exponential ARCH model. Translated to the case of the EGARCH model (2) and (33) such that gj (zt−j ) = αj zt−j + ψj (|zt−j | − E|zt−j |), j = 1, . . . , q, where not all αj and ψj equal zero, these existence conditions imply that if the error process {zt } has all moments and p 2 β < 1 in (33), then all moments for the EGARCH process {εt } exist. j=1 j For example, if {zt } is a sequence of independent standard normal variables then the restrictions on βj , j = 1, . . . , p, are necessary and sufficient for the existence of all moments simultaneously. This is different from the family (12) of GARCH models considered in Section 3.2. For those models, the moment conditions become more and more stringent for higher and higher even moments. The expressions for moments of the first-order EGARCH process can be found in He et al. (2002); for the more general case, see He (2000).

4.2 Stylized facts and the first-order EGARCH model In Section 3.9 we considered the capability of first-order GARCH models to characterize certain stylized facts in financial time series. It is instructive to do the same for EGARCH models. For the first-order EGARCH model, the decay of autocorrelations of squared observations is faster than exponential in the beginning before it slows down towards an exponential rate; see He et al. (2002). Thus it does not appear possible to use a standard EGARCH(1,1) model to characterize processes with very slowly decaying autocorrelations. Malmsten and Teräsvirta (2004) showed that the symmetric EGARCH(1,1)

36

T. Teräsvirta

model with normal errors is not sufficiently flexible either for characterizing series with high kurtosis and slowly decaying autocorrelations. As in the standard GARCH case, assuming normal errors means that the first-order autocorrelation of squared observations increases quite rapidly as a function of kurtosis for any fixed β1 before the increase slows down. Analogously to GARCH, the observed kurtosis/autocorrelation combinations cannot be reached by the EGARCH(1,1) model with standard normal errors. The asymmetry parameter is unlikely to change things much. Nelson (1991) recommended the use of the so-called Generalized Error Distribution (GED) for the errors. It contains the normal distribution as a special case but also allows heavier tails than the ones in the normal distribution. Nelson (1991) also pointed out that a t-distribution for the errors may imply infinite unconditional variance for {εt }. As in the case of the GARCH(1,1) model, an error distribution with fatter tails than the normal one helps to increase the kurtosis and, at the same time, lower the autocorrelations of squared observations or absolute values. Because of analytical expressions of the autocorrelations for k > 0 given in He et al. (2002) it is possible to study the existence of the Taylor property in EGARCH models. Using the formulas for the autocorrelations of {|εt |k }, k > 0, it is possible to find parameter combinations for which these autocorrelations peak in a neighbourhood of k = 1. A subset of first-order EGARCH models thus has the Taylor property. This subset is also a relevant one in practice in the sense that it contains EGARCH processes with the kurtosis of the magnitude frequently found in financial time series. For more discussion on stylized facts and the EGARCH(1,1) model, see Malmsten and Teräsvirta (2004).

4.3 Stochastic volatility The EGARCH equation may be modified by replacing gj (zt−j ) by gj (st−j ) where {st } is a sequence of continuous unobservable independent random variables that are often assumed independent of zt at all lags. Typically in applications, p = q = 1 and g1 (st−1 ) = δst−1 where δ is a parameter to be estimated. This generalization is called the autoregressive stochastic volatility (SV) model, and it substantially increases the flexibility of the EGARCH parameterization. For evidence of this, see Malmsten and Teräsvirta (2004) and Carnero et al. (2004). A disadvantage is that model evaluation becomes more complicated than that of EGARCH models because the estimation does not yield residuals. Several articles in this Handbook are devoted to SV models.

An Introduction to Univariate GARCH Models

37

5 Comparing EGARCH with GARCH The standard GARCH model is probably the most frequently applied parameterization of conditional heteroskedasticity. This being the case, it is natural to evaluate an estimated EGARCH model by testing it against the corresponding GARCH model. Since the EGARCH model can characterize asymmetric responses to shocks, a GARCH model with the same property, such as the GJR-GARCH or the smooth transition GARCH model, would be a natural counterpart in such a comparison. If the aim of the comparison is to choose between these models, they may be compared by an appropriate model selection criterion as in Shephard (1996). Since the GJR-GARCH and the EGARCH model of the same order have equally many parameters, this amounts to comparing their maximized likelihoods. If the investigator has a preferred model or is just interested in knowing if there are significant differences in the fit between the two, the models may be tested against each other. The testing problem is a non-standard one because the two models do not nest each other. Several approaches have been suggested for this situation. Engle and Ng (1993) proposed combining the two models into an encompassing model. If the GARCH model is an GJRGARCH(p, q) one (both models can account for asymmetries), this leads to the following specification of the conditional variance: ln ht =

q 

{α∗j zt−j

+

ψj∗ (|zt−j |

− E|zt−j |)} +

j=1

+ ln(α0 +

q  j=1

{αj + ωj I(εt−j )}ε2t−j +

p  j=1 p 

βj∗ ln ht−j βj ht−j ).

(34)

j=1

Setting (αj , ωj ) = (0, 0), j = 1, . . . , q, and βj = 0, j = 1, . . . , p, in (34) yields an EGARCH(p, q) model. Correspondingly, the restrictions (α∗j , ψj∗ ) = (0, 0), j = 1, . . . , q, and βj∗ = 0, j = 1, . . . , p, define the GJR-GARCH(p, q) model. Testing the models against each other amounts to testing the appropriate restrictions in (34). A Lagrange Multiplier test may be constructed for the purpose. The test may also be viewed as another misspecification test and not only as a test against the alternative model. Another way of testing the EGARCH model against GARCH consists of forming the likelihood ratio statistic despite the fact that the null model is not nested in the alternative. This is discussed in Lee and Brorsen (1997) and Kim et al. (1998). Let M0 be the EGARCH model and M1 the GARCH one, and let the corresponding log-likelihoods be LT (ε; M0 , θ0 ) and LT (ε; M1 , θ1 ), respectively. The test statistic is LR = 2{LT (ε; M1 , θ1 ) − LT (ε; M0 , θ0 )}.

(35)

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The asymptotic null distribution of (35) is unknown but can be approximated by simulation. Assuming that the EGARCH model is the null model and that θ0 is the true parameter, one generates N realizations of T observations from this model and estimates both models and calculates the value of (35) using each realization. Ranking the N values gives an empirical distribution with which one compares the original value of (35). The true value of θ0 is unknown, but the approximation error due to the use of θ0 as a replacement vanishes asymptotically as T → ∞. If the value of (35) exceeds the 100(1 − α)% quantile of the empirical distribution, the null model is rejected at significance level α. Note that the models under comparison need not have the same number of parameters, and the value of the statistic can also be negative. Reversing the roles of the models, one can test GARCH models against EGARCH ones. Chen and Kuan (2002) proposed yet another method based on the pseudoscore, whose estimator under the null hypothesis and assuming the customary regularity conditions is asymptotically normally distributed. This result forms the basis for a χ2 -distributed test statistic; see Chen and Kuan (2002) for details. Results of small-sample simulations in Malmsten (2004) indicate that the pseudo-score test tends to be oversized. Furthermore, the Monte Carlo likelihood ratio statistic seems to have consistently higher power than the encompassing test, which suggests that the former rather than the latter should be applied in practice.

6 Final Remarks and Further Reading The literature on univariate GARCH models is quite voluminous, and it is not possible to incorporate all developments and extensions of the original model in the present text. Several articles of this Handbook provide detailed analyses of various aspects of GARCH models. Modern econometrics texts contain accounts of conditional heteroskedasticity. A number of surveys of GARCH models exist as well. Bollerslev et al. (1994), Diebold and Lopez (1995), Palm (1996), and Guégan (1994) (Chapter 5) survey developments till the early 1990s; see Giraitis et al. (2006) for a very recent survey. Shephard (1996) considers both univariate GARCH and stochastic volatility models. The focus in Gouriéroux (1996) lies on both univariate and multivariate ARCH models. The survey by Bollerslev et al. (1992) also reviews applications to financial series. The focus in Straumann (2004) is on estimation in models of conditional heteroskedasticity. Theoretical results on time series models with conditional heteroskedasticity are also reviewed in Li et al. (2002). Engle (1995) contains a selection of the most important articles on ARCH and GARCH models up until 1993.

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Multivariate GARCH models are not included in this article. There exists a recent survey by Bauwens et al. (2006), and these models are also considered in Silvennoinen and Teräsvirta (2008). Acknowledgement This research has been supported by Jan Wallander’s and Tom Hedelius’s Foundation, Grant No. P2005-0033:1. A part of the work for the chapter was done when the author was visiting Sonderforschungsbereich 649 at the Humboldt University Berlin. Comments from Changli He, Marcelo Medeiros and Thomas Mikosch (editor) are gratefully acknowledged. Any shortcomings and errors are the author’s sole responsibility.

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He, C., Teräsvirta, T. and Malmsten, H. (2002): Moment Structure of a Family of FirstOrder Exponential GARCH Models. Econometric Theory 18, 868–885. He, C., Silvennoinen, A. and Teräsvirta, T. (2006): Parameterizing Unconditional Skewness in Models for Financial Time Series. Unpublished paper, Stockholm School of Economics. Hentschel, L. (1995): All in the Family. Nesting Symmetric and Asymmetric GARCH Models. Journal of Financial Economics 39, 71–104. Hong, E.P. (1991): The Autocorrelation Structure for the GARCH-M Process. Economics Letters 37, 129–132. Kim, S., Shephard, N. and Chib, S. (1998): Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economic Studies 65, 361–393. Kim, T.-H. and White, H. (2004): On More Robust Estimation of Skewness and Kurtosis. Finance Research Letters 1, 56–73. Klaassen, F. (2002): Improving GARCH Volatility Forecasts with Regime-Switching GARCH. Empirical Economics 27, 363–394. Lamoureux, C.G. and Lastrapes, W.G. (1990): Persistence in Variance, Structural Change and the GARCH Model. Journal of Business and Economic Statistics 8, 225–234. Lange, T. and Rahbek, A. (2008): An introduction to regime switching time series. In: Andersen, T. G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 871–887. Springer, New York. Lanne, M. and Saikkonen, P. (2005): Nonlinear GARCH Models for Highly Persistent Volatility. Econometrics Journal 8, 251–276. Lee, J.-H. and Brorsen, B.W. (1997): A Non-Nested Test of GARCH vs. EGARCH Models. Applied Economics Letters 4, 765–768. Li, C.W. and Li, W.K. (1996): On a Double Threshold Autoregressive Heteroskedasticity Time Series Model. Journal of Applied Econometrics 11, 253–274. Li, W.K., Ling, S. and McAleer, M. (2002): Recent Theoretical Results for Time Series Models with GARCH Errors. Journal of Economic Surveys 16, 245–269. Lindner, A.M. (2008): Stationarity, mixing, distributional properties and moments of GARCH(p,q)–processes. In: Andersen, T. G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 43–69. Springer, New York. Linton, O. (2008): Semiparametric and nonparametric ARCH modelling. In: Andersen, T. G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handook of Financial Time Series, 156–167. Springer, New York. Malmsten, H. (2004): Valuating Exponential GARCH Models. SSE/EFI Working Paper Series in Economics and Finance, Stockholm School of Economics 564. Malmsten, H. and Teräsvirta, T. (2004): Stylized Facts of Financial Time Series and Three Popular Models of Volatility. SSE/EFI Working Paper Series in Economics and Finance, Stockholm School of Economics 563. Mandelbrot, B. (1963): The Variation of Certain Speculative Prices. Journal of Business 36, 394–419. Mandelbrot, T. and Taylor, H. (1967): On the Distribution of Stock Price Differences. Operations Research 15, 1057–1062. Mikosch, T. and Stărică, C. (2004): Nonstationarities in Financial Time Series, the LongRange Dependence, and the IGARCH Effects. Review of Economics and Statistics 66, 378–390. Nelson, D.B. (1991): Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica 59, 347–370. Nelson, D.B. and Cao, C.Q. (1992): Inequality Constraints in the Univariate GARCH Model. Journal of Business and Economic Statistics 10, 229–235. Palm, F.C. (1996): GARCH Models of Volatility. In: Maddala, G.S. and Rao, C.R. (Eds.): Handbook of Statistics 14: Statistical Methods in Finance 14, 209–240. Elsevier, Amsterdam. Pantula, S.G. (1986): Modeling the Persistence of Conditional Variances: A Comment. Econometric Reviews 5, 71–74.

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Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)–Processes Alexander M. Lindner

Abstract This paper collects some of the well known probabilistic properties of GARCH(p, q) processes. In particular, we address the question of strictly and of weakly stationary solutions. We further investigate moment conditions as well as the strong mixing property of GARCH processes. Some distributional properties such as the tail behaviour and continuity properties of the stationary distribution are also included.

1 Introduction Since their introduction by Engle (1982), autoregressive conditional heteroskedastic (ARCH) models and their extension by Bollerslev (1986) to generalised ARCH (GARCH) processes, GARCH models have been used widely by practitioners. At a first glance, their structure may seem simple, but their mathematical treatment has turned out to be quite complex. The aim of this article is to collect some probabilistic properties of GARCH processes. Let (εt )t∈Z be a sequence of independent and identically distributed (i.i.d.) random variables, and let p ∈ N = {1, 2, . . .} and q ∈ N0 = N ∪ {0}. Further, let α0 > 0, α1 , . . . , αp−1 ≥ 0, αp > 0, β1 , . . . , βq−1 ≥ 0 and βq > 0 be nonnegative parameters. A GARCH(p, q) process (Xt )t∈Z with volatility process (σt )t∈Z is then a solution to the equations Xt = σt εt , σt2 = α0 +

t ∈ Z, p  i=1

2 αi Xt−i +

(1) q 

2 βj σt−j ,

t ∈ Z,

(2)

j=1

Alexander M. Lindner Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstrasse 14, D-38106 Braunschweig, e-mail: [email protected]

T.G. Anderson et al., Handbook of Financial Time Series, DOI: 10.1007/978-3-540-71297-8_2, © Springer-Verlag Berlin Heidelberg 2009

43

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where the process (σt )t∈Z is non-negative. The sequence (εt )t∈Z is referred to as the driving noise sequence. GARCH(p, 0) processes are called ARCH(p) processes. The case of a GARCH(0, q) process is excluded since in that case, the volatility equation (2) decouples from the observed process Xt and the driving noise sequence. Note that in some articles (including the original paper by Bollerslev (1986)) the definition of p and q for GARCH processes is interchanged and the process defined in (1) with volatility given by (2) is referred to as GARCH(q, p) rather than GARCH(p, q). It is a desirable property that σt should depend only on the past innovations (εt−h )h∈N , i.e. be measurable with respect to the σ-algebra generated by (εt−h )h∈N . If this condition holds, we shall call the GARCH(p, q) process causal. Then Xt is measurable with respect to the σ-algebra σ(εt−h : h ∈ N0 ) generated by (εt−h )h∈N0 . Also, σt is independent of (εt+h )h∈N0 , and Xt is independent of σ(εt+h : h ∈ N), for fixed t. Often the requirement of causality is added to the definition of GARCH processes. However, since we shall be mainly interested in strictly stationary solutions which turn out to be automatically causal for GARCH processes, we have dropped the requirement at this point. The requirement that all the coefficients α1 , . . . , αp and β1 , . . . , βq are nonnegative ensures that σt2 is non-negative, so that σt can indeed be defined as the square root of σt2 . The parameter constraints can be slightly relaxed to allow for some negative parameters, but such that σt2 will still be nonnegative, see Nelson and Cao (1992). In the present paper, we shall however always assume non-negative coefficients. The paper is organized as follows: in Section 2 we collect the criteria under which strictly stationary and weakly stationary solutions to the GARCH equations exist. The ARCH(∞) representation for GARCH processes is given in Section 3. In Section 4, we focus on conditions ensuring finiteness of moments, and give the autocorrelation function of the squared observations. Section 5 is concerned with the strong mixing property and an application to the limit behaviour of the sample autocorrelation function when sufficiently high moments exist. In Section 6 we shortly mention the tail behaviour of stationary solutions and their continuity properties. GARCH processes indexed by the integers are addressed in Section 7. Finally, some concluding remarks are made in Section 8. For many of the results presented in this paper, it was tried to give at least a short sketch of the proof, following often the original articles, or the exposition given by Straumann (2005).

2 Stationary Solutions Recall that a sequence (Yt )t∈Z of random vectors in Rd is called strictly stationary, if for every t1 , . . . , tk ∈ Z, the distribution of (Yt1 +h , . . . , Ytk +h )

Probabilistic Properties of GARCH

45

does not depend on h for h ∈ N0 . When speaking of a strictly stationary GARCH(p, q) process, we shall mean that the bivariate process (Xt , σt )t∈N0 is strictly stationary.

2.1 Strict stationarity of ARCH(1) and GARCH(1, 1) Now suppose that (p, q) = (1, 1) or that (p, q) = (1, 0), that (εt )t∈Z is i.i.d., and that (Xt , σt )t∈Z satisfy (1), (2). Hence we have a GARCH(1, 1)/ARCH(1) process, whose volatility process satisfies 2 2 2 σt2 = α0 + β1 σt−1 + α1 σt−1 ε2t−1 = α0 + (β1 + α1 ε2t−1 )σt−1 ,

(3)

where β1 := 0 if q = 0. Denoting At = β1 + α1 ε2t ,

Bt = α0 ,

2 and Yt = σt+1 ,

(4)

2 it follows that (Yt )t∈Z = (σt+1 )t∈Z is the solution of the random recurrence equation Yt = At Yt−1 + Bt , where (At , Bt )t∈Z is i.i.d. As we shall see, every strictly stationary solution (σt2 )t∈Z of (3) can be expressed as an appropriate function of the driving noise sequence (εt )t∈Z , so that stationarity of (σt2 )t∈Z implies stationarity of (σt2 , εt )t∈Z and hence of (Xt , σt ). Thus, the question of existence of strictly stationary solutions of the GARCH(1, 1) process can be reduced to the study of strictly stationary solutions of (3). Since we will need multivariate random reccurence equations for the treatment of higher order GARCH processes, we give their definition already in Rd . So let d ∈ N, and suppose (At , Bt )t∈Z is an i.i.d. sequence, where At is a (d × d)-random matrix and Bt is a d-dimensional random vector. The difference equation

Yt = At Yt−1 + Bt ,

t ∈ Z,

(5)

is then called a random recurrence equation (with i.i.d. coefficients), where the solution (Yt )t∈Z is a sequence of d-dimensional random vectors. Every such solution then satisfies Yt = At Yt−1 + Bt = At At−1 Yt−2 + At Bt−1 + Bt = · · · ⎛ ⎞

k k i−1



⎝ = At−i Yt−k−1 + At−j ⎠ Bt−i i=0

i=0

(6)

j=0

−1 for all k ∈ N0 , with the usual convention that j=0 At−j = 1 for the product over an empty index set. Letting k → ∞, itis reasonable to hope k that for a stationary solution, limk→∞ i=0 At−i Yt−k−1 = 0 a.s. and

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 i−1



k that i=0 j=0 At−j Bt−i converges almost surely as k → ∞. In the GARCH(1, 1) and ARCH(1) case, this is indeed the case: let At , Bt and Yt as in (4). By (6), we have

k k i−1



2 2 At−i σt−k + α0 At−j . σt+1 = Yt = i=0

i=0 j=0

 i−1 Since this is a sum of non-negative components, it follows that ∞ i=0 j=0 At−j k converges almost surely for each t, and hence that i=0 At−i converges al2 most  surelyto 0 as k → ∞. Hence if (σt )t∈Z is strictly stationary, then k 2 i=0 At−i σt−k converges in distribution and hence in probability to 0 as k → ∞. So in the ARCH(1) and GARCH(1, 1) case, there is at most one strictly stationary solution (σt2 )t∈Z = (Yt−1 )t∈Z , given by ⎛ ⎞ ∞ i−1 

⎝ At−j ⎠ Bt−i , t ∈ Z. (7) Yt := i=0

j=0

On the other hand, it is clear that if (7) converges a.s. for some and hence all t ∈ Z, where (At , Bt )t∈Z are the i.i.d. coefficients of the random recurrence equation (5) in Rd , then Yt , defined by (7), defines a strictly stationary solution of (5). We have seen that existence of a strictly stationary GARCH(1, 1)/ARCH(1) k process implies almost sure convergence of i=0 A−i to 0 as k → ∞. For the converse, we cite the following result: Proposition 1 (Goldie and Maller (2000), Theorem 2.1) )t∈Z be i.i.d. in R × R. Suppose that P (B0 = 0) < 1, Let d = 1 and (At , Bt n P (A0 = 0) = 0, that i=0 A−i converges almost surely to zero as n → ∞, and that  log q P|B0 | (dq) < ∞, (8) T (log q) A (1,∞) y of |B0| and TA (y) := 0 P (|A0 | < where P|B0 | denotes the distribution  ∞ i−1 e−x ) dx for y ≥ 0. Then i=0 j=0 At−j Bt−i converges almost surely absolutely for every t ∈ Z. In the GARCH(1, 1) / ARCH(1) case,  we have B0 = α0 > 0 and (8) clearly ∞ i−1 holds. Observe that i=0 j=0 At−j Bt−i converges trivially almost surely ∞ if P (A0 = 0) > 0, in which case also i=0 At−i = 0 a.s. Hence we see that a strictly stationary solution of GARCH(1, 1) / ARCH(1) exists if and only k if i=0 A−i converges almost surely to 0 as k → ∞. If P (A0 = 0) > 0 this is clearly the case,  so suppose that β1 > 0 or that P (ε20 > 0) = 1. ∞ Denoting Wt := log At , i=0 A−i = 0 a.s. is then equivalent to the almost

Probabilistic Properties of GARCH

47

 sure divergence to −∞ of the random walk Sn := ni=0 W−n . If EW0+ < ∞, then it is well known that Sn → −∞ if and only if EW0+ < EW0− ≤ ∞, i.e. either EW0− = ∞ or E|W0 | < ∞ with EW0 < 0. Furthermore, Sn cannot diverge almost surely to −∞ as n → ∞ if EW0− < EW0+ = ∞. Observe that in the GARCH(1, 1) case we have β1 > 0, so that W0 ≥ log β1 > −∞, hence EW0− < ∞, and it follows that there exists a strictly stationary solution of the GARCH(1, 1) process if and only if E log(β1 +α1 ε20 ) < 0. In the ARCH(1) case, however, EW0− = ∞ can happen. If EW0− = ∞, it is known from Kesten and Maller (1996) and Erickson (1973), that Sn → −∞ a.s. if and only if  x dP (W0+ ≤ x) < ∞. − (0,∞) E(W0 ∧ x) With W0 = log α1 + log ε20 , the latter condition can be easily seen to be independent of α1 > 0. Summing up, we have the following characterisation of stationary solutions of the GARCH(1, 1) and ARCH(1) equations. For the GARCH(1, 1) case, and for the ARCH(1) case with E log+ (ε20 ) < ∞ this is due to Nelsen (1990). The ARCH(1) case with E log+ (ε20 ) = ∞ was added by Klüppelberg et al. (2004). Here, as usual, for a real number x we set log+ (x) = log(max(1, x)), so that log+ (ε20 ) = (log ε20 )+ . Theorem 1 (Nelsen (1990), Theorem 2, Klüppelberg et al. (2004), Theorem 2.1) (a) The GARCH(1, 1) process with α0 , α1 , β1 > 0 has a strictly stationary solution if and only if −∞ < E log(β1 + α1 ε20 ) < 0.

(9)

This solution is unique, and its squared volatility is given by σt2 = α0

∞ i−1 

(β1 + α1 ε2t−1−j ).

(10)

i=0 j=0

(b) The ARCH(1) process with β1 = 0 and α1 , α0 > 0 has a strictly stationary solution if and only if one of the following cases occurs: (i) P (ε0 = 0) > 0. (ii) E| log ε20 | < ∞ and E log ε20 < − log α1 , i.e. (9) holds. (iii) E(log ε20 )+ < ∞ and E(log ε20 )− = ∞. (iv) E(log ε20 )+ = E(log ε20 )− = ∞ and 



∞

P (log ε20

x 0

−1

x

< −y) dy

dP (log ε20 ≤ x) < ∞.

(11)

0

In each case, the strictly stationary solution is unique, and its squared volatility is given by (10).

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Observe that condition (9) depends on ε20 , α1 and β1 , while conditions (i), (iii) and (iv) in the ARCH case depend on ε20 only. Example 1 (a) Suppose that (εt )t∈Z is i.i.d. with Eε20 ∈ (0, ∞), and suppose that either β1 > 0 (GARCH(1, 1)) or that E| log ε20 | < ∞. Since E log(β1 + α1 ε20 ) ≤ log E(β1 + α1 ε20 ) = log(β1 + E(ε20 ) α1 ) by Jensen’s inequality, a sufficient condition for a strictly stationary solution to exist is that E(ε20 ) α1 + β1 < 0. Now suppose that ε0 is standard normally distributed. If β1 = 0, then  ∞ 2 4 E log(α1 ε20 ) = log α1 + √ log(x)e−x /2 dx = log(α1 ) − (CEM + log(2)), 2π 0  1 where CEM := limN →∞ N n=1 n − log(N ) ≈ 0.57721566 is the EulerMascheroni constant. Hence, the ARCH(1) process with standard normal noise has a strictly stationary solution if and only α1 < 2 exp(CEM ) ≈ 3.562. Since limβ1 ↓0 E log(β1 + α1 ε20 ) = E log(α1 ε20 ), it follows that for every α1 < 2 exp(CEM ) there exists some β(α1 ) > 0 such that the GARCH(1, 1) process with parameters α0 , α1 and β1 ∈ (0, β(α1 )) and standard normal innovations has a strictly stationary solution. In particular, strictly stationary solutions of the GARCH(1, 1) process with α1 + β1 > 1 do exist. However, observe that while α1 may be bigger than 1, β1 < 1 is a necessary condition for a strictly stationary solution to exist. For normal noise, E(log(β1 + α1 ε20 )) can be expressed in terms of confluent and generalised hypergeometric functions, which in turn can be calculated numerically. See Nelsen (1990), Theorem 6, for details. (b) Consider the ARCH(1) process with α1 > 0, and  let (εt )t∈Z be i.i.d. such that the distribution of ε0 has atoms at ± 2 − E2 (2) with mass 1/4 each, and an absolutely continuous component  ∞ with density fε (x) = (4|x|(log |x|)2 )−1 1(−1/e,1/e) (x). Here, En (x) = 1 e−xt /tn dt denotes the  1/e exponential integral, and it holds E2 (2) ≈ 0.0375. Since −1/e fε (x) dx =  −1 (2y 2 )−1 dy = 1/2, fε indeed defines a probability distribution. Moreover, −∞ since ε0 is symmetric, we have Eε0 = 0 and Eε20

1 = 2

 0

1/e

1 1 x dx+ (2−E2 (2)) = 2 (log x) 2 2



−1

−∞

1 e2y dy + (2−E2 (2)) = 1. 2 y 2

The absolutely continuous component of log ε20 can be easily seen to have density x → (2x2 )−1 1(−∞,−1) (x), so that E(log ε20 )− = ∞. Since E(log ε20 )+ < ∞, the ARCH(1) process with α1 > 0 and the given distribution of the (εt )t∈Z has a unique strictly stationary solution by Case (iii) of the previous

Probabilistic Properties of GARCH

49

Theorem. (c) Let (εt )t∈Z be i.i.d. with marginal density ⎧ ⎨ (2|x|(log |x|)3/2 )−1 , |x| > e, fε (x) = (4|x|(log |x|)2 )−1 , 0 < |x| < 1/e, ⎩ 0, else. Then the density of log ε20 is given by ⎧ −3/2 , x > 1, ⎨x flog ε2 (x) = (2x2 )−1 , x < −1, ⎩ 0, x ∈ [−1, 1]. We conclude that E(log ε20 )+ = E(log ε20 )− = ∞, and it is easily checked that (11) is satisfied. Hence, a unique strictly stationary solution of the ARCH(1) process with driving noise (εt )t∈Z exists.

2.2 Strict stationarity of GARCH(p, q) For higher order GARCH processes, one has to work with multidimensional random recurrence equations. Consider a GARCH(p, q) process (Xt )t∈Z with volatility (σt )t∈Z and driving noise sequence (εt )t∈Z . Let p := max(p, 2), q := max(q, 2) and define the random ( p + q − 1)–vectors Yt and Bt by 2 2 2 2  Yt = (σt+1 , . . . , σt− p+2 , Xt , . . . , Xt− q+2 )

(12)

and Bt = (α0 , 0, . . . , 0) ∈ Rp+q−1 , respectively. Further, let βq+1 = β2 = 0 if q ≤ 1, and α2 = 0 if p = 1, and define the random ( p + q − 1) × ( p + q − 1)-matrix At by ⎞ ⎛ β1 + α1 ε2t β2 · · · βq−1 βq α2 · · · αp−1 αp ⎜ 1 0 ··· 0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜ 0 1 ··· 0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜ .. ⎟ .. .. .. .. .. .. . . .. ⎟ ⎜ . . . . . . . . . ⎟ ⎜ ⎟. 0 0 · · · 1 0 0 · · · 0 0 (13) At = ⎜ ⎟ ⎜ 2 ⎟ ⎜  0 · · · 0 0 0 · · · 0 0 t ⎟ ⎜ ⎜ 0 0 ··· 0 0 1 ··· 0 0 ⎟ ⎟ ⎜ ⎜ .. ⎟ .. .. .. . . .. .. . . .. ⎝ . . . . . ⎠ . . . . 0 0 ··· 0 0 0 ··· 1 0 These matrices where introduced by Bougerol and Picard (1992a). It is then easy to see that each strictly stationary solution of the GARCH equations

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(1), (2) gives rise to a strictly stationary solution of the random recurrence equation (5) with Yt , Bt and At as defined in (12) and (13), and vice versa. Observe that for p = q = 1 and for (p, q) = (1, 0), the random recurrence equation with At and Bt as in (12) and (13) differs from the one with At and Bt as in (4). In fact, the former is a random recurrence equation in R3 , while the latter is one-dimensional. Strict stationarity of multivariate random recurrence equations is studied in terms of the top Lyapunov exponent. Let  ·  be any vector norm in Rd . For a matrix M ∈ Rd×d , the corresponding matrix norm M  is defined by M  :=

sup x∈R,x=0

M x . x

Definition 1 Let (An )n∈Z be an i.i.d. sequence of d × d random matrices, such that E log+ A0  < ∞. Then the top Lyapunov exponent associated with (An )n∈Z is defined by   1 γ := inf E A0 A−1 · · · A−n  . n∈N0 n+1 Furstenberg and Kesten (1960) showed that 1 log A0 A−1 · · · A−n  n→∞ n + 1

γ = lim

(14)

almost surely, and an inspection of their proof shows that γ is independent of the chosen vector norm (hence matrix norm). The existence of stationary solutions of random recurrence equations can be described neatly in terms of strict negativity of the associated top Lyapunov exponent. Namely, Bougerol and Picard (1992b) have shown that so called irreducible random recurrence equations with i.i.d. coefficients (At , Bt )t∈Z , such that E log+ A0  < ∞ and E log+ B0  < ∞, admit a nonanticipative strictly stationary solution if and only if the top Lyapunov exponent associated with (At )t∈Z is strictly negative. Here, nonanticipative means that Yt is independent of (At+h , Bt+h )h∈N for each t. For GARCH(p, q) cases, it is easier to exploit the positivity of the coefficients in the matrix At rather than to check that the model is irreducible. The result is again due to Bougerol and Picard: Theorem 2 (Bougerol and Picard (1992a), Theorem 1.3) Let (εt )t∈Z be an i.i.d. sequence of random variables such that E(log ε20 )+ < ∞. Let α0 , . . . , αp , β1 , . . . , βq be GARCH(p, q) parameters, and let the ( p + q− 1)× ( p + q− 1) random matrices At as well as the ( p + q− 1)–vectors Bt be defined as in (13) and (12), respectively. Then the corresponding GARCH(p, q) process admits a strictly stationary solution if and only if the top Lyapunov exponent γ associated with the sequence (At )t∈Z is strictly negative. This solution is unique, and the random vector Yt defined in (12) satisfies (7).

Probabilistic Properties of GARCH

51

The fact that every strictly stationary solution must be unique and of the form (7) follows with a refined argument similar to the GARCH(1, 1) case, using that every element in the vectors Yt and in the matrices At must be non-negative. In particular this shows that every strictly stationary solution must be causal (the argument here does not require the assumption of finite log-moments). Further, existence of a strictly stationary solution implies limk→∞ A0 A−1 · · · A−k  = 0 a.s. Since (An )n∈Z is i.i.d. and E log+ A0  < ∞, this in turn implies strict negativity of the top Lyapunov exponent γ (see Bougerol and Picard (1992b), Lemma 3.4). That γ < 0 implies convergence of (7) can be seen from the almost sure convergence in (14), which implies  ⎛ ⎞   k−1  

γk/2  ⎝ ⎠ A B t−j t−k  ≤ Ct e    j=0 for some random variable Ct . Hence, the series (7) converges almost surely and must be strictly stationary. That strict negativity of the top Lyapunov exponent implies convergence of (7) and hence the existence of strictly stationary solutions is true for a much wider class of random recurrence equations, see e.g. Kesten (1973), Vervaat (1979), Brandt (1986) or Bougerol and Picard (1992b). Due to its importance, we state the observation made after Theorem 2 again explicitly: Remark 1 A strictly stationary solution to the GARCH equations (1) and (2) is necessarily unique and the corresponding vector Yt defined in (12) satisfies (7). In particular, every strictly stationary GARCH process is causal. For matrices, it may be intractable to obtain explicit expressions for the top Lyapunov exponent and hence to check whether it is strictly negative or not. Often, one has to use simulations based on (14) to do that. If the noise sequence has finite variance, however, Bollerslev gave a handy sufficient condition for the GARCH process to have a strictly stationary solution, which is easy to check (part (a) of the following theorem). Bougerol and Picard showed that the boundary values in this condition can still be attained under certain conditions, and they have also given a necessary condition for strictly stationary solutions to exist: Corollary 1 (Bollerslev (1986), Theorem 1, Bougerol and Picard (1992a), Corollaries 2.2, 2.3) Let (εt )t∈Z be the driving noise sequence of a GARCH(p, q) process, and suppose that 0 <Eε20 < ∞.Then the following hold: p q (a) If E(ε20 ) i=1 αi + j=1 βj < 1, then the GARCH(p, q) process admits a unique strictly stationary solution. (b) If P (ε0 = 0) = 0, ε0 hasunboundedsupport, p, q ≥ 2 and α1 , . . . , αp > 0, β1 , . . . , βq > 0, and E(ε20 ) pi=1 αi + qj=1 βj = 1, then the GARCH(p, q) process admits a unique strictly stationary solution.

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 (c) If qj=1 βj ≥ 1, then no strictly stationary solution of the GARCH(p, q) process exists. For the proof of Corollary 1, one may assume that Eε20 = 1. The general result then follows by an easy transformation. If Eε20 = 1, Bougerol and Picard (1992a) prove (b) by showing that the spectral radius ρ(E(A0 )) of the matrix E(A0 ) is equal to 1. Recall that the spectral radius ρ(C) of a square matrix C is defined by ρ(C) = sup {|λ| : λ eigenvalue of C}. Since A0 is almost surely not bounded, neither has zero columns nor zero rows, and has non-negative entries, it follows from Theorem 2 of Kesten and Spitzer (1984) that γ < log ρ(E(A0 )) = 0. The proofs of (a) and (c) are achieved by similar reasoning, using estimates between the top Lyapunov exponent and the spectral radius. In particular, in case (a) one has γ ≤ log ρ(E(A0 )) < 0. p qFor real data one often estimates parameters αi and βj such that i=1 αi + j=1 βj is close to one, when assuming noise with variance 1. In analogy to the integrated ARMA (ARIMA) p process, q Engle and Bollerslev (1986) call GARCH processes for which i=1 αi + j=1 βj = 1 integrated GARCH(p, q) processes, or IGARCH(p, q) processes, for short. Observe that Corollary 1(b) shows that IGARCH processes may have a strictly stationary solution, unlike ARIMA processes where a unit root problem occurs. Remark 2 Let ε0 , p, q and α1 , . . . , αp , β1 , . . . , βq be as in Corollary 1(b). Then there exists δ > 0 such that for all α i ≥ 0, βj ≥ 0 with | αi − αi | < δ  (i = 1, . . . , p) and |βj − βj | < δ (j = 1, . . . , q), the GARCH(p, q) process with parameters α0 , α 1 , . . . , α p , β1 , . . . , βq and noise sequence (εt )t∈Z admits a unique strictly stationary solution. In particular, p there exist q strictly stationi + j=1 βj > 1. This ary GARCH(p, q) processes for which E(ε20 ) i=1 α follows immediately from Definition 1 and Theorem 2, since for the parameters of Corollary 1(b), the top Lyapunov exponent γ is strictly negative.

2.3 Ergodicity Let Y = (Yt )t∈Z be a strictly stationary time series of random vectors in Rk . Then Y can be seen as a random element in (Rk )Z , equipped with its Borel-σalgebra B((Rk )Z ). Let the backshift operator ΦBS : (Rk )Z → (Rk )Z be given by ΦBS ((zi )i∈Z ) = (zi−1 )i∈Z . Then the time series (Yt )t∈Z is called ergodic if ΦBS (Λ) = Λ for Λ ∈ B((Rk )Z ) implies P (Y ∈ Λ) ∈ {0, 1}. See e.g. Ash and Gardner (1975) for this and further properties of ergodic time series. In particular, it is known that if (gn )n∈Z is a sequence of measurable functions

Probabilistic Properties of GARCH

53

gn : (Rk )Z → Rd such that gn−1 = gn ◦ ΦBS and Y = (Yt )t∈Z is strictly stationary and ergodic with values in Rk , then (gn (Y ))n∈Z is also strictly stationary and ergodic (see e.g. Brandt et al. (1990), Lemma A 1.2.7). Since the sequence (At , Bt )t∈Z is i.i.d. and hence strictly stationary and ergodic for a GARCH process, it follows that every strictly stationary GARCH process is ergodic, since it can be expressed via (7). This is due to Bougerol and Picard (1992a), Theorem 1.3.

2.4 Weak stationarity Recall that a time series (Zt )t∈Z of random vectors in Rd is called weakly stationary or wide-sense stationary, if EZt 2 < ∞ for all t ∈ Z, E(Zt ) ∈ Rd is independent of t ∈ Z, and the covariance matrices satisfy Cov(Zt1 +h , Zt2 +h ) = Cov(Zt1 , Zt2 ) for all t1 , t2 , h ∈ Z. Clearly, every strictly stationary sequence which satisfies EZ0 2 < ∞ is also weakly stationary. For causal GARCH processes, we shall see that the converse is true also, i.e. that every causal weakly stationary GARCH process is also strictly stationary. Let (Xt , σt ) be a GARCH process such that σt is independent of εt , which is in particular satisfied for causal solutions. Then if P (ε0 = 0) < 1, it follows from (1) and the independence of σt and εt that for given r ∈ (0, ∞), E|Xt |r < ∞ if and only if E|εt |r < ∞ and Eσtr < ∞. Suppose Eε20 ∈ (0, ∞), and that (Xt , σt ) is a GARCH(p, q) process such that Eσt2 = Eσt2 < ∞ for all t, t ∈ Z. Then (2) shows that E(σ02 ) = α0 +

p  i=1

αi E(σ02 ) E(ε20 ) +

q 

βj E(σ02 ).

j=1

Hence we see that a necessarycondition  for a causal weakly stationary sop q lution to exist is that E(ε20 ) i=1 αi + j=1 βj < 1. Now suppose that (σt )t∈Z is a causal weakly stationary solution, and for simplicity assume that Eε20 = 1. With Yt , Bt and At as in (12) and (13), Yt must satisfy (6). Note ∞ i−1 that then i=0 j=0 At−j Bt−i converges a.s. to the strictly stationary   k solution by Corollary 1. By (6), this implies that A i=0 t−i Yt−k−1 converges almost surely to some finite random variable as k → ∞. If this limit can be seen to be 0, then it follows that the weakly stationary solution must coincide with the strictly stationary. As remarked after Corollary 1, the spectral radius of E(A0 ) is less than 1. Hence there is some N ∈ N such that (EA0 )N  = E(A 0 · · · A−N +1 )  < 1. By  causality and weak stationarity, k Y A this implies that E t−k−1 converges to 0 as k → ∞, and since i=0 t−i

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k

each of the components of i=0 At−i Yt−k−1 is positive, Fatou’s lemma shows that its almost sure limit must be 0, so that every causal weakly stationary solution is also strictly  stationary. Conversely, if (Yt )t∈Z is a strictly stationary solution and E(ε20 ) pi=1 αi + qj=1 βq < 1 with Eε20 = 1 for simplic    i−1 ity, it follows from (EA0 )N  < 1 that ∞ i=0 E j=0 At−j Bt−i is finite, and since each of its components is positive, this implies that EYt  < ∞ for the strictly stationary solution. Summing up, we have the following characterisation of causal weakly stationary solutions, which was derived by Bollerslev (1986). Theorem 3 (Bollerslev (1986), Theorem 1) Let (εt )t∈Z be such that Eε20 < ∞. Then the GARCH(p, q) process  (Xt , σt )t∈Z p 2 admits a causal weakly stationary solution if and only E(ε ) 0 i=1 αi + q j=1 βj < 1. In that case, the causal weakly stationary solution is unique and coincides with the unique strictly stationary solution. It holds E(σt2 ) =

1 − E(ε20 )

p

α0

i=1 αi −

q

j=1 βj

,

E(Xt2 ) = E(σt2 )E(ε20 ).

(15)

3 The ARCH(∞) Representation and the Conditional Variance Often it can be helpful to view a GARCH(p, q) process as an ARCH process of infinite order. In particular, from the ARCH(∞) representation one can easily read off the conditional variance of Xt given its infinite past (Xs : s < t). Originally, Engle (1982) and Bollerslev (1986) defined ARCH and GARCH processes in terms of the conditional variance. Equation (18) below then shows that this property does hold indeed, so that the definition of GARCH processes given here is consistent with the original one of Engle and Bollerslev. Theorem 4 (Bollerslev (1986), pp. 309–310) Let (Xt , σt )t∈Z be a strictly stationary p GARCH(p, q q) process driven by (εt )t∈Z , such that Eε20 < ∞ and E(ε20 ) i=1 αi + j=1 βj < 1. Then there is a sequence (ψj )j∈N0 of real constants such that ψ0 > 0, ψj ≥ 0 for all j, ∞ ψ < ∞, and j=0 j ∞  2 σt2 = ψ0 + ψi Xt−i . (16) i=1

The constants are determined by

Probabilistic Properties of GARCH

55

α 0q , 1 − j=1 βj p ∞ i  j i=1 αi z  ψj z = , q 1 − j=1 βj z j j=1 ψ0 =

z ∈ C,

|z| ≤ 1.

(17)

In particular, σt2 is measurable with respect to the infinite past (Xs : s ≤ t − 1), and the conditional expectation and variance of Xt given (Xs : s < t) are given by E(Xt |Xs : s < t) = E(ε0 ) σt

and

V (Xt |Xs : s < t) = V (ε0 ) σt2 ,

(18)

respectively. For example, if (εt )t∈Z is i.i.d. standard normal, then conditionally on (Xs : s < t), Xt is N (0, σt2 ) distributed, since σt2 is a Borel function of (Xs : s < t). ARCH(∞) models were introduced in more generality by Robinson (1991). The explicit expression in (16) can be found in Bollerslev (1986) or Nelson and Cao (1992). It can be derived defining St := σt2 − E(σt2 ),

Zt := Xt2 − E(Xt2 ),

t ∈ Z.

(19)

Then (2) is equivalent to St −

q  j=1

βj St−j =

p 

αi Zt−i .

(20)

i=1

This is an ARMA equation  for (St )t∈Z such that supt∈Z E|Zt | < ∞ and q β < 1, this ARMA equation is causal, E(St ) = E(Zt ) = 0. Since ∞ j=1 j and it follows that St = j=1 ψj Zt−j where (ψj )j∈N are given by (17). An easy calculation prevails that ψj ≥ 0, and resubstituting σt2 and Xt2 in this ARMA equation shows (16). Hence σt is measurable with respect to the σalgebra generated by (Xs : s < t), while εt is independent of this σ-algebra by causality. This then implies (18). In the literature there exist many other examples of ARCH(∞) models apart from GARCH(p, q). For more information and references regarding ARCH(∞) models, see Giraitis et al. (2006) and (2008).

4 Existence of Moments and the Autocovariance Function of the Squared Process It is important to know whether the stationary solution has moments of higher order. For example, in Theorem 3, we have seen that the pstrictly stationary solution has finite second moments if and only if E(ε20 ) i=1 αi +

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q

2 2 j=1 βj < 1, and we have given an explicit expression for Eσt and EXt . However, one is also interested in conditions ensuring finiteness of moments of higher order, the most important case being finiteness of Eσt4 and EXt4 . For the GARCH(1, 1) process with normal innovations, a necessary and sufficient condition for such moments to exist has been given by Bollerslev (1986), and extended by He and Teräsvirta (1999b) to general noise sequences. Ling (1999) and Ling and McAleer (2002) give a necessary and sufficient condition for moments of higher order to exist. For ARCH(p) processes, a necessary and sufficient condition for higher order moments to exist was already obtained earlier by Milhøj (1985). Observe that if P (ε0 = 0) < 1, then by independence of Xt and σt for strictly stationary and hence causal solutions, the m’th moment of Xt = σt εt exists if and only Eσtm < ∞ and E|εt |m < ∞. Hence we shall only be concerned with moment conditions for σt2 . In most cases, εt will be a symmetric distribution, so that the odd moments of εt and hence Xt will be zero. The main concern is hence on even moments of GARCH processes.

4.1 Moments of ARCH(1) and GARCH(1, 1) The following theorem gives a complete characterisation when the (possible fractional) moment of a GARCH(1, 1) or ARCH(1) process exists: Theorem 5 (Bollerslev (1986), Theorem 2, and He and Teräsvirta (1999b), Theorem 1) Let (Xt , σt ) be a strictly stationary GARCH(1, 1) or ARCH(1) process as in (1), (2). Let m > 0. Then the (fractional) m’th moment E(σt2m ) of σt2 exists if and only if E(β1 + α1 ε20 )m < 1. (21) If m is a positive integer and this condition is satisfied, and μj := E(σt2j ) denotes the j’th moment of σt2 , then μm can be calculated recursively by μm = (1 − E(β1 +

α1 ε20 )m )−1

m−1  j=0

 m m−j α0 E(β1 + α1 ε20 )j μj . j

(22)

The (2m)’th moment of Xt is given by E(Xt2m ) = μm E(ε2m 0 ). That condition (21) is necessary and sufficient for finiteness of E(σt2m ) (m ∈ (0, ∞)) can be easily seen from representation (10): for if E(β1 + α1 ε20 )m < 1 and m ∈ [1, ∞), then Minkowski’s inequality shows that

Probabilistic Properties of GARCH

(E(σt2m ))1/m ≤ α0

57 ∞ 

(E(β1 + α1 ε20 )m )i/m < ∞,

i=0

and for m < 1 one uses similarly E(U + V )m ≤ EU m + EV m for posi tive random variables U, V . Conversely, if E(σt2m ) < ∞, then E i−1 j=0 (β1 + 2 m α1 εt−1−j ) must converge to 0 as i → ∞, which can only happen if (21) holds. Finally, if m is an integer and (21) holds, then (22) follows easily by raising (2) to the m’th power and taking expectations. ! m−j m Example 2 For an integer m, E(σt2m ) is finite if and only if j=0 m j β1 αj1 Eε2j t < 1. If εt is standard normally distributed, this means that j ∞    m m−j j

β1 α1 (2i − 1) < 1. j j=0 i=1

For example, the fourth moment of σt exists if and only if β12 + 2β1 α1 + 3α21 < 1. As an immediate consequence of Theorem 5, one sees that GARCH processes do not have finite moments of all orders if ε0 has unbounded support, which is a first indication that GARCH processes will generally have heavy tails: Corollary 2 Let (Xt , σt : t ∈ Z) be a strictly stationary GARCH(1, 1) or ARCH(1) process and assume that P (α1 ε20 + β1 > 1) > 0. Then there is r ≥ 1, such that Eσ02r = E|X0 |2r = ∞.

4.2 Moments of GARCH(p, q) For GARCH processes of higher order, Ling (1999) and Ling and McAleer (2002) give necessary and sufficient conditions for even moments of σt to be finite. In order to state their result, we need the notion of the Kronecker product of two matrices. For an (m × n)-matrix C = (cij )i=1,...,m,j=1,...,n and a (p × r)-matrix D, the Kronecker product C ⊗ D is the (mp × nr)-matrix given by ⎛ ⎞ c11 D · · · c1n D ⎜ ⎟ C ⊗ D = ⎝ ... . . . ... ⎠ . cm1 D · · · cmn D See e.g. Lütkepohl (1996) for elementary properties of the Kronecker product. We then have:

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Theorem 6 (Ling and McAleer (2002), Theorem 2.1) Let (Xt , σt )t∈Z be a strictly stationary GARCH(p, q) process as in (1), (2), and assume that α1 + β1 > 0. Let At be the ( p + q − 1) × ( p + q − 1) matrix of (13). Let m ∈ N. Then the m’th moment of σt2 is finite if and only if the spectral radius of the matrix E(A⊗m ) is strictly less than 1. t Originally, Ling and McAleer (2002) formulated their result in terms of the spectral radius of a matrix corresponding to another state space representation of GARCH processes than the At -matrix defined in (13). The proof, however, is quite similar. We shortly sketch the argument: Suppose that ρ(E(A⊗m )) = lim supn→∞ (E(A⊗m ))n 1/n < 1. Then there is t t ⊗m n n λ ∈ (0, 1) such that (E(At ))  ≤ λ for large enough n, so that the supremum of all elements of (E(A⊗m ))n decreases exponentially as n → ∞. The t  same is then true for all elements of (E(A⊗m ))n for every m ∈ {1, . . . , n}. t Now take the m’th Kronecker power of the representation (7) for the vector Yt defined in (12). For example, for m = 2, one has (since Bt = Bt−i in (12)) ⎛⎛ ⎞ ⎞ ⎛⎛ ⎞ ⎞ i

i

∞ ∞  1 −1 2 −1  ⎝⎝ At−j1 ⎠ Bt ⎠ ⊗ ⎝⎝ At−j2 ⎠ Bt ⎠ Yt⊗2 = i1 =0 i2 =0

=

∞ ∞  

⎛ ⎝

i1 =0 i2 =i1

+

∞ i 1 −1  i1 =1 i2 =0

j1 =0 i

1 −1

⎠⎝ A⊗2 t−j1

j1 =0

⎛ ⎝

⎞⎛

i

2 −1

j2 =0

j2 =0 i

2 −1

(Id ⊗ At−j2 )⎠ Bt⊗2

j2 =i1

⎞⎛ ⎠⎝ A⊗2 t−j2

⎞

i

1 −1

⎞ (At−j1 ⊗ Id)⎠ Bt⊗2 ,

j1 =i2

where Id denotes the ( p + q − 1) × ( p + q − 1) identity matrix. Taking expectations and using the exponential decay of the elements, which are all non-negative, this then shows that E(Yt⊗m ) is finite, and hence that E(σt2m ) < ∞. The converse is established along similar lines: finiteness of E(σt2m ) implies finiteness of E(Yt⊗m ). Using the fact that all appearing matrices vectors have non-negative entries, this then implies finiteness of ∞ and ⊗m ))i B0⊗m as argued by Ling and McAleer (2002), and making i=0 (E(At use α1 + β1 > 0, this can be shown to imply finiteness of ∞of the assumption ⊗m i )) , showing that ρ(E(A⊗m )) < 1. t i=0 (E(At ) is less than To check whether the spectral radius of the matrix E(A⊗m t 1 or not may be tedious or only numerically achievable. A simple sufficient condition for the existence of moments can however be obtained by developing the ARCH(∞) representation (16) into a Volterra series expansion, as described by Giraitis et al. (2006) and (2008). Accordingly, a sufficient condition moment of σt2 in an ARCH(∞) process to exist is that ∞ for the m’th 2m 1/m )) < 1. This was shown by Giraitis et al. (2000) for j=1 ψj (E(|ε0 | m = 2 and observed to extend to hold for general m ≥ 1 by Giraitis et al. (2006). With (17), this gives for the GARCH(p, q) process:

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59

Proposition 2 (Giraitis et al. (2006), Theorem 2.1) Let (Xt , σt )t∈Z be a strictly stationary GARCH(p, q) process as in (1), (2), let m ∈ [1, ∞), and suppose that 0 < E|ε0 |2m < ∞. Then 

m p αi i=1 q E|ε0 |2m < 1 1 − j=1 βj is a sufficient condition for E(σ02m ) < ∞.  Observe that 0 < E|ε0 |2m < ∞ implies qj=1 βj < 1 by Corollary 1(c), so that the expressions in the condition above are well-defined. In econometrics, the kurtosis is often seen as an indicator for tail heaviness. Recall that the kurtosis KR of a random variable R with ER4 < ∞ is defined ER4 by KR = (ER 2 )2 . If (Xt , σt ) is a stationary GARCH process which admits finite fourth moment, then if follows from Jensen’s inequality that EXt4 = E(ε4t )E(σt4 ) ≥ E(ε4t )(E(σt2 ))2 = Kε0 (E(Xt2 ))2 , so that KX0 ≥ Kε0 . This shows that the kurtosis of the stationary solution is always greater or equal than the kurtosis of the driving noise sequence, giving another indication that GARCH processes lead to comparatively heavy tails. While Theorem 6 gives a necessary and sufficient condition for even moments to exist, it does not give any information about the form of the moment. The most important higher order moment is the fourth moment of σt , and an elegant method to determine Eσt4 was developed by Karanasos (1999). To illustrate it, suppose that (Xt , σt )t∈Z is a strictly stationary GARCH(p, q) process as in (1), (2), such that E(σt4 ) < ∞, and denote w := Eε20 , v := Eε40 , f := Eσ04 and g := Eσ02 =

1−w

α0 q , i=1 αi − j=1 βj

p

where we used (15). Then w, v, and g are known and we want to determine f . For i ∈ N, denote further 2 2 λi := E(σt2 Xt−i ) and ci := E(σt2 σt−i ).

Since E(Xt2 |εt−h : h ∈ N) = wσt2 , it further holds for i ∈ N, 2 2 wλi = E(Xt2 Xt−i ), wci = E(Xt2 σt−i ), and f = E(Xt2 σt2 )/w = E(Xt4 )/v.

Then, taking expectations in each of the equations

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Xt2 σt2 = Xt2

α0 +

=

2 σt−j

α0 +

2 σt2 Xt−j

=

2 Xt−j

2 αi Xt−i +

i=1 p 

2 σt2 σt−j

p 

α0 +

2 βi σt−i

i=1 q 

2 αi Xt−i

i=1 p 

q 

+

2 αi Xt−i

+

2 βi σt−i

i=1 q 

i=1

,

,

j = 1, . . . , q,

2 βi σt−i

,

j = 1, . . . , p,

i=1

one obtains wf = α0 wg +

p 

wαi λi +

i=1

q 

(23)

wβi ci ,

i=1

cj = α0 g + (wαj + βj )f +

j−1 

(wαj−i + βj−i )ci

i=1

+

p−j  i=1

αj+i λi +

q−j 

βj+i ci ,

j = 1, . . . , q,

(24)

i=1

λj = α0 wg + (vαj + wβj )f +

j−1  (wαj−i + βj−i )λi i=1

+

p−j  i=1

wαj+i λi +

q−j 

wβj+i ci ,

j = 1, . . . , p,

(25)

i=1

where αi = 0 for i > p and βi = 0 for i > q. Substituting cq from (24) and λp from (25) into (23), one obtains a system of (p + q − 1) equations for the unknown variables (f, c1 , . . . , cq−1 , λ1 , . . . , λp−1 ). See Karanasos (1999), Theorem 3.1, for more information. For another approach to obtain necessary conditions for the fourth moment to exist and to obtain its structure, we refer to He and Teräsvirta (1999a), Theorem 1.

4.3 The autocorrelation function of the squares If the driving noise process of a strictly and weakly stationary GARCH process has expectation Eε0 = 0, then EXt = E(ε0 ) E(σt ) = 0, and for h ∈ N it follows from (18) that E(Xt Xt−h ) = E E(Xt Xt−h |Xs : s < t) = E(Xt−h E(ε0 )σt ) = 0, so that (Xt )t∈Z is (weak) White Noise (provided Eε20 = 0), i.e. a weakly stationary sequence whose elements are uncorrelated. This uncorrelatedness is however not preserved in the squares of the GARCH process. Rather do

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61

the squares (Xt2 )t∈Z satisfy an ARMA equation. This was already observed by Bollerslev (1986), (1988). More precisely, we have: Theorem 7 (Bollerslev (1986), Section 4) Let (Xt , σt )t∈Z be a strictly stationary GARCH(p, q) process such that Eσ04 < ∞, Eε40 < ∞ and Var(ε20 ) > 0. Define ut := Xt2 − (Eε2t )σt2 = (ε2t − E(ε2t ))σt2 ,

t ∈ Z.

(26)

Then (ut )t∈Z is a White Noise sequence with mean zero and variance E(σ04 ) Var(ε20 ), and St := σt2 − and Wt := Xt2 −

1 − (Eε20 )

α p 0 i=1

αi −

q j=1

βj

,

α0 Eε20   , 1 − (Eε20 ) pi=1 αi − qj=1 βj

t ∈ Z,

t ∈ Z,

satisfy the causal ARMA(max(p, q), p − 1) and causal ARMA(max(p, q), q) equations 

max(p,q)

St −

i=1

((Eε20 )αi + βi )St−i =

p 

αi ut−i ,

t ∈ Z,

i=1

and 

max(p,q)

Wt −

i=1

((Eε20 )αi + βi )Wt−i = ut −

q 

βj ut−j ,

t ∈ Z,

j=1

respectively. Here, αi = 0 for i > p and βj = 0 for j > q. In particular, the autocovariance and autocorrelation functions of (σt2 )t∈Z and that of (Xt2 )t∈Z are those of the corresponding ARMA processes. The fact that (ut )t∈Z is White Noise follows in complete analogy to the White Noise property of (Xt )t∈Z by using (18). The ARMA representations then follow by inserting (26) into (2), and they are causal by Theorem 3. Observe that the ARMA equation for (St )t∈Z is actually an ARMA(max(p, q), p − 1)-equation driven by (ut−p )t∈Z , where p := min{j ∈ {1, . . . , p} : αj = 0}. For general expressions for the autocoviarance functions of ARMA processes, see Brockwell and Davis (1991), Section 3.3.

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5 Strong Mixing Mixing conditions describe some type of asymptotic independence, which may be helpful in proving limit theorems, e.g. for the sample autocorrelation function or in extreme value theory. There exist many types of mixing conditions, see e.g. Doukhan (1994) for an extensive treatment. For GARCH processes, under weak assumptions one has a very strong notion of mixing, namely β-mixing, which in particular implies strong mixing: let Y = (Yt )t∈Z be a strictly stationary time series in Rd , defined on a probability space 0 (Ω, F , P ). Denote by F−∞ the σ-algebra generated by (Ys : s ≤ 0) and by ∞ Ft the σ-algebra generated by (Ys : s ≥ t), and for k ∈ N let (SM)

αk

(SM)

βk

:=

:=

sup

0 C∈F−∞ ,D∈Fk∞

|P (C ∩ D) − P (C)P (D)|,

J I   1 sup |P (Ci ∩ Dj ) − P (Ci )P (Dj )|, 2 i=1 j=1 (SM)

where in the definition of βk the supremum is taken over all pairs of 0 finite partitions {C1 , . . . , CI } and {D1 , . . . , DJ } of Ω such that Ci ∈ F−∞ (SM) (SM) for each i and Dj ∈ Fk∞ for each j. The constants αk and βk are the α-mixing coefficients and β-mixing coefficients, respectively, and (Yt )t∈Z is (SM) called strongly mixing (or α-mixing) if limk→∞ αk = 0, and β-mixing (or (SM) absolutely regular) if limk→∞ βk = 0. It is strongly mixing with geometric (SM) rate if there are constants λ ∈ (0, 1) and c such that αk ≤ cλk for every k, i.e. if αk decays at an exponential rate, and β-mixing with geometric rate is defined similarly. Since (SM)

αk

≤

1 (SM) β , 2 k

β-mixing implies strong mixing. Based on results of Mokkadem (1990), Boussama (1998) showed that GARCH processes are beta mixing with geometric rate under weak assumptions, see also Boussama (2006). The proof hereby relies on mixing criteria for Markov chains as developed by Feigin and Tweedie (1985), see also Meyn and Tweedie (1996). Observe that the sequence Y = (Yt )t∈N0 of random vectors + q −1 defined by (12) defines a discrete time Markov chain with state space Rp+ . Boussama (1998) then shows that under suitable assumptions on the noise sequence this Markov chain is geometrically ergodic, i.e. there is a constant λ ∈ (0, 1) such that lim λ−n pn (y, ·) − π(·)T V = 0.

n→∞

Probabilistic Properties of GARCH

63

+ q−1 + q −1 Here, pn (y, E) for y ∈ Rp+ and E ∈ B(Rp+ ) denotes the n-step transition probability from y to E, i.e.

pn (y, E) = P (Yn ∈ E|Y0 = y), π denotes the initial distribution of Y0 which is chosen to be the stationary one, and  · T V denotes the total variation norm of measures. Since geometric ergodicity implies β-mixing of (Yt )t∈Z with geometric rate, using the causality it can be shown that this in turn implies β-mixing of (σt , εt )t∈Z and hence of (Xt )t∈Z . Originally, the results in Boussama (1998) and (2006) are stated under the additional assumption that the noise sequence has finite second moment, but an inspection of the proof shows that it is sufficient to suppose that E|ε0 |s < ∞ for some s > 0. The next Theorem gives the precise statements. See also Basrak et al. (2002), Corollary 3.5, and Mikosch and Straumann (2006), Theorem 4.5 and Proposition 4.10. Theorem 8 (Boussama (1998), Théorème 3.4.2) Let (Xt , σt )t∈Z be a strictly stationary GARCH(p, q) process as in (1), (2), and suppose the noise sequence is such that ε0 is absolutely continuous with Lebesgue density being strictly positive in a neighbourhood of zero, and such that there exists some s ∈ (0, ∞) such that E|ε0 |s < ∞. Let Yt be defined as in (12). Then (Yt )t∈Z is β-mixing with geometric rate. In particular, (σt2 )t∈Z , (Xt2 )t∈Z and (Xt )t∈Z are β-mixing and hence strongly mixing with geometric rate. An important application of strong mixing is the asymptotic normality of the sample autocovariance and autocorrelation function, under suitable moment conditions. Recall that the sample autocovariance function of a time series (Zt )t∈Z based on observations Z1 , . . . , Zn is defined by γZ,n (h) :=

n−h 1  (Zt − Z n )(Zt+h − Z n ), n t=1

h ∈ N0 ,

n where Z n := n1 t=1 Zt denotes the sample mean. Similarly, the sample autocorrelation function is given by ρZ,n (h) :=

γZ,n (h) , γZ,n (0)

h ∈ N0 .

If now (Zt )t∈Z is a strictly stationary strongly mixing time series with geometric rate such that E|Zt |4+δ < ∞ for some δ > 0, then for each h ∈ N0 , also (Zt Zt+h )t∈Z is strongly mixing with geometric rate and 2+δ/2 E|Z < ∞. Then a central limit theorem applies, showing that t Zt+h | √  n n j=1 (Zt Zt+h − E(Zt Zt+h )) converges in distribution to a mean zero normal random variable as n → ∞, see e.g. Ibragimov and Linnik (1971), Theorem 18.5.3. More generally, using the Cramér-Wold device, one can show

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√  that the vector ( n nj=1 (Zt Zt+h − E(Zt Zt+h )))h=0,...,m converges for every m ∈ N to a multivariate normal distribution. Standard arguments as presented in Brockwell and Davis (1991), Section 7.3, then give multivariate asymptotic normality of the sample autocovariance function and hence of the autocorrelation function via the delta method. Applying these results to the GARCH process, we have: Corollary 3 Suppose that (Xt , σt )t∈Z is a strictly stationary GARCH process whose noise sequence (εt )t∈Z is such that ε0 is absolutely continuous with Lebesgue density being strictly positive in a neighbourhood of zero. (a) If there is δ > 0 such that E|Xt |4+δ < ∞, then the sample autocovariance and sample autocorrelation function of (Xt )t∈Z are asymptotically normal with rate n1/2 , i.e. for every m ∈ N there exists a multivariate nor√ mal random vector (V0 , . . . , Vm ) with mean zero such that ( n(γn,X (h) − γ√ X (h)))h=0,...,m converges in distribution to (V0 , . . . , Vm ) as n → ∞, and ( n(ρn,X (h)−ρX (h)))h=1,...,m converges to (γX (0))−1 (Vh −ρX (h)V0 )h=1,...,m as n → ∞. Here, γX and ρX denote the true autocovariance and autocorrelation function of (Xt )t∈Z , respectively. (b) If there is δ > 0 such that E|Xt |8+δ < ∞, then the sample autocovariance and sample autocorrelation functions of (Xt2 )t∈Z are asymptotically normal with rate n1/2 . The above statement can for example be found in Basrak et al. (2002), Theorems 2.13 and 3.6. In practice one often estimates GARCH processes with parameters which are close to IGARCH. Hence the assumption on finiteness of E|Xt |4+δ is questionable. Indeed, in cases when EXt4 = ∞, one often gets convergence of the sample autocovariance and autocovariance functions to √ stable distributions, and the rate of convergence is different from n. For the ARCH(1) case, this was proved by Davis and Mikosch (1998), extended by Mikosch and Stărică (2000) to the GARCH(1, 1) case, and by Basrak et al. (2002) to general GARCH(p, q). See also Davis and Mikosch (2008).

6 Some Distributional Properties In this section we shortly comment on two other properties of the strictly stationary solution, namely tail behaviour and continuity properties. We have already seen that the kurtosis of a GARCH process is always greater than or equal to the kurtosis of the driving noise sequence. Furthermore, Corollary 2 shows that under any reasonable assumption, a GARCH(1, 1) process will never have moments of all orders. Much more is true. Based on Kesten’s (Kesten (1973)) powerful results on the tail behaviour of random recurrence equations (see also Goldie (1991) for a simpler proof in dimension 1), one can deduce that GARCH processes have Pareto tails under weak assumptions. For the ARCH(1) process this was proved by de Haan et al. (1989), for

Probabilistic Properties of GARCH

65

GARCH(1, 1) by Mikosch and Stărică (2000), and for general GARCH(p, q) processes by Basrak et al. (2002). For a precise statement of these results, we refer to Corollary 1 in the article of Davis and Mikosch (2008) in this volume. For example, for a GARCH(1, 1) process with standard normal noise, it holds for the stationary solutions (Xt , σt )t∈Z , lim x2κ P (σ0 > x) = cσ ,

x→∞

lim x2κ P (|X0 | > x) = cσ E(|ε0 |2κ ),

x→∞

lim x2κ P (X0 > x) =

x→∞

cσ E(|ε0 |2κ ). 2

Here, κ is the unique solution in (0, ∞) to the equation E(α1 ε20 + β1 )κ = 1, and cσ is a strictly positive constant. Regarding continuity properties of stationary solutions of GARCH(p, q) processes, we shall restrict us to the case of GARCH(1, 1) and ARCH(1). Observe that in that case, the strictly stationary solution satisfies the random recurrence equation 2 . σt2 = α0 + (β1 + α1 ε2t−1 )σt−1 2 Hence if ε0 is absolutely continuous, so is log(β1 + α1 ε2t−1 ) + log σt−1 by in2 dependence of εt−1 and σt−1 , and we conclude that σt must be absolutely continuous. It follows that absolute continuity of ε0 leads to absolute continuity of the stationary σt and hence of the stationary Xt . Excluding the case when ε20 is constant, i.e. when the distribution of σt2 is a Dirac measure, one might wonder whether the stationary distribution σt will always be absolutely continuous, regardless whether ε0 is absolutely continuous or not. For stationary distributions of the related continuous time GARCH processes (COGARCH) introduced by Klüppelberg et al. (2004), this is indeed the case, see Klüppelberg et al. (2006). For the discrete time GARCH(1, 1) process, the author is however unaware of a solution to this question. At least there is the following positive result which is an easy consequence of Theorem 1 of Grincevicius (1980):

Theorem 9 (Grincevicius (1980), Theorem 1) Let (Xt , σt )t∈Z be a strictly stationary GARCH(1, 1) or ARCH(1) process. Then σ0 is continuous with respect to Lebesgue measure, i.e. cannot have atoms, unless σ0 is degenerate to a constant, i.e. unless ε20 is constant. Consequently, X0 does not have atoms unless ε20 is constant or ε0 has an atom at zero. Actually, Grincevičius’ result applies more general situations, but in the  to i−1 2 GARCH case says that if σ02 = α0 ∞ 1 + α1 ε−j ) has an atom, then i=1 j=1 (β ∞ there must exist a sequence (Sn )n∈N0 such that n=1 P (α0 +(β1 +α1 ε2n )Sn =

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Sn−1 ) > 0. By the i.i.d. assumption on (εn )n∈Z , this can be seen to happen only if ε20 is constant.

7 Models Defined on the Non-Negative Integers We defined a GARCH process as a time series indexed by the set Z of integers. This implies that the process has been started in the infinite past. It may seem more natural to work with models which are indexed by the non-negative integers N0 . Let (εt )t∈N0 be a sequence of i.i.d. random variables, and p ∈ N, q ∈ N0 . Further, let α0 > 0, α1 , . . . , αp−1 ≥ 0, αp > 0, β1 , . . . , βq−1 ≥ 0 and βq > 0 be non-negative parameters. Then by a GARCH(p, q) process indexed by N0 , we shall mean a process (Xt )t∈N0 with volatility process (σt )t∈N0 which is a solution to the equations t ∈ N0 , p q   2 2 2 αi Xt−i + βj σt−j , σt = α0 +

(27)

Xt = σt εt ,

i=1

t ≥ max(p, q).

(28)

j=1

The process is called causal if additionally σt2 is independent of (εt+h )h∈N0 for t = 0, . . . , max(p, q). By (28), the latter independence property then easily extends to hold for all t ∈ N0 . Recall that every strictly stationary GARCH(p, q) process indexed by Z is causal by Remark 1. When restricting such a process to N0 , it is clear that we obtain a causal strictly stationary GARCH process indexed by N0 . Conversely, suppose that (Xt , σt )t∈N0 is a strictly stationary GARCH process indexed by N0 . Like any strictly stationary process indexed by N0 , it can be extended to a strictly stationary process (Xt , σt )t∈Z , see Kallenberg (2002), Lemma 10.2. With εt = Xt /σt for t < 0 (observe that σt2 ≥ α0 ), one sees that also (Xt , σt , εt )t∈Z is strictly stationary. Hence (εt )t∈Z must be i.i.d., and (27) and (28) continue to hold for t ∈ Z. Since (Xt , σt )t∈Z is strictly stationary, it is causal, and hence so is (Xt , σt )t∈N0 . We have seen that there is an easy correspondence between strictly stationary GARCH processes defined on the integers and strictly stationary GARCH processes defined on N0 . This justifies the restriction to GARCH processes indexed by Z, which are mathematically more tractable. Furthermore, strictly stationary GARCH processes indexed by N0 are automatically causal.

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67

8 Conclusion In the present paper we have collected some of the mathematical properties of GARCH(p, q) processes (Xt , σt )t∈Z . The existence of strictly and weakly stationary solutions was characterised, as well as the existence of moments. The GARCH process shares many of the so called stylised features observed in financial time series, like a time varying volatility or uncorrelatedness of the observations, while the squared observations are not uncorrelated. The autocorrelation of the squared sequence was in fact seen to be that of an ARMA process. Stationary solutions of GARCH processes have heavy tails, since they are Pareto under weak assumptions. On the other hand, there are some features which are not met by the standard GARCH(p, q) process, such as the leverage effect, to name just one. In order to include these and similar effects, many different GARCH type models have been introduced, such as the EGARCH model by Nelson (1991), or many other models. We refer to the article by Teräsvirta (2008) for further information regarding various extensions of GARCH processes. Acknowledgement I would like to thank Richard Davis and Thomas Mikosch for careful reading of the paper and many valuable suggestions.

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Klüppelberg, C., Lindner, A. and Maller, R. (2004): A continuous-time GARCH process driven by a Lévy process: stationary and second order behaviour. Journal of Applied Probability 41, 601–622. Klüppelberg, C., Lindner, A. and Maller, R. (2006): Continuous time volatility modelling: GARCH versus Ornstein models. In: Kabanov, Yu., Liptser, R. and Stoyanov, J. (Eds.): From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, 393–419. Springer, Berlin New York Heidelberg. Ling, S. (1999): On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. Journal of Applied Probability 36, 688–705. Ling, S. and McAleer, M. (2002): Necessary and sufficient moment conditions for the GARCH(r,s) and asymmetric power GARCH(r,s) models. Econometric Theory 18, 722–729. Lütkepohl, H. (1996): Handbook of Matrices. Wiley, Chichester. Meyn, S.P. and Tweedie, R.L. (1996): Markov Chains and Stochastic Stability. 3rd edition. Springer, Berlin Heidelberg New York. Milhøj, A. (1985): The moment structure of ARCH processes. Scandinavian Journal of Statistics 12, 281–292. Mikosch, T. and Stărică, C. (2000): Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. The Annals of Statistics 28, 1427–1451. Mikosch, T. and Straumann, D. (2006): Stable limits of martingale transforms with application to the estimation of GARCH parameters. The Annals of Statistics 34, 493–522. Mokkadem, A. (1990): Propriétés de mélange des processus autorégressifs polynomiaux. Annales de l’Institut Henri Poincaré. Probabilités et Statistique 26, 219–260. Nelsen, D.B. (1990): Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334. Nelson, D.B. (1991): Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370. Nelson, D.B. and Cao, C.Q. (1992): Inequality constraints in the univariate GARCH model. Journal of Business and Economic Statistics 10, 229–235. Robinson, P.M. (1991): Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84. Straumann, D. (2005): Estimation in Conditionally Heterscedastic Time Series Models. Lecture Notes in Statistics 181, Springer, Berlin. Teräsvirta, T. (2008): An introduction to univariate GARCH models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 17–42. Springer, New York. Vervaat, W. (1979): On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Advances in Applied Probability 11, 750–783.

ARCH(∞) Models and Long Memory Properties Liudas Giraitis, Remigijus Leipus and Donatas Surgailis

Abstract ARCH(∞)-models are a natural nonparametric generalization of the class of GARCH(p, q) models which exhibit a rich covariance structure (in particular, hyperbolic decay of the autocovariance function is possible). We discuss stationarity, long memory properties and the limit behavior of partial sums of ARCH(∞) processes as well as some of their modifications (linear ARCH and bilinear models).

1 Introduction A random process (usually interpreted as financial log-return series) (rt ) = (rt )t∈Z is said to satisfy the ARCH(∞) equations if there exists a sequence of standard (zero mean and unit variance) iid random variables (εt ) and a deterministic sequence bj ≥ 0, j = 0, 1, . . . , such that rt = σt εt ,

σt2 = b0 +

∞ 

2 bj rt−j ,

t ∈ Z.

(1)

j=1

Liudas Giraitis Department of Economics, Queen Mary, University of London, Mile End Road, London E1 4NS, U.K., e-mail: [email protected] Remigijus Leipus Vilnius University, Naugarduko 24, Vilnius 03225, Lithuania, and Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania, e-mail: [email protected] Donatas Surgailis Vilnius University, Naugarduko 24, Vilnius 03225, Lithuania, and Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania, e-mail: [email protected]

T.G. Anderson et al., Handbook of Financial Time Series, DOI: 10.1007/978-3-540-71297-8_3, © Springer-Verlag Berlin Heidelberg 2009

71

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Moreover, we always assume that (rt ) is causal, that is, for any t, rt has a representation as a measurable function of the present and past values εs , s ≤ t. The last property implies that εt is independent of rs , s < t and therefore rt have zero conditional mean and a (typically random) conditional variance σt2 : E[rt |rs , s < t] = 0, Var[rt |rs , s < t] = σt2 . The class of ARCH(∞) processes includes the parametric ARCH and GARCH models of (Engle (1982)) and (Bollerslev (1986)) (see review by (Teräsvirta (2008))). For instance, the GARCH(p, q) process rt = σt εt ,

σt2 = α0 +

p  i=1

2 βi σt−i +

q 

2 αj rt−j ,

(2)

j=1

can be written as (1) with σt2 = (1 − β(1))−1 α0 + (1 − β(L))−1 α(L)rt2 ,

(3)

where α(L) = α1 L + · · · + αq Lq , β(L) = β1 L + · · · + βp Lp and L stands for the back-shift operator, Lj Xt = Xt−j . Equation (3) yields the ARCH(∞) representation of the GARCH(p, q) model with positive exponentially decaying weights bj defined by the generating function α(z)/(1 − β(z)) =  ∞ i −1 α0 . i=1 bi z ; b0 = (1 − β(1)) The ARCH(∞) process was introduced by Robinson (1991) and later studied in Kokoszka and Leipus (2000), Giraitis et al. (2000) (see also the review papers Giraitis et al. (2006), Berkes et al. (2004)). In contrast to GARCH(p, q), an ARCH(∞) process can have autocovariances Cov(rk2 , r02 ) decaying to zero at the rate k −γ with γ > 1 arbitrarily close to 1. That is, the squares rt2 of an ARCH(∞) process with finite fourth moment have short memory in the sense of absolutely summable autocovariances. Numerous empirical studies (Dacorogna et al. (1993), Ding et al. (1993), Baillie et al. (1996), Ding and Granger (1996), Breidt et al. (1998), Andersen et al. (2001)) confirm that the sample autocorrelations of absolute powers of returns series and volatilities are non-negligible for very large lags, which is often referred to as the long memory phenomenon of asset returns. The last fact can be alternatively explained by structural changes in GARCH or linear models (Granger and Hyung (2004), Mikosch and Stărică (2000) and (2003), Liu (2000), Leipus et al. (2005)). Several stationary ARCH-type models were proposed to capture the long memory and other empirical “stylized facts” of asset returns. The long memory of the squared process, in the sense of powerlaw decay of the autocovariance function, was rigorously established for some stochastic volatility models (Harvey (1998), Robinson (2001), Robinson and Zaffaroni (1997) and (1998), Surgailis and Viano (2002)), the Linear ARCH (LARCH) model (Giraitis et al. (2000) and (2004)) and the bilinear model (Giraitis and Surgailis (2002)).

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A stationary process (X t ) with finite variance is said to have covariance long memory if the series t∈Z |Cov(X0 , Xt )| diverges; otherwise (Xt ) exhibits covariance short memory. We also say that a stationary process (Xt ) has distributional long memory (respectively, distributional short memory)   [nτ ] 1 −1 An (X − B ) : τ ∈ [0, 1] , if its normalized partial sum process t n t=1 converges, in the sense of weak convergence of the finite dimensional distributions, as n → ∞, to a random process (Z(τ ))τ ∈[0,1] with dependent increments (respectively, to a random process with independent increments). Covariance long memory and distributional long memory are related though generally different notions. These and other definitions of long memory (long-range dependence) can be found in Beran (1994), Cox (1984), Giraitis and Surgailis (2002) and other papers.

2 Stationary ARCH(∞) Process 2.1 Volterra representations A formal recursion of (1) yields the following Volterra series expansion of rt2 : rt2

=

ε2t σt2

=

ε2t b0

 1+

∞ 

∞ 

 bj1 · · · bjk ε2t−j1 · · · ε2t−j1 −···−jk . (4)

k=1 j1 ,··· ,jk =1

By taking the expectation on both sides and using the independence of the εt ’s, one obtains ∞  ∞ " k #  = bj Ert2 = b0 1 + k=1

Let B =

∞

j=1 bj .

j=1

1−

b 0∞

j=1 bj

.

The condition (5)

B<1

is necessary and sufficient for the existence of a unique stationary solution of (1) with Ert2 < ∞, see Kokoszka and Leipus (2000), Giraitis et al. (2000), Zaffaroni (2000). In a similar way, λ1/2 B < 1,

λ := Eε40

(6)

is sufficient for the existence of a stationary solution (rt ) with finite fourth moment Ert4 < ∞. However, condition (6) is not necessary for the existence of such a solution. In the case of GARCH(1,1) (see (12) below) condition (6) 1

[s] denotes the integer part of s.

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translates into αλ1/2 + β < 1, while a fourth order stationary solution to (12) exists under the weaker condition (α + β)2 + α2 (λ − 1) < 1,

(7)

see Karanasos (1999), He and Teräsvirta (1999) or Davis and Mikosch (2008). A relevant sufficient and necessary condition in the general case of ARCH(∞) can be obtained by centering the innovations in the (nonorthogonal) representation (4), i.e. by replacing the ε2j ’s by κζj + 1 = ε2j , where the standardized ζj = (ε2j − Eε2j )/κ, κ2 = Var(ε20 ) have zero mean and unit variance. This leads to the following orthogonal Volterra representation of (rt2 ) (Giraitis and Surgailis (2002)): rt2 = μ + κμ

∞ 



gt−s1 hs1 −s2 · · · hsk−1 −sk ζs1 · · · ζsk ,

(8)

k=1 sk <···<s2 <s1 ≤t

where μ = Ert2 = b0 /(1 − B), hj = κgj , j ≥ 1, and where gj , j ≥ 0, are the coefficients of the generating function ∞ 

gj z j =

−1  ∞  1− bi z i .

j=0

Let H 2 =

∞ j=1

(9)

i=1

h2j . The series (8) converges in mean square if and only if B < 1,

H < 1.

(10)

Condition (10) is sufficient and necessary for the existence of a stationary ARCH(∞) solution with finite fourth moment (Giraitis and Surgailis (2002)). By orthogonality, it also follows that Cov(rt2 , r02 )

2 2

=κ μ

= κ2 μ2

∞ 



k=1 sk <···<s1 ≤0 ∞  

H 2(k−1)

gs gt−s

s≤0 2 2

=

g−s1 gt−s1 h2s1 −s2 · · · h2sk−1 −sk

κ μ 1 − H2

k=1

∞ 

gs gs+t .

(11)

s=0

For the GARCH(1,1) process specified as rt = εt σt ,

2 2 σt2 = α0 + αrt−1 + βσt−1

condition (10) is equivalent to (7), gj = α(α + β)j−1 , and (8) becomes

(12)

ARCH(∞) Models and Long Memory Properties ∞   rt2 = μ + μκζt 1 + (ακ/γ)k

+μ

(ακ/γ)k



γ t−sk ζs1 · · · ζsk



sk <···<s1
k=1 ∞ 

75



γ t−sk ζs1 · · · ζsk ,

(13)

sk <···<s1
k=1

where γ = α + β, μ = α0 /(1 − γ). Representation (13) yields the covariance function of the squared GARCH(1,1) process as a function of the parameters α0 , α, β, λ, which was first obtained in Teräsvirta (1996). An alternative approach to the problem of the existence of fourth order stationary solution of ARCH(∞) was discussed in Kazakevičius et al. (2004). Doukhan et al. (2006) discuss vector-valued ARCH(∞) process.

2.2 Dependence structure, association, and central limit theorem Equation (11) can be applied to study summability and decay properties of the covariance function of (rt2 ) (Giraitis and Surgailis (2002)). Summability of the bj ’s implies summability of the gj ’s in (9) which in turn implies by (11) that ∞  Cov(rk2 , r02 ) < ∞. (14) k=−∞

≥ 0 for all k, which follows from (11) and also from (Note that the associativity property of (rt2 ), see below.) Therefore, the squares (rt2 ) of a stationary solution of ARCH(∞) with finite fourth moment have covariance short memory. Giraitis et al. (2000), Giraitis and Surgailis (2002) also prove that hyperbolic decay bj ∼ Cj −γ with γ > 1 implies2 Cov(rk2 , r02 )

Cov(rk2 , r02 )  k −γ .

(15)

Thus, even though condition (10) implies (14), it allows for a hyperbolic rate of decay in (15), with γ > 1 arbitrarily close to 1. The last property is called intermediate memory (Brockwell and Davis (1991), p. 465). A class of ARCH(∞) processes with intermediate memory was discussed in Davidson (2004). Further insight into the dependence properties of (rt2 ) with finite fourth moment can be obtained from the moving average representation

2

xk ∼ yk means xk /yk → 1 while xk  yk means that there are positive constants C1 and C2 such that C1 yk < xk < C2 yk for all k large enough.

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rt2 = Ert2 +

∞ 

gj νt−j ,

(16)

j=0

where (gj )j≥0 are defined in (9) and νt = σt2 (ε2t − Eε2t ) = rt2 − σt2 are martingale differences, satisfying Eνt2 < ∞, E[νt |rs , s < t] = 0. Representation (16) is a direct consequence of (8), from which νt can also be expressed as a Volterra series in the standardized variables ζs , s < t. Note that (16) yields the same covariance formula as (11). In the literature, (16) is sometimes obtained without sufficient justification, by treating (νt ) as “innovations” and ∞ 2 formally inverting the equation rt2 − j=1 bj rt−j = b0 + νt , see the discussion in Davidson (2004). It is important to realize that the definition of (νt ) per se assumes that (rt ) is a causal solution of (1) and the martingale property of (νt ) implies Eσt2 < ∞, or B < 1, thereby excluding the IGARCH case (see below), for which the sum of the coefficients B = 1. On the other hand, even if the fourth moment is finite as in (16), the νt ’s are not independent, meaning that “higher order” dependence and distributional properties of (16) may be very different from the usual moving average in iid random variables. Squared ARCH(∞) processes have the important property of association. A stochastic process (Xt ) is said to be associated (or positively correlated) if Cov(f (Xt1 , . . . , Xtn ), g(Xt1 , . . . , Xtn )) ≥ 0 holds for any coordinate nondecreasing functions f, g : Rn → R and any t1 , . . . , tn , n = 1, 2, . . . . In particular, the autocovariance function of an associated process is nonnegative. Association is a very strong property, under which uncorrelatedness implies independence similarly to the Gaussian case. A well-known result due to Newman and Wright (1981) says that if (Xt ) is strictly stationary, associated, and t∈Z Cov(X0 , Xt ) < ∞ then its partial sums process converges to a standard Brownian motion in the Skorokhod space D[0, 1] with the sup-topology. It is well known that independent random variables are associated, and that this property is preserved by coordinate-nondecreasing (nonlinear) transformations. In particular, the squared ARCH(∞) process of (4) is a coordinate-nondecreasing transformation of the iid sequence (ε2t ): since all bj ’s are nonnegative, rt2 can only increase if any of the ε2s , s ≤ t on the right-hand side of (4) is replaced by a larger quantity. Therefore the squared ARCH(∞) process (4) is associated. The same conclusion holds for any nondecreasing function of (rt2 ), in particular, for fractional powers (|rt |δ ) with arbitrary δ > 0. An immediate consequence from (14), the association property and the above mentioned Newman-Wright theorem is the following functional central limit theorem for the squares of ARCH(∞):   [nτ ]  n−1/2 (rt2 − Ert2 ) t=1

τ ∈[0,1]

→D[0,1] (σW (τ ))τ ∈[0,1] ,

(17)

ARCH(∞) Models and Long Memory Properties

77

where (W (τ )) is a standard Brownian motion and σ 2 equals the sum in (14). According to our terminology, the squares (rt2 ) of a stationary solution of ARCH(∞) with finite fourth moment have distributional short memory. This result is quite surprising given the rather complicated nonlinear structure of ARCH(∞). Giraitis et al. (2000) obtained a similar result by using finite memory approximations to ARCH(∞).

2.3 Infinite variance and integrated ARCH(∞) One of the surprising features of ARCH equations is the fact that they may admit a stationary solution which may have arbitrarily heavy power tails, even if the iid “shocks” εt are light-tailed, e.g. N (0, 1) (see Klivečka and Surgailis (2007)). Bougerol and Picard (1992) obtained sufficient and necessary conditions for the existence of a causal stationary solution of GARCH(p, q), possibly with infinite variance, in terms of the top Lyapunov exponent γ of an associated vector stochastic recurrence equation. The GARCH(1, 1) case was first discussed by Nelson (1990). In general, γ is not known explicitly in terms of the coefficients of the GARCH(p, q), the only exception being the GARCH(1,1) case where γ = E log(αε2 + β) (Nelson (1990)). The tail behavior of a stationary solution of GARCH(p, q) process was discussed in Basrak et al. (2002), Mikosch and Stărică (2000) (see also Davis and Mikosch (2008) and Lindner (2008) in this volume). Sufficient conditions for the existence of stationary ARCH(∞) processes without moment conditions were obtained in Kazakevičius and Leipus (2002). They observed that the volatility can be written as

∞  2 , σt2 = b0 1 + σt,n n=1

the convergence of the series being equivalent to the existence of a stationary solution rt = εt σt , where 2 = σt,n

n 



k=1

i1 ,··· ,ik ≥1 i1 +···+ik =n

bi1 bi2 · · · bik ε2t−i1 ε2t−i1 −i2 · · · ε2t−i1 −···−ik ,

n ≥ 1,

2 σt,n = 0, n ≤ 0, satisfy the recurrence equation

2 = ε2t−n σt,n

n 

2 bi σt,n−i ,

n ≥ 1.

i=1

The stationarity condition of Kazakevičius and Leipus (2002) (which applies also to the case Eε20 = ∞) essentially reduces to the condition

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R > 1, where

(18)

$ −1  2 R = lim sup n σ0,n

constitutes the (nonrandom) convergence radius of the random power series ∞ 2 n σ z . In the GARCH(p, q) case, − log R = γ coincides with the top n=1 0,n Lyapunov exponent and condition (18) translates to the condition γ < 0 of Bougerol and Picard (1992). Similarly to GARCH case, the convergence radius R cannot be explicitly determined in terms of the coefficients bj of ARCH(∞) and therefore the above result does not provide a constructive answer to the question as to the existence of an infinite variance stationary solution to ARCH(∞) except for the case B = 1, see below. Kazakevičius and Leipus (2002) also proved the uniqueness of the stationary solution under condition (18) and some additional condition on the coefficients bj which is satisfied, for example, if bj ultimately decrease monotonically. An important class of ARCH(∞) processes with infinite variance are Integrated ARCH(∞), or IARCH(∞), defined as a solution to (1) with B=1

(19)

(recall that Eε2t = 1). Forthe GARCH(p, p q) process, condition (19) becomes q the unit root condition α + i i=1 j=1 βj = 1. Thus, the IGARCH(∞) model is a generalization of the Integrated GARCH(p, q) model introduced in Engle and Bollerslev (1986) in order to explain the so-called IGARCH effect of return data when the estimated parameters of the GARCH(p, q) sum up to a value close to 1. Bougerol and Picard (1992) proved that in the IGARCH(p, q) case, the Lyapunov exponent γ < 0 and therefore equation (1) has a stationary solution. The above result was extended to the IARCH(∞) equation in Kazakevičius and Leipus (2003), under the additional assumption that the coefficients bj in (1) decay exponentially. The last condition is crucial in the proof of R > 1 in Kazakevičius and Leipus (2003), and is also satisfied in the case of the IGARCH(p, q) discussed in Bougerol and Picard (1992). Kazakevičius and Leipus (2003) also proved that if the exponential decay condition of the bj ’s is not satisfied (as in the FIGARCH case), then R = 1. However, contrary to the GARCH situation, condition (18) is not necessary for the existence of a stationary solution to ARCH(∞); see Giraitis et al. (2006). An interesting example of an IARCH process is the FIGARCH process defined by σt2 = b0 + (1 − (1 − L)d )rt2 , (20) rt = σt εt , where b0 > 0 and (1 − L)d , 0 < d < 1, is the fractional differencing operator. This model, introduced by Baillie et al. (1996) in order to capture the long memory effect in volatility, allows for a hyperbolic decay of the coefficients bj which are positive, summable, and satisfy the unit root condition (19).

ARCH(∞) Models and Long Memory Properties

79

However, the FIGARCH equation has no stationary solution with Ert2 < ∞. In general the question of the existence of a stationary solution to (20) with infinite variance remains open. But under additional assumptions on the distribution of εt (cf. (20)) it has been shown in Douc et al. (2008) that a non-zero stationary solution exists (related arguments were used also in Robinson and Zaffaroni (2006)). See Giraitis et al. (2000), Kazakevičius and Leipus (2003), Mikosch and Stărică (2000) and (2003), Davidson (2004) for a discussion of the controversies surrounding the FIGARCH.

3 Linear ARCH and Bilinear Model The Linear ARCH (LARCH) model, introduced by Robinson (1991), is defined by the equations rt = σt εt ,

σt = α +

∞ 

βj rt−j ,

(21)

j=1

where (εt ) is an iid sequence, with zero mean and unit variance, and the #1/2 " ∞ 2 β < ∞. The particular case coefficients βj satisfy B2 = j=1 j σt = (1 − L)−d rt corresponds to the LARCH equation with rt = σt εt , FARIMA(0, d, 0) coefficients. The main advantage of LARCH is that it allows for modelling of long memory as well as some characteristic asymmetries (the “leverage effect”). Both these properties cannot be modeled by the classical ARCH(∞) with finite fourth moment. Condition B2 < ∞ is weaker than the assumption B < ∞ for the ARCH(∞) model (1). Neither α nor the βj ’s are assumed positive and, unlike (1), σt (not σt2 ), is a linear combination of the past values of rt , rather than their squares. Note that σt2 = Var[rt |rs , s < t] is the conditional variance of the causal (rt ). Contrary to ARCH(∞), in the LARCH model σt may be negative or vanish and so it lacks some of the usual volatility interpretation. Long memory properties of the LARCH model were studied in Giraitis et al. (2000) and (2004). Similarly as in the ARCH(∞) case, it is easy to show that a second order strictly stationary solution (rt ) to (21) exists if and only if (22) B2 < 1, in which case it can be represented by the convergent orthogonal Volterra series

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rt = σt εt ,

∞   σt = α 1 +

 βj1 · · · βjk εt−j1 · · · εt−j1 −···−jk . (23)

∞ 

k=1 j1 ,··· ,jk =1

Whence, or directly from the LARCH equations (21), relation Cov(σt , σ0 ) =

∞ α2  βj βt+j 1 − B22 j=1

(24)

immediately follows. Note (24) coincides with the covariance of the linear filter with coefficients βj (α/(1 − B22 ))1/2 . In particular, if βj ∼ cj d−1

(j → ∞)

(25)

for some c > 0, 0 < d < 1/2, the sequence (σt ) has covariance long memory similar to FARIMA(0, d, 0) models. The proof of long memory of (rt2 ) is more involved and uses the combinatorial formalism of diagrams, see Giraitis et al. (2000). As is shown in this paper, under some additional boundedness condition on B2 in (22), the squared LARCH process of (21), (25) exhibits both covariance and distributional long memory, since Cov(rk2 , r02 ) ∼ Ck 2d−1 ,

k → ∞,

(26)

and the normalized partial sum process of (rt2 ) tends to a fractional Brownian motion with Hurst parameter H = d + 1/2 ∈ (1/2, 1). Similar results under increasingly stringent bounds on B2 in (22) were proved in Giraitis et al. (2000) for arbitrary powers (rtk ), k = 2, 3, . . . . Berkes and Horvàth (2003) and Beran (2006) obtained the limit distribution of sums of general nonlinear functions of LARCH process. The above results suggest certain similarities between the long memory properties of LARCH and a linear moving average process with coefficients βj in (25). In fact, the first term of the expansion (23) (corresponding to k = 1) is exactly the linear process ∞ j=1 βj εt−j , up to constant α. The nonlinearity of the LARCH appears when analyzing the behavior of higher-order multiple  sums in (23). It turns out that every term j1 ,··· ,jk βj1 · · · βjk εt−j1 · · · εt−jk behaves similarly as the first (linear) term and contributes to the asymptotic constant C in (26), although these “contributions decay geometrically” with k. A natural generalization of the LARCH model is the heteroscedastic bilinear equation (Giraitis and Surgailis (2002)): X t = c0 +

∞  j=1

cj Xt−j + σt ζt ,

σt = a0 +

∞  j=1

aj Xt−j ,

(27)

ARCH(∞) Models and Long Memory Properties

81

where (ζt ) are standard iid, with zero mean and unit variance, and aj , cj , j ≥ 0, are real (not necessary nonnegative) coefficients. Clearly, aj = 0, j ≥ 1, in (27) gives a (linear) AR(∞) equation, while cj = 0, j ≥ 0, results in the LARCH equation (21). Moreover, the ARCH(∞) model in (1) also turns out to be a special case of (27), by putting Xt = rt2 , ζt = (ε2t − Eε20 )/ Var(ε20 ). Equation (27) appears naturally when studying the class of processes with the property that the conditional mean μt = E[Xt |Xs , s < t] is a linear combination of Xs , s < t, and the conditional variance σt2 = Var[Xt |Xs , s < t] is the square of a linear combination of Xs , s < t, as it is in the case of a causal solution (Xt ) of (27):  σt2

=

a0 +

∞ 

2 aj Xt−j

,

μt = c0 +

j=1

∞ 

cj Xt−j .

j=1

The bilinear model (27) in the cases c0 = 0 and c0 = 0 has different properties. The first case (to which ARCH(∞) reduces) does not essentially allow for long memory, see Giraitis and Surgailis (2002). The second case reduces to a linear filter of LARCH. Indeed, let ∞νt = σt ζt , then (Xt ) in (27) satisfies the autoregressive equation Xt = j=1 cj Xt−j + νt . By inverting the last equation, one obtains ∞  Xt = gj νt−j , (28) ∞

j=0

 j where G(z) = (1 − C(z)) = j=0 gj z j , C(z) = ∞ j=1 cj z . With (28) in mind, equation (27) can be rewritten as the LARCH equation −1

νt = σt ζt ,

σt = a0 +

∞ 

hj νt−j

(29)

j=1

∞ with coefficients hj given by H(z) = j=1 hj z j = A(z)G(z). Under condition  ∞ H 2 = j=1 h2j < 1 (and some additional assumptions on the coefficients aj , cj , see Giraitis and Surgailis (2002)), equation (29) has a unique stationary causal solution (νt ) given by a convergent Volterra series which can be substituted into (28), yielding a corresponding solution of the bilinear equation (27), viz. X t = a0

∞ 



gt−s1 hs1 −s2 · · · hsk−1 −sk ζs1 · · · ζsk .

(30)

k=1 sk <···<s1 ≤t

∞ The fact that the conditional mean μt = j=1 gj νt−j and the conditional ∞ variance σt = a0 + j=1 hj νt−j of the stationary process in (30) admit moving average representations in the martingale difference sequence (νt ) suggests that (μt ) and/or (σt ) may exhibit covariance long memory, provided the filter coefficients decay slowly:

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gj ∼ C1 j d1 −1 ,

hj ∼ C2 j d2 −1 ,

(31)

with some Ci = 0, 0 < di < 1/2, i = 1, 2. The above mentioned paper presents concrete examples of generating functions of the form C(z) = 1 − P1 (z)(1 − z)d1 , A(z) = P2 (z)(1 − z)d1 −d2 , where Pi (z), i = 1, 2 satisfy some root conditions, for which the corresponding G(z), H(z) satisfy (31). Consequently, the process (Xt ) in (27) may exhibit double long memory (i.e. long memory both in the conditional mean and in the conditional variance). Let us finally mention, that heteroscedastic models with non-zero conditional mean (combinations of the type ARMA-ARCH) have been studied in the econometrics literature; see e.g. Beran (2006), Ling and Li (1998), Li et al. (2002), Teyssière (1997). The last paper also discusses econometric aspects of double long memory.

References Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2001): The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc. 96, 45–55. Baillie, R.T., Bollerslev, T. and Mikkelsen, H.O. (1996): Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econometrics 74, 3–30. Baillie, R.T., Chung, C.-F. and Tieslau, M.A. (1996): Analysing inflation by the fractionally integrated ARFIMA-GARCH model. J. Appl. Econometrics 11, 23–40. Basrak, B., Davis, R.A. and Mikosch, T. (2002): Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–116. Beran, J. (1994): Statistics for Long-Memory Processes. Chapman and Hall, New York. Beran, J. (2006): Location estimation for LARCH processes. J. Mult. Anal. 97, 1766–1782. Berkes, I., Horvàth, L. and Kokoszka, P.S. (2004): Probabilistic and statistical properties of GARCH processes. Fields Inst. Commun. 44, 409–429. Berkes, I. and Horvàth, L. (2003): Asymptotic results for long memory LARCH sequences. Ann. Appl. Probab. 13, 641–668. Bollerslev, T. (1986): Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327. Bougerol, P. and Picard, N. (1992): Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52, 115–127. Breidt, F.J., Crato, N. and de Lima, P. (1998): On the detection and estimation of long memory in stochastic volatility. J. Econometrics 83, 325–348. Brockwell, P.J. and Davis, R.A. (1991): Time Series: Theory and Methods. Springer, New York. Cox, D.R (1984): Long-range dependence: a review. In: David, H.A. and David, H.T. (Eds.): Statistics: An Appraisal. Proc. 50th Anniversary Conference, 55–74. Iowa State University Press. Dacorogna, M.M., Müller, U.A., Nagler, R.J., Olsen, R.B. and Pictet, O.V. (1993): A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. J. Intern. Money and Finance 12, 413–438. Davidson, J. (2004): Moment and memory properties of linear conditional heteroscedasticity models, and a new model. J. Business and Economic Statist. 22, 16–29. Davis, R.A. and Mikosch, T. (2008): Extreme value theory for GARCH processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 186–200. Springer, New York.

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Ding, Z., Granger, C.W.J. and Engle, R.F. (1993): A long memory property of stock market returns and a new model. J. Emp. Finance 1, 83–106. Ding, Z. and Granger, C.W.J. (1996): Modeling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185–215. Douc, R., Roueff, F. and Soulier, P. (2008): On the existence of some ARCH(∞)–processes. Stochastic Processes and Their Applications 118, 755–761. Doukhan, P., Teyssière, G. and Winant, P. (2006): An LARCH(∞) vector valued process. In: Bertail, P., Doukhan, P. and Soulier, P. (Eds.): Dependence in Probability and Statistics. Lecture Notes in Statistics 187, 307–320. Springer, New York. Engle, R.F. (1982): Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008. Engle, R.F. and Bollerslev, T. (1986): Modelling the persistence of conditional variances. Econometric Reviews 27, 1–50. Giraitis, L., Kokoszka, P. and Leipus, R. (2000): Stationary ARCH models: dependence structure and Central Limit Theorem. Econometric Th. 16, 3-22. Giraitis, L., Robinson, P.M. and Surgailis, D. (2000): A model for long memory conditional heteroskedasticity. Ann. Appl. Probab. 10, 1002–1024. Giraitis, L., Leipus, R., Robinson, P.M. and Surgailis, D. (2004): LARCH, leverage and long memory. J. Financial Econometrics 2, 177–210. Giraitis, L. and Surgailis, D. (2002): ARCH-type bilinear models with double long memory. Stoch. Process. Appl. 100, 275–300. Giraitis, L., Leipus, R. and Surgailis, D. (2006): Recent advances in ARCH modelling. In: Teyssière, G. and Kirman, A. (Eds.): Long Memory in Economics, 3–38. Springer, New York. Granger, C.W.J. and Hyung, N. (2004): Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. J. Empirical Finance 11, 399–421. Harvey, A. (1998): Long memory in stochastic volatility. In: Knight, J. and Satchell, S. (Eds.): Forecasting Volatility in the Financial Markets 307–320. Butterworth & Heineman. He, C. and Teräsvirta, T. (1999): Fourth moment structure of the GARCH(p, q) process. Econometric Th. 15, 824–846. Karanasos, M. (1999): The second moment and the autocovariance function of the squared errors of the GARCH model. J. Econometrics 90, 63–76. Kazakevičius, V. and Leipus, R. (2002): On stationarity in the ARCH(∞) model. Econometric Th. 18, 1–16. Kazakevičius, V. and Leipus, R. (2003): A new theorem on existence of invariant distributions with applications to ARCH processes. J. Appl. Probab. 40, 147–162. Kazakevičius, V., Leipus, R. and Viano, M.-C. (2004): Stability of random coefficient autoregressive conditionally heteroskedastic models and aggregation schemes. J. Econometrics 120, 139–158. Klivečka, A. and Surgailis, D. (2007): GARCH(1,1) process can have arbitrarily heavy power tails. Lithuan. Math. J. 47, 196–210. Kokoszka, P. and Leipus, R. (2000): Change-point estimation in ARCH models. Bernoulli 6, 513–539. Leipus, R., Paulauskas, V. and Surgailis, D. (2005): Renewal regime switching and stable limit laws. J. Econometrics 129, 299–327. Li, W.K., Ling, S. and McAleer, M. (2002): Recent theoretical results for time series models with GARCH errors. J. Economic Surv. 16, 245–269. Lindner, A.M. (2008): Stationarity, mixing, distributional properties and moments of GARCH(p,q)-processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 43–69. Springer, New York. Ling, S. and Li, W.K. (1998): On fractionally integrated autoregressive moving average time series models with conditional heteroskedasticity. J. Amer. Statist. Assoc. 92, 1184–1193.

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Liu, M. (2000): Modeling long memory in stock market volatility. J. Econometrics 99, 139–171. Mikosch, T. and Stărică, C. (2000): Is it really long memory we see in financial returns? In: Embrechts, P. (Ed.): Extremes and Integrated Risk Management, 149–168. Risk Books, London. Mikosch, T. and Stărică, C. (2003): Long-range dependence effects and ARCH modeling. In: Doukhan, P., Oppenheim, G. and Taqqu, M. S. (Eds.): Theory and Applications of Long-Range Dependence, 539–459. Birkhäuser, Boston. Mikosch, T. and Stărică, C. (2000): Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, 1427–1451. Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6, 318–334. Newman, C.M. and Wright, A.L. (1981): An invariance principle for certain dependent sequences. Ann. Probab. 9, 671–675. Robinson, P.M. (1991): Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47, 67–84. Robinson, P.M. (2001): The memory of stochastic volatility models. J. Econometrics 101, 195–218. Robinson, P.M. and Zaffaroni, P. (1997): Modelling nonlinearity and long memory in time series. Fields Inst. Commun. 11, 161–170. Robinson, P.M. and Zaffaroni, P. (1998): Nonlinear time series with long memory: a model for stochastic volatility. J. Statist. Plan. Infer. 68, 359–371. Robinson, P.M. and Zaffaroni, P. (2006): Pseudo-maximum likelihood estimation of ARCH(1) models. Ann. Statist. 34, 1049–1074. Surgailis, D. and Viano, M.-C. (2002): Long memory properties and covariance structure of the EGARCH model. ESAIM: Probability and Statistics 6, 311–329. Teräsvirta, T. (2008): An introduction to univariate GARCH models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 17–42. Springer, New York. Teräsvirta, T. (1996): Two stylized facts and the GARCH(1,1) model. Stockholm School of Economics. SSE/EFI Working Paper Series in Economics and Finance 96. Teyssière, G. (1997): Double long-memory financial time series. Preprint, http://greqam.univ-mrs.fr/pdf/working papers/1997/97B01S.pdf Zaffaroni, P. (2000): Stationarity and memory of ARCH(∞) models. Econometric Theory 20, 147–160.

A Tour in the Asymptotic Theory of GARCH Estimation Christian Francq and Jean-Michel Zakoïan

Abstract The main estimation methods of the univariate GARCH models are reviewed. A special attention is given to the asymptotic results and the quasi-maximum likelihood method.

1 Introduction In comparison with other volatility models (e.g. the standard stochastic volatility model) GARCH models are simple to estimate, which has greatly contributed to their popularity. The volatility being a function of the past observations, the likelihood function has an explicit form which makes it easy to handle. A variety of alternative estimation methods can also be considered. Least squares and quasi-maximum likelihood estimations in ARCH models were considered in the seminal paper Engle (1982). The asymptotic properties of the quasi-maximum likelihood estimator (QMLE) received broad interest in the last 20 years. Pioneering work established consistency and asymptotic normality under strong assumptions on the parameter space and the true parameter value. The problem of finding weak assumptions for the consistency and asymptotic normality of the QMLE in GARCH models has attracted a lot of attention in the statistics literature. The first papers limited their scope to ARCH (see Weiss (1986)) or GARCH(1,1) models (see Lee and Hansen (1994), Lumsdaine (1996)). See Berkes and Horváth (2003), Berkes and Horváth (2004), Berkes et al. (2003), Francq and Zakoïan (2004), Hall and Yao (2003), for recent references on the QMLE of general GARCH(p, q) Christian Francq University Lille III, EQUIPPE-GREMARS, Domaine du Pont de bois, BP 60149, 59653 Villeneuve d’Ascq Cedex, France, e-mail: [email protected] Jean-Michel Zakoïan University Lille III, EQUIPPE-GREMARS, and CREST, 15 Bd G. Péri, 92245 Malakoff cedex, France, e-mail: [email protected]

T.G. Anderson et al., Handbook of Financial Time Series, DOI: 10.1007/978-3-540-71297-8_4, © Springer-Verlag Berlin Heidelberg 2009

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models. See Straumann (2005) for a recent comprehensive monograph on the estimation of GARCH models. Numerous GARCH-type models have been introduced and it is simply not possible to consider the estimation of all of them. In this article we limit ourselves to the standard GARCH(p, q) model given by the equations √ t = ht ηt p (1) q ht = ω0 + i=1 α0i 2t−i + j=1 β0j ht−j , t ∈ Z = {0, ±1, . . . } where ω0 > 0,

α0i ≥ 0 (i = 1, . . . , q),

β0j ≥ 0 (j = 1, . . . , p),

{ηt , t ∈ Z} are iid random variables such that Eη12 = 1. We assume that 1 , . . . , n are observations from the process (t , t ∈ Z), assumed to be a strictly stationary, ergodic and nonanticipative solution of Model (1). Conditions for stationarity are obtained (see Lindner (2008)) from the vector representation z t = bt + A0t−1 z t−1 ,

(2)

where, for p ≥ 2 and q ≥ 2, z t = (ht , . . . , ht−p+1 , 2t−1 , . . . , 2t−q+1 ) ∈ Rp+q−1 , bt = (ω, 0, . . . , 0) ∈ Rp+q−1 , ⎛  ⎞ τt β0p α02:q−1 α0q ⎜ Ip−1 0 0 0 ⎟ ⎟, A0t = ⎜ ⎝ ξt 0 0 0 ⎠ 0 0 Iq−2 0 with τt = (β01 + α01 ηt2 , β02 , . . . , β0p−1 ) ∈ Rp−1 , ξt = (ηt2 , 0, . . . , 0) ∈ Rp−1 , α02:q−1 = (α02 , . . . , α0q−1 ) ∈ Rq−2 . A nonanticipative solution (t ) of model (1) is such that t is a measurable function of the (ηt−i , i ≥ 0). Bougerol and Picard (1992) showed that the model has a (unique) strictly stationary non anticipative solution if and only if γ(A0 ) < 0, where γ(A0 ) is the top Lyapunov exponent of the sequence (A0t ), that is

A Tour in the Asymptotic Theory of GARCH Estimation

γ(A0 ) = lim

t→∞

1 log A0t A0t−1 . . . A01  t

87

a.s.

where · denotes any norm on the space of the (p+q−1)×(p+q−1) matrices. In addition, the strictly stationary solution is ergodic as a measurable function of the p(ηt−i , i ≥ 0). Let us mention two important consequences of γ(A0 ) < 0: (i) j=1 β0j < 1, and (ii) for some s > 0, E|1 |2s < ∞ (see Lemma 2.3 in Berkes et al. (2003) for the proof). The latter property is crucial to avoid unnecessary moment conditions in the proof of the asymptotic properties of the QMLE. Throughout the orders p and q are assumed to be known. The vector of parameters is denoted by θ = (θ1 , . . . , θp+q+1 ) = (ω, α1 , . . . , αq , β1 , . . . , βp ) and it belongs to a parameter space Θ ⊂]0, +∞[×[0, ∞[p+q . The true parameter value θ0 = (ω0 , α01 , . . . , α0q , β01 , . . . , β0p ) is unknown. We review the main estimation methods, with special attention to the quasi-maximum likelihood method. The focus will be on asymptotic results rather than on small-sample and numerical issues. We start, in Section 2, by considering the Least-Squares estimator (LSE) for ARCH(q) models, which is simple to compute but requires high moment assumptions. Then we turn to QMLE in Section 3. Section 4 is devoted to efficiency issues. In Section 5 we consider alternative estimators. Finally we discuss in Section 6 the case where some GARCH coefficients are equal to zero, putting the true parameter value on the boundary of the parameter space.

2 Least–Squares Estimation of ARCH Models In this section we assume p = 0. The LSE is obtained from the AR(q) representation for 2t : q  α0i 2t−i + ut , (3) 2t = ω0 + i=1

2t − ht

(ηt2 − 1)ht .

where ut = = The sequence (ut , Ft−1 )t is thus a martingale difference when E21 = Eh1 < ∞ and Ft−1 denotes the σ-field generated by {ηu , u < t}. Let 0 , . . . , 1−q denote arbitrary initial values. Introducing the !  = 1, 2t−1 , . . . , 2t−q , we get from (3) vector Zt−1 Y = Xθ0 + U where

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⎛

⎞

 Zn−1 ⎜ ⎟ X = ⎝ ... ⎠ ,

Z0

⎛

⎞ 2

n ⎜ .. ⎟ Y = ⎝ . ⎠,

⎞ un ⎜ ⎟ U = ⎝ ... ⎠ .

21

⎛

u1



When X X is non-singular (which can be shown to hold, a.s. for large enough n, under Assumption A3 given below) the LSE of θ0 is thus given by: ω, α ˆ1 , . . . , α ˆ q ) = (X  X)−1 X  Y. θˆnLS = (ˆ The LSE of s20 = Var(u1 ) is % &2 q n   1 1 Y − X θˆnLS 2 = 2 − ω sˆ2n = ˆ− α ˆ i 2t−i . n−q−1 n − q − 1 t=1 t i=1 It is worth mentioning that the LSE is asymptotically equivalent to the YuleWalker estimator of the AR(q) model (3) (see Chapter 8 in Brockwell and Davis (1991)) and to the Whittle estimator studied by Giraitis and Robinson (2001), Mikosch and Straumann (2002), Straumann (2005). The consistency and asymptotic normality of the LSE require some additional assumptions. For identifiability we assume that the distribution of ηt is centered and nondegenerate, i.e. P (η12 = 1) = 1. If E(41 ) < +∞, the LSE can be shown to be strongly consistent (see Bose and Mukherjee (2003)): θˆnLS → θ0 ,

sˆ2n → s20 ,

a.s. as n → ∞.

If, in addition E(81 ) < +∞ the estimator of θ0 is asymptotically normal (see also Bose and Mukherjee (2003)); more precisely, ( ' √ LS d n(θˆn − θ0 ) → N 0, (Eη14 − 1)A−1 BA−1 , where

A = Eθ0 (Zq Zq ),

(4)

B = Eθ0 (h2q+1 Zq Zq )

are non-singular matrices. Note that the vector Zq does not depend on the initial values 0 , . . . , 1−q . Consistent estimators of the matrices A and B are straightforwardly obtained by replacing the theoretical moments by empirical ones. In the framework of linear regression models, it is well known that for heteroscedastic observations the ordinary LSE is outperformed by the quasigeneralized least squares estimator (QGLSE); see e.g. Hamilton (1994) Chapter 8. In our framework the QGLSE is defined by ˆ −1 X  ΩY, ˆ θˆnQGLS = (X  ΩX) ˆ is a consistent estimator of Ω = Diag(h−2 , . . . , h−2 ). If θˆLS is where Ω n n 1 ˆ can be obtained by replacing ht by computed in a first step, then Ω

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 ω ˆ + qi=1 α ˆ i 2t−i in Ω. Then the two-stage least squares estimator θˆnQGLS is consistent and asymptotically normal ( ' √ QGLS d n(θˆn − θ0 ) → N 0, (Eη14 − 1)J −1 ,

 J = Eθ0 (h−2 q+1 Zq Zq ),

(5)

under the moment assumption E41 < ∞ when all the ARCH coefficients are strictly positive, and under a slightly stronger moment assumption in the general case; see Bose and Mukherjee (2003), Gouriéroux (1997). The moment conditions can be made explicit using the vector representation (2). It is shown in Chen and An (1998) that E(41 ) < +∞ if and only if ρ {E(A01 ⊗ A01 )} < 1 where ⊗ denotes the Kronecker product and ρ(A) the spectral radius of a square matrix A. More generally, if'Eη12m < (∞ for ⊗m some positive integer m then E(2m 1 ) < +∞ if and only if ρ E(A01 ) < 1, ⊗m where A01 stands for the kronecker product of A01 by itself m times. As can be seen in Table 1, the moment conditions imply strong reduction of the admissible parameter space. It is the main advantage of the QMLE to avoid such restrictions. Table 1 Conditions for strict stationarity and for the existence of moments of the ARCH(1) model when ηt follows a N (0, 1) or Student distributions normalized in such a way that Eη12 = 1 (Stν stands for a normalized Student distribution with ν degrees of freedom) Normal St3 St5 St9

Strict stationarity α01 < 3.562 α01 < 7.389 α01 < 4.797 α01 < 4.082

E2t < ∞ α01 < 1 α01 < 1 α01 < 1 α01 < 1

E4t < ∞ E8t < ∞ α01 < 0.577 α01 < 0.312 No No α01 < 0.333 No α01 < 0.488 α01 < 0.143

Note that, as in the case of linear regression models, the QGLSE is at least −1 as efficient as the LSE. Indeed, setting D = hq+1 A−1 Zq − h−1 Zq , the q+1 J matrix  −1 Eθ0 DD = A−1 Eθ0 (h2q+1 Zq Zq )A−1 + J −1 Eθ0 (h−2 q+1 Zq Zq )J

− A−1 Eθ0 (Zq Zq )J −1 − J −1 Eθ0 (Zq Zq )A−1 = A−1 BA−1 − J −1 is semi-positive definite.

3 Quasi–Maximum Likelihood Estimation Gaussian quasi-maximum likelihood estimation has become a very popular method for GARCH models. The basic idea of this approach is to maximize

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the likelihood function written under the assumption that the noise (ηt ) is Gaussian. Since t is then Gaussian conditionally on the past ’s and σ’s, the likelihood function factorizes under a very tractable form which is maximized to produce the QMLE. The Gaussianity of the noise is inessential for the asymptotic properties of the QMLE. We start by considering the case of pure GARCH models, corresponding to the practical situation where a GARCH is estimated on the log-returns.

3.1 Pure GARCH models 2 Conditionally on initial values 0 , . . . , 1−q , σ ˜02 , . . . , σ ˜1−p , let us define recursively q p   2 ˜t2 (θ) = ω + αi 2t−i + βj σ ˜t−j σ ˜t2 = σ i=1

j=1

for t = 1, . . . , n. Due to the initial values, the sequence (˜ σt2 ) is not stationary, but can be viewed (see Francq and Zakoïan (2004)) as an approximation of the strictly stationary, ergodic and nonanticipative solution of σt2 = σt2 (θ) = ω +

q 

αi 2t−i +

i=1

p 

2 βj σt−j

∀t,

j=1

 under the assumption pj=1 βj < 1. Note that σt2 (θ0 ) = ht . The Gaussian quasi-likelihood of the observations 1 , . . . , n is the function   2t  exp − 2 . 2˜ σt 2π˜ σt2 t=1 n

˜ n (θ) = L

1

A QMLE of θ0 is defined as any measurable solution θˆnQML of ˜ n (θ) = arg min ˜ln (θ), θˆnQML = arg max L θ∈Θ

where

n 

(6)

θ∈Θ

2 and ˜t = ˜t (θ) = t2 + log σ ˜t2 . σ ˜ t t=1 q p i Let Aθ (z) = i=1 αi z and Bθ (z) = 1 − j=1 βj z j with Aθ (z) = 0 if q = 0 and Bθ (z) = 1 if p = 0. The paper Berkes et al. (2003) was the first one where the GARCH(p, q) QMLE was captured in a mathematically rigorous way under weak conditions. Several technical assumptions made in Berkes et al. (2003) were relaxed ˜ln (θ) = n−1

˜t ,

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by Francq and Zakoïan (2004) and Straumann (2005). The two latter papers show that, under the following assumptions A1: A2: A3: A4:

θ0 ∈ Θ and Θ is compact,  p γ(A0 ) < 0 and ∀ θ ∈ Θ, j=1 βj < 1, 2 ηt has a non-degenerate distribution with Eη12 = 1, if p > 0, Aθ0 (z) and Bθ0 (z) have no common root, Aθ0 (1) = 0, and α0q + β0p = 0,

the QMLE is strongly consistent, θˆnQML → θ0 ,

a.s. as n → ∞.

(7)

Note that in A2 the condition for strict stationarity is imposed on the true value of the parameter only. To show where Assumptions A1-A4 are used we present the scheme of proof of (7), the reader being referred to Francq and Zakoïan (2004) and Straumann (2005) for a detailed proof. pFrom the second part of A2 and the compactness of Θ, we have supθ∈Θ i=1 βj < 1. This inequality is used to show that almost surely σ ˜t2 (θ) − σt2 (θ) → 0 uniformly in θ ∈ Θ as t → ∞, and to show that the initial values do not matter asymptotically: ) ) ) ) lim sup )˜ln (θ) − ln (θ)) = 0 a.s. n→∞ θ∈Θ

where ln (θ) is a stationary ergodic sequence defined by replacing σ ˜t2 (θ) by σt2 (θ) in ˜ln (θ). Then the first condition in A2 and the ergodic theorem show that ˜ln (θ) converges a.s. to the asymptotic criterion Eθ0 l1 (θ). For any random variable X, let X + = max(X, 0) and X − = max(−X, 0). Note that Eθ0 l+ 1 (θ) can be equal to +∞, but Eθ0 l− (θ) is always finite (because inf ω > 0) θ∈Θ 1 s (θ ) is also finite (because under A2 we have E h < ∞ for some and Eθ0 l+ 0 θ0 t 1 s > 0, see Berkes et al. (2003) and Nelson (1990) for a proof of this result). Under the identifiability assumptions A3 and A4, Eθ0 l1 (θ) ≥ Eθ0 l1 (θ0 ) with equality if and only if θ = θ0 . These are the main arguments to show that (7) holds. The rest of the proof does not require additional assumptions. Notice that the condition Eη1 = 0 is not required. The assumption that Eη12 = 1 is made for identifiability reasons and is not restrictive provided Eη12 < ∞; see Berkes and Horváth (2003). The identifiability condition A4 excludes that all coefficients α0i be zero when p > 0, as well as the overidentification of both orders p and q. However, other situations where some coefficients α0i or β0j vanish are allowed. This is worth-noting since it is no longer the case for the asymptotic normality (AN). Indeed, the main additional assumption required for the AN is that θ0 ◦

belongs to the interior Θ of Θ. The case where θ0 belongs to the boundary of Θ will be considered below. Following Francq and Zakoïan (2004), under Assumptions A1-A4 and A5:

◦

θ0 ∈ Θ ,

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A6:

Eη14 < ∞,

the QMLE is asymptotically normal; more precisely √

( ' d n(θˆnQML − θ0 ) → N 0, (Eη14 − 1)J −1 ,

J = Eθ0

1 ∂σ12 ∂σ12 (θ0 ). (8) σ14 ∂θ ∂θ

It is shown in Francq and Zakoïan (2004) that σ1−2 (∂σ12 /∂θ) admits moments of any order. For simplicity we give the arguments in the case. GARCH(1,1) ∞ 2 We have σt2 = ω + α2t−1 + βσt−1 = ω(1 − β)−1 + α i=0 β i 2t−i−1 . Thus σ1−2 (∂σ12 /∂ω) and σ1−2 (∂σ12 /∂α) are bounded, and therefore admit moments of any order. We have already seen that the strict stationarity condition A2 implies the  existence of some s ∈ (0, 1) such that Eθ0 |1 |2s < ∞. Using ∞ ∂σt2 /∂βj = k=1 kβ k−1 (ω + α2t−1−k ), σt2 ≥ ω + β k (ω + α2t−1−k ), and the elementary inequality x/(1 + x) ≤ xs for all x ≥ 0, we obtain for any d > 0  ∞ !     1  kβ k ω + α2   1 ∂σt2    t−k−1   ≤ !   σ 2 ∂β  k ω + α2   β ω + β t d t−k−1 k=1 d % & ! s/d    ∞ k 2    β ω + αt−k−1 1   ≤ k  β ω   k=1

≤

1 ω s/d β

d

∞ " !s #1/d  Eθ0 ω + α21 k|β|sk/d < ∞,

(9)

k=1

d

where Xd = E|X|d for any random variable X. The idea of exploiting the inequality x/(1 + x) ≤ xs for all x > 0 is due to Boussama (1998). Finally σ1−2 (∂σ12 /∂θ) admits moments of any order, and J is well defined. The identifiability assumptions A3 and A4 entail the invertibility of J (see (ii) of the proof of Theorem 2.2 in Francq and Zakoïan (2004)). The consistency (7) of the QMLE, Assumption A5 and a Taylor expansion of ∂˜ln (·)/∂θ yield

∗  ∂ 2˜ln (θij ) √  QML √ ∂˜ln (θˆnQML ) √ ∂˜ln (θ0 ) = n + n θˆn − θ0 0= n ∂θ ∂θ ∂θi ∂θj ∗ where the θij are between θˆnQML and θ0 . The AN in (8) is then obtained by showing that n √ ∂˜ln (θ0 ) 1  1 ∂σ 2 ∂σt2 = √ n (1 − ηt2 ) 4 t (θ0 ) + oP (1) ∂θ n t=1 σt ∂θ ∂θ ' ( d → N 0, (Eη14 − 1)J ,

and

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n  t=1

93

∂2 ˜ ∗ t (θij ) → J(i, j) in probability. ∂θi ∂θj

(11)

The convergence (10) follows from the central limit theorem for martingale differences given by Billingsley (1961). To show (11), a new Taylor expansion and already given arguments are employed. It is worth-noting that no moment assumption is required for the observed process. This is particularly interesting for financial series, for which the existence of fourth and even second-order moments is questionable. The moment assumption A6 on the iid process (ηt ) is obviously necessary for the existence of the variance of the Gaussian distribution in (8). In the ARCH case we find the same asymptotic variance as for the QGLSE; see (5). Note also that the iid assumption on (ηt ) can be replaced by a martingale difference assumption at the price of additional moment assumptions; see Escanciano (2007). Tables 2 and 3 illustrate the asymptotic accuracy obtained from using the LSE and the QMLE for several ARCH(1) models with Gaussian errors and parameter ω0 = 1. When a sequence of random vectors Xn converges in law to a Gaussian distribution, we denote by Varas (Xn ) the variance of this Gaussian distribution. In view of (4), the asymptotic variance of the LSE in Table 2 is explicitly given by √ Varas { n(θˆnLS − θ0 )} = 2A−1 BA−1 , 

where A=

Eθ0 21 2 Eθ0 1 Eθ0 41 1



 ,

B=

Eθ0 σ24 Eθ0 σ24 21 Eθ0 σ24 21 Eθ0 σ24 41

 ,

with Eθ0 21 =

ω0 , 1 − α01

Eθ0 41 = 3Eθ0 σ14 =

3ω02 (1 + α01 ) . (1 − 3α201 )(1 − α01 )

The other terms of the matrix B are obtained using σ24 = (ω0 + α01 21 )2 and computing the moments of order 6 and 8 of 21 . For an ARCH(1) model, the asymptotic variance of the QMLE is given by ⎛ ⎞ 21 1 √ QML 2 2 2 2 − θ0 )} = 2J −1 , J = Eθ0 ⎝ (ω0 +α201 1 ) (ω0 +α401 1 ) ⎠ , Varas { n(θˆn 1 1 (ω0 +α01 21 )2 (ω0 +α01 21 )2

but it seems impossible to obtain J explicitly as a function of θ0 = (ω0 , α01 ) . For this reason, the asymptotic variance in Table 3 is approximated by 2Jˆ−1 , where ⎛ ⎞ 2t N 1  2 2 1 2 2 ⎝ (ω0 +α201 t ) (ω0 +α401 t ) ⎠ , Jˆ−1 = t t N t=1 2 2 2 2 (ω0 +α01 t )

(ω0 +α01 t )

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and 1 , . . . , N is a simulation of length N = 10, 000 of the ARCH(1) model with parameter θ0 and the N (0, 1) distribution for ηt . Due to the moment conditions the asymptotic variance of the LSE does not exist for α01 > 0.312 (see Table 1). Even when α01 is sufficiently small so that all moments exist up to a sufficiently large order, the asymptotic accuracy is much better with the QMLE than with the LSE.

Table 2 Asymptotic covariance matrix of the LSE of an ARCH(1) model α01 √ ˆLS Varas { n(θn − θ0 )}



0.1 0.2 0.3      3.98 −1.85 8.03 −5.26 151.0 −106.5 −1.85 2.15 −5.26 5.46 −106.5 77.6

Table 3 Approximation of the asymptotic variance of an ARCH(1) QMLE α01 QM L * as {√n(θˆn Var − θ0 )}



0.1 0.5 0.95      3.46 −1.34 4.85 −2.15 6.61 −2.83 −1.34 1.87 −2.15 3.99 −2.83 6.67

In passing we mention that Jensen and Rahbek (2004a) considers the QMLE α ˆ in ARCH(1) models of the form ht = ω0 + α01 2t−1 , when the scale parameter ω0 is known. In Jensen and Rahbek (2004a), (2004b), consistency and AN of α ˆ are established even when α01 is outside the strict stationarity region. Although the assumption that ω0 is known does not correspond to any realistic situation, these results are interesting from a theoretical point of view.

3.2 ARMA–GARCH models Assuming that the log-returns follow a GARCH model may be found restrictive. The autocorrelations of certain log-returns are incompatible with a GARCH model, and lead practitioners to specify the conditional mean. In this section we limit ourselves to ARMA specifications with GARCH errors. The GARCH process is not directly observed and the observations, which represent log-returns, are now denoted by r1 , . . . , rn . The (rt ) process satisfies an ARMA(P, Q)-GARCH(p, q) model of the form ⎧ P Q ⎨ rt − c0 = √ i=1 a0i (rt−i − c0 ) + t − j=1 b0j t−j t = ht ηt p ⎩ q ht = ω0 + i=1 α0i 2t−i + j=1 β0j ht−j

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where (ηt ) and the coefficients ω0 , α0i and β0j are defined as in (1), and where c0 , a0i and b0j are real parameters. If one allows for an ARMA part, one considerably extends the range of applications, but this approach also entails serious technical difficulties in the proof of asymptotic results. References for the estimation of ARMA-GARCH processes are Francq and Zakoïan (2004), Ling and Li (1997), Ling and Li (1998), Ling and McAleer (2003). In Francq and Zakoïan (2004) it is shown that the consistency of the QMLE holds under assumptions similar to the pure GARCH case. In particular, the observed process does not need a finite variance for the QMLE to be consistent. However the assumption Eη1 = 0 is required. The extension of the AN is more costly in terms of moments. This is not very surprising since in the case of pure ARMA models with iid innovations, the QMLE is asymptotically normal only when these innovations admit second-order moments; see Brockwell and Davis (1991). With GARCH innovations the AN is established in Francq and Zakoïan (2004) under a fourthmoment condition on the observed process or equivalently on the GARCH process.

4 Efficient Estimation An important issue is the possible efficiency loss of the QMLE, resulting from the use of an inappropriate Gaussian error distribution. In practice, the true error distribution is if course unknown and the MLE cannot be computed. However, it is interesting to consider the MLE in comparison with the QMLE, as a gauge of (in)efficiency. In particular we will see that, contrary to common belief, the QMLE can be efficient even if the underlying error distribution is not Gaussian. In this section we limit ourselves to pure GARCH models. The proof of the results of this section can be found in Francq and Zakoïan (2006). See also Berkes and Horváth (2004), Straumann (2005) for results in a more general setting. We assume that the error process (ηt ) is iid, endowed with a positive density f which is known. Conditionally on initial values, the likelihood is given by   n

1 t . Ln,f (θ) = Ln,f (θ; 1 , . . . , n ) = f σ ˜ σ ˜t t=1 t A MLE of θ is defined as any measurable solution θˆnML of θˆnML = arg max Ln,f (θ). θ∈Θ

(13)

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Recall that f is supposed to be positive. Assume that f is derivable and write g(y) = yf  (y)/f (y). The following conditions on the smoothness of f and g are introduced: A7: A8:

There is a δ1 > 0 such that supy∈R |y|1±δ1 f (y) < ∞; There exist 0 < C0 , δ2 < ∞ such that |g(y)| ≤ C0 (|y|δ2 + 1) for all y ∈ (−∞, ∞).

Such conditions are obviously satisfied for the standard normal distribution. For the Student distribution with ν degree of freedom, we have f (x) = K(y 2 + ν)−(1+ν)/2 where K is a positive constant and g(y) = −y 2 (1 + ν)/(y 2 + ν). Assumptions A7 and A8 are thus satisfied with ν > 0, for 0 < δ1 ≤ min{ν, 1} and δ2 ≥ 0. Under A1, A2, A4, A7, A8 the ML estimator is strongly consistent, θˆnML → θ0 , a.s. as n → ∞. It should be noted that no moment assumption is needed for the iid process (ηt ). For the QMLE, it was crucial to assume the existence of the first two moments, and an assumption such as Eη12 = 1 was required for identifiability reasons. Here, because the density √ f is fixed, there is no identification problem. For instance, the volatility ht can not be multiplied by a positive constant c = 1 and the noise ηt with density f can not be changed in the new noise ηt∗ = ηt /c, because the density of ηt∗ would not be f . Obviously when the assumption Eη12 = 1 is relaxed, ht is no more the conditional variance of t given the past, but as in Berkes and Horváth (2004), one can interpret ht as a conditional scaling parameter of t . The assumption that the density f is entirely known is clearly not realistic for the applications. Straumann (2005) considers the situation where the density f belongs to a known class of densities parameterized by a nuisance parameter ν, for instance a normalized Student distribution Stν with ν degrees of freedom and unit variance. Berkes and Horváth (2004) consider a very general framework in which the function f involved in the definition (13) is not necessarily the true density of ηt . Under some regularity assumptions, Straumann (2005) and Berkes and Horváth (2004) showed that this (non Gaussian) QMLE converges almost surely to  (14) θ0∗ = (dω, dα1 , . . . , dαq , β1 , . . . , βp ) , d > 0. When the density f is misspecified and non Gaussian, d is generally not equal to 1 and θˆnML is inconsistent. For the asymptotic normality of the MLE, it is necessary to strengthen the smoothness assumptions in A7 and A8. Assume that g is twice derivable and let g (0) (y) = g(y), g (1) (y) = g  (y) and g (2) (y) = g  (y). There is 0 < C0 < ∞ and 0 ≤ κ < ∞ such that |y k g (k) (y)| ≤ C0 (|y|κ + 1) for all y ∈ (−∞, ∞) and such that E|η1 |κ < ∞ for k = 0, 1, 2. 2 A10: I˜f = {1 + g(y)} f (y)dy < ∞, and limy→±∞ y 2 f  (y) = 0.

A9:

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The assumptions on the density f are mild and are satisfied for various standard distributions, such as (i) the standard Gaussian distribution, for any δ1 ∈ (0, 1], δ2 ≥ 2 and κ ≥ 2; (ii) the Student distribution with parameter ν > 0, for δ1 ≤ min{ν, 1}, δ2 ≥ 0 and κ < ν; (iii) the density displayed in (16) below with δ1 ≤ 2a, δ2 ≥ 2 and κ ≥ 2. If A1, A2, A4, A5 and A7-A10 hold, then

 √  ML 4 −1 d ˆ , as n → ∞. (15) n θn − θ0 → N 0, J I˜f It is worth-noting that, contrary to the QMLE (see Berkes and Horváth √ (2003)), the MLE can be n-consistent even when Eη14 = ∞. The asymptotic distributions in (8) and (15) allow to quantify the efficiency loss due to the use of Gaussian likelihood. The asymptotic variances differ only by a scaling factor, which is independent of the GARCH orders and coefficients. Interestingly, the QMLE is not always inefficient when the error distribution is not normal. More precisely, under the assumptions required for (8) and (15), the QMLE has the same asymptotic variance as the MLE when the density of ηt is of the form  ∞ aa exp(−ay 2 )|y|2a−1 , a > 0, Γ (a) = f (y) = ta−1 exp(−t)dt. Γ (a) 0 (16) Figure 1 displays the graph of this density for different values of a. When the density f does not belong to this family of distributions, the QMLE is asymptotically inefficient in the sense that

# # √ " ˆQML √ " ˆML 4 4 J −1 − θ0 − Varas n θn − θ0 = Eη1 − 1 − Varas n θn I˜f is positive definite. Table 4 illustrates the loss of efficiency of the QMLE in the case of the Student distribution with ν degrees of freedom (rescaled so that they have the required unit variance). The Asymptotic Relative Efficiency (ARE) of the MLE with respect to the QMLE is (for ν > 3) " #−1 √  √  = ARE = Varas n θˆnQML − θ0 Varas n θˆnQML − θ0

ν(ν − 1) . ν(ν − 1) − 12

An efficient estimator can be constructed from the QMLE in two steps. The method consists, in a first step, √ of running one Newton-Raphson iteration with the QMLE, or√any other n-consistent preliminary estimator θ˜n of θ0 , as starting point: n(θ˜n − θ0 ) = OP (1). The second step does not require any optimization procedure. Let Iˆn,f be any weakly consistent estimator of If (θ0 ). Then the sequence (θ¯n ) defined by

C. Francq and J.-M. Zakoïan

0.6

0.7

98

0.0

0.1

0.2

0.3

0.4

0.5

a=1/8 a=1/4 a=1/2 a=1 a=2

−3

−2

−1

0

1

2

3

Fig. 1 Graph of the density defined by (16) for several values of a > 0. When ηt has a density of this form the QMLE has the same asymptotic variance as the MLE.

−1 1 ∂ log Ln,f (θ˜n ) θ¯n = θ¯n,f = θ˜n + Iˆn,f n ∂θ

has the same asymptotic distribution (15) as the MLE, under the same Table 4 ARE of the MLE with respect to the QMLE when the density f of ηt is the normalized Student distribution with ν degrees of freedom and unit variance: f (y) =   ν/ν − 2fν (y ν/ν − 2), where fν denotes the standard Student density with ν degrees of freedom ν 5 ARE 2.5

6 1.66

7 1.4

8 1.27

9 1.2

10 1.15

20 1.03

30 ∞ 1.01 1

assumptions. In concrete situations, f is unknown and θ¯n is not feasible. A feasible estimator is obtained by replacing the unknown error density f by an estimator, which can be obtained from the standardized residuals ηˆt = t /σt (θˆnQML ), t = 1, . . . , n. A non parametric kernel density estimator fˆ

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can for instance be used. An issue is whether θ¯n,fˆ is an adaptive estimator, in the sense that it inherits the asymptotic optimality properties of θ¯n,f . Adaptive estimation in GARCH models has been studied by several authors; see e.g Drost and Klaassen (1997), Engle and González-Rivera (1991), Ling and McAleer (2003), Linton (1993). From these references, adaptiveness holds in the sense that the volatility parameters can be estimated up to a scale parameter, with the same asymptotic precision as if the error distribution were known; see Drost and Klaassen (1997). However, adaptive estimation of all GARCH coefficients is not possible. Efficiency losses of the QMLE and semi-parametric estimators, with respect to the MLE, are quantified in González-Rivera and Drost (1999) and illustrated numerically in Drost and Klaassen (1997), Engle and González-Rivera (1991).

5 Alternative Estimators It is known that parameter estimation is not standard for ARMA models with infinite variance innovations; see Mikosch et al. (1995). Indeed, with the notation of Section 3.2, the score vector t ∂t /∂ϑ has a finite variance I when E21 < ∞ and the t are iid. In the presence of conditionally heteroscedastic innovations, or more generally when the t are not iid, the existence of fourth-order moments is required for the existence of I. Thus the moment condition Er14 < ∞ seems necessary for the asymptotic normality of the LSE of the ARMA-GARCH models defined by (12). Similarly, it can be shown that the variance of the quasi-maximum likelihood score vector may not exist when Er12 = +∞. We have seen in Section 3.2 that moment conditions are not needed for the consistency of the QMLE. For statistical inference, consistency is however not sufficient, and the asymptotic distribution of the estimator is generally required. The asymptotic distributions of the LSE and QMLE are unknown when Er14 = +∞. Sections 5.1 and 5.2 present alternative estimators which require less moment assumptions on the observed process rt . The estimators defined in Sections 5.3 and 5.4 allow one to reduce the moment assumptions on the iid process (ηt ). Section 5.5 is devoted to the Whittle estimator. It will be seen that this estimator is less attractive for GARCH models than for ARMA models. The moment estimators mentioned in Section 5.6 seem particularly interesting to allow for GARCH-type effects without imposing a fully specified model. To save space we only present the main ideas of these estimation methods. The precise assumptions and asymptotic variance matrices can be found in the corresponding references.

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5.1 Self–weighted LSE for the ARMA parameters To estimate the ARMA parameters ϑ0 = (c0 , a01 , . . . a0P , b01 , . . . , b0Q ) of the ARMA-GARCH model (12), Ling (2003) considered the self-weighted LSE (SWL) defined by L ϑˆSW = arg min n−1 n ϑ∈Ψ

n 

ωt2 ˜2t (ϑ),

t=1

where the weights ωt are positive measurable functions of rt−1 , rt−2 , . . . , Ψ is a compact subspace of RP +Q+1 , and ˜t (ϑ) are the ARMA residuals computed for the value ϑ of the ARMA parameter and with fixed initial values. Take  −1−1/s k |rt−k | with E|r1 |2s < ∞ and s ∈ (0, 1). for instance ωt−1 = 1 + t−1 k=1 It can be shown that there exist constants K > 0 and ρ ∈ (0, 1) such that

) ) t−1 t−1   ) ∂˜t ) k )≤K |˜ t | ≤ K (1 + |ηt |) 1 + ρ |rt−k | and )) ρk |rt−k |. ∂ϑi ) k=1

It follows that



|ωt ˜t | ≤ K (1 + |ηt |) 1 +

∞ 

k

k=1



)

) ∞  ) ∂˜t ) 1+1/s k ) ) ≤K 1+ ρ , )ωt k ρ . ∂ϑi )

1+1/s k

k=1

Thus

k=1

∞

4 )2 )  ) 2 ∂˜ t )) 2 4 1+1/s k ) ≤ K E (1 + |η1 |) k ρ < ∞, E )ωt ˜t ∂ϑi ) k=1

which entails a finite variance for the SWL score vector ωt2 ˜t ∂˜t /∂ϑ. Ling √ L − ϑ0 ), allowing (2006) then deduced the asymptotic normality of n(ϑˆSW n 2 for the case Er1 = ∞.

5.2 Self–weighted QMLE To obtain an AN estimator of the parameter ϕ0 = (ϑ0 , θ0 ) in the ARMAGARCH model (12) under mild moment assumptions on the observed process, Ling (2006) proposed the self-weighted QMLE Q ϕˆSW n

= arg min n ϕ∈Φ

−1

n  t=1

ωt ˜t (ϕ),

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where ˜t (ϕ) = ˜2t (ϑ)/˜ σt2 (ϕ) + log σ ˜t2 (ϕ) with obvious notations. To understand the principle of this estimator, let us note that the minimized criterion generally converges to a limit criterion l(ϕ) = Eϕ ωt t (ϕ) satisfying + σt2 (ϕ0 ) σt2 (ϕ) {t (ϑ) − t (ϑ0 )}2 + 2 − 1 + Eϕ0 ωt l(ϕ) − l(ϕ0 ) = Eϕ0 ωt log 2 σt (ϕ0 ) σt (ϕ) σt2 (ϕ) 2ηt σt (ϕ0 ){t (ϑ) − t (ϑ0 )} . + Eϕ0 ωt σt2 (ϕ) The last expectation (when it exists) is zero because ηt is centered and is independent of the other random variables involved in the expectation. From the inequality x − 1 ≥ log x, we have + + σ 2 (ϕ0 ) σ 2 (ϕ0 ) σ 2 (ϕ) σ 2 (ϕ) + t2 − 1 ≥ Eϕ0 ωt log 2t + log t2 . Eϕ0 ωt log 2t σt (ϕ0 ) σt (ϕ) σt (ϕ0 ) σt (ϕ) Hence under the usual identifiability assumptions, l(ϕ) ≥ l(ϕ0 ) with equality if and only if ϕ = ϕ0 . Note that the orthogonality between ηt and the weight ωt is essential. Q Ling (2006) showed consistency and AN of ϕˆSW under the assumption n s E|r1 | < ∞ for some s > 0.

5.3 Lp–estimators The weighted estimators of the previous sections require the moment assumption Eη14 < ∞. Practitioners often claim that financial series do not admit (even low-order) moments. In GARCH processes an infinite variance can be obtained either by relaxing the parameters constraint or by allowing an infinite variance for ηt . In the GARCH(1,1) case the two sets of assumptions α01 + β01 ≥ 1 i) : or ii) : Eη12 = ∞ Eη12 = 1 imply an infinite variance for t . Under i), and the strict stationarity assumption, the asymptotic distribution of the QLME is generally Gaussian (see Section 3), whereas the usual estimators have non standard asymptotic distributions or are even non-consistent under ii); see Berkes and Horváth (2003), Hall and Yao (2003), Mikosch and Straumann (2002). It is therefore of interest to define alternative estimators enjoying a Gaussian asymptotic distribution under ii), or even under the more general situation where both α01 + β01 > 1 and Eη12 = ∞ are allowed for. Note that a GARCH model is generally defined under the standardization Eη12 = 1. When the existence of Eη12 is relaxed, one can identify the GARCH coefficients by imposing that the median of η12 be τ = 1. In the framework of

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ARCH(q) models, Horváth and Liese (2004) consider Lp -estimators, including the L1 -estimator ) ) q n ) )   ) 2 L1 −1 2 ) ˆ θn = arg min n ωt )t − ω − αi t−1 ) , ) ) θ t=1

i=1

p

where, for example, ωt−1 = 1 + i=1 2t−i + 4t−i . When ηt2 has a density, continuous and positive around its median τ = 1, the consistency and asymptotic normality of θˆnL1 is shown in Horváth and Liese (2004), without any moment assumption.

5.4 Least absolute deviations estimators In the framework of ARCH and GARCH models, Peng and Yao (2003) studied several least absolute deviations estimators. An interesting specification is the following θˆnLAD = arg min n−1 θ

n  ) ) )log 2t − log σ ˜t2 (θ)) .

(17)

t=1

With this estimator it is convenient to define the GARCH parameters under the condition that the median of η12 is equal to 1. It entails a reparametrization of standard GARCH models. Consider, for instance, a GARCH(1,1) model with parameters ω0 , α01 and β01 , and a Gaussian noise ηt . Since the median √ of η12 is τ = 0.4549..., the median of the square of ηt∗ = ηt / τ is 1, and the model is rewritten as t = σt ηt∗ ,

2 σt2 = τ ω0 + τ α01 2t−1 + β01 σt−1 .

It is interesting to note that the error terms log ηt∗2 = log 2t − log σ ˜t2 (θ) are iid with median 0 when θ = θ0 . Intuitively, this is the reason why it is not necessary to use weights in the sum (17). Under the moment assumption E21 < ∞ and certain regularity assumptions, it is shown in Peng and Yao (2003) that there exists a local solution of (17) which is weakly consistent and AN, with the standard rate of convergence n1/2 . This convergence holds even in the case of heavy-tailed errors : no condition on the moments of η1 beyond Eη12 = 1 is imposed.

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5.5 Whittle estimator Whittle estimation is a standard method for ARMA models, working in the spectral domain of the process; see Brockwell and Davis (1991), Section 10.8 for further details. It is well known that, under the moment assumption E41 < ∞, the square of a GARCH(p, q) model satisfies an ARMA(p ∧ q, q) model φθ0 (L)2t = ω0 + ψθ0 (L)ut ,

(18)

where L denotes the lag operator, φθ0 (z) = 1 −

p∧q 

(α0i + β0i )z i ,

ψθ0 (z) = 1 −

i=1

p 

β0i z i ,

ut = (ηt2 − 1)σt2 .

i=1

Thus, the spectral density of 2t is ) )2 Eu2t )ψθ0 (e−iλ )) . fθ0 (λ) = 2π |φθ0 (e−iλ )|2 Denote by γˆ2 (h) the sample autocovariance of 2t at lag h. At the Fourier frequencies λj = 2πj/n ∈ (−π, π], the periodogram , n -# ", n  + 1, . . . , , γˆ2 (h)e−ihλj , j ∈ J = − In (λj ) = 2 2 |h|
can be considered as a non parametric estimator of 2πfθ0 (λj ). Let ut (θ) =

( φθ (L) ' 2 t − ωφ−1 θ (1) . ψθ (L)

It can be shown that Eu21 (θ) =

Eu21 (θ0 ) 2π



π

−π

fθ0 (λ) dλ ≥ Eu21 (θ0 ) fθ (λ)

with equality if and only if θ = θ0 ; see Brockwell and Davis (1991) Proposition 10.8.1. In view of this inequality, it seems natural to consider the so-called Whittle estimator 1  In (λj ) . θˆnW = arg min n fθ (λj ) θ j∈J

For ARMA models with iid innovations the Whittle estimator has the same asymptotic behavior as the QMLE and LSE. For GARCH processes the Whittle estimator has still the same asymptotic behavior as the LSE, but simulations studies indicate that the Whittle estimator, for normal and student noises (ηt ), is less accurate than the QMLE. Moreover Giraitis and Robinson

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(2001), Mikosch and Straumann (2002), Straumann (2005) have shown that consistency requires the existence of E41 , and asymptotic normality requires E81 < ∞.

5.6 Moment estimators A sequence (t ) is called weak white noise if the t ’s are centered and uncorrelated, but not necessarily independent. In contrast, a sequence of centered and independent random variables is sometimes called strong white noise. The GARCH process is a leading example of weak white noise, but there exist numerous other examples of weak white noises satisfying (18). Consider for example the process vt = ηt ηt−1 where (ηt ) is iid N (0, 1). This process is clearly weak white noise. Straightforward computations show that vt2 satisfies a weak MA(1) representation of the form vt2 = 1 + ut + θut−1 , where (ut ) is weak white noise. Although (vt ) does not belong to the class of the strong GARCH models defined by (1), it can be called weak GARCH, in the sense that (vt ) is a white noise and (vt2 ) satisfies an ARMA model. The ARMA representations (18) of these weak GARCH models are estimated in Francq and Zakoïan (2000) by LS, under moment and mixing conditions, but without imposing a particular parametric model for (t ). The generalized method of moment (GMM) approach is particularly relevant (see Rich et al. (1991)) to estimate ARCH models without assuming strong assumptions on the noise (ηt ). To finish this non exhaustive list of alternative GARCH estimators, let us mention the existence of Bayesian estimators, using Monte Carlo integration with importance sampling for the computation of the posterior expectations; see Geweke (1989).

6 Properties of Estimators when some GARCH Coefficients are Equal to Zero To obtain the AN of the QMLE of GARCH models, a crucial assumption is that the true parameter vector has strictly positive components. When some components are equal to zero, the parameter, which is constrained to have nonnegative components, lies at the boundary of the parameter space and then, Assumption A5 in Section 3.1 is not satisfied. This assumption is a serious limitation to the estimation theory of GARCH. Indeed it could be particularly useful to derive the asymptotic distribution of the QMLE of a GARCH(p, q) model when, for instance, the underlying process is a GARCH(p − 1, q), or a GARCH(p, q − 1) process. Tests of the significance of the coefficients and tests of conditional homoscedasticity constitute typical

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situations where we have to study the QMLE when the parameter is at the boundary. In this section we study the asymptotic behaviour of the QMLE for GARCH processes, when the true parameter may have zero coefficients. We first see, by means of an elementary example, why the asymptotic distribution of the QMLE cannot be Gaussian when one or several GARCH coefficients are equal to zero.

6.1 Fitting an ARCH(1) model to a white noise The QMLE of an ARCH(1) model is obtained by minimizing the criterion ln (ω, α) = n−1

n 

t (ω, α),

t (ω, α) =

t=2

2t + log σt2 , σt2

where σt2 = ω + α2t−1 . In absence of constraints on the coefficients, the value of σt2 could be negative (this is the case when α < 0, ω > 0 and 2t−1 > −ω/α). In such a situation, t (ω, α), and thus the objective function ln (ω, α), are not defined. This is the reason why the minimization is made under the constraints ω > 0 and α ≥ 0. The QMLE estimator (ˆ ωn , α ˆ n ) = arg min ln (ω, α) ω>0, α≥0

then satisfies α ˆ n ≥ 0 almost surely, for all n. When the process is a white noise, then α01 = 0 and with probability one √ √ n(ˆ αn − α01 ) = nˆ αn ≥ 0, ∀n. √ In this case n(ˆ αn − α01 ) cannot converge in law to any non-degenerate Gaussian distribution N (m, s2 ) with s2 > 0. Indeed '√ ( ' ( lim P n(ˆ αn − α01 ) < 0 = 0 whereas P N (m, s2 ) < 0 > 0. n→∞

For the same reason, when the true value of a general GARCH parameter has zero components, the asymptotic distribution cannot be Gaussian, for the QMLE or for any other estimator which takes into account the positivity constraints.

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6.2 On the need of additional assumptions To prove the existence of the information matrix involved in the asymptotic distribution of the QMLE, we have to show that the variance of the vector σt−2 (θ0 )∂σt2 (θ0 )/ ∂θ, and the expectation of the matrix  2  1 ∂σt (θ0 ) ∂σt2 (θ0 ) Jt = 4 σt (θ0 ) ∂θ ∂θ are finite. A bound for these norms can be shown to be of the form Kc−1 or Kc−2 , where K is a constant and c > 0 is the smallest component of θ0 . Obviously, the proof breaks down when one or several components of θ0 are equal to zero. To see this technical problem more clearly, let us consider the ARCH(1) example. If ω0 α01 > 0 then the expectation of Jt is finite because     1 1 21 ω0−2 ω0−1 α−1 01 EJt = E ≤ , ω0−1 α−1 α−2 (ω0 + α01 21 )2 21 41 01 01 where the last inequality has to be taken componentwise. However, if α01 = 0   1 1 21 EJt = 2 E 21 41 ω0 is finite when E41 < ∞ only. Such extra moment assumptions seem necessary for ARCH models and for the GARCH(1,1), but can sometimes be avoided for more complex GARCH models. Consider for example a strictly GARCH(p, q) process with stationary p α01 > 0 and β01 > 0. Then, because j=1 β0j < 1, the following ARCH(∞) ∞ expansion holds σt2 (θ0 ) = c0 + j=1 b0j 2t−j with c0 > 0 and b0j > 0 for all j; see Giraitis et al. (2008) for a review on ARCH(∞) models. Similar expansions hold for the derivatives ∂σt2 /∂θi . Thus every term 2t−j appearing in the numerator of this ratio {∂σt2 /∂θ}/σt2 is also present in the denominator. In such a situation the moment assumption E41 < ∞ is not necessary for the existence of EJt .

6.3 Asymptotic distribution of the QMLE on the boundary For simplicity, let us take a parameter space of the form Θ = [ω, ω] × [0, α1 ] × · · · × [0, β p ]

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where ω > 0 and α1 , . . . , β p > 0. We assume that A11:

θ0 ∈ (ω, ω) × [0, α1 ) × · · · × [0, β p ),

allowing for zero GARCH coefficients, but excluding the case where θ0 is on the upper boundary of Θ. When Θ is not a product of intervals, Assumption A11 must be modified appropriately. We now define the "local" parameter space Λ = Λ(θ0 ) = Λ1 × · · · × Λp+q+1 , where Λ1 = R, and, for i = 2, . . . , p + q + 1, Λi = R if θ0i = 0 and Λi = [0, ∞) √ if θ0i = 0. In view of the positivity constraints, the random vector n(θˆn − θ) belongs to Λ with probability one. We already know that the QMLE is consistent under A1–A4, even when θ0 is on the boundary of Θ. If in addition A6, A11 and either A12:

E61 < ∞

or alternatively A12’:

σt2 (θ0 ) = c0 +

∞

2 j=1 b0j t−j

with b0j > 0 for all j ≥ 1

hold, then √ QML d n(θˆn − θ0 ) → λΛ := arg inf {λ − Z} J {λ − Z} , λ∈Λ

(19)

! with Z ∼ N 0, (Eη14 − 1)J −1 . ◦

When θ0 ∈Θ , we have Λ = Rp+q+1 !and we retrieve the standard result because λΛ = Z ∼ N 0, (Eη14 − 1)J −1 . When θ0 is on the boundary, the √ asymptotic distribution of n(θˆnQML − θ0 ) is more complex than a Gaussian. This is the law of the projection of the Gaussian vector Z on the convex cone Λ. The reader is referred to Andrews (1999) for similar results on a general framework, and to Francq and Zakoïan (2007) for the proof of (19). For fitting ARCH(q) models, Jordan (2003) allows a parameter belonging to the boundary of a non compact set, and a DGP which is not necessarily an ARCH process, but requires in particular the moment assumption E81 < ∞.

6.4 Application to hypothesis testing An important consequence of the non Gaussian behavior of the QMLE is that the Wald and Likelihood-Ratio (LR) tests do not have the standard χ2 asymptotic distribution. As an illustration consider the ARCH(2) case with θ0 = (ω0 , 0, 0). We have ⎧ ⎛ ⎛ ⎞ ⎞⎫ (Eη14 + 1)ω02 −ω0 −ω0 ⎬ Z1 ⎨ −ω0 1 0 ⎠ Z = ⎝ Z2 ⎠ ∼ N 0, (Eη14 − 1)J −1 = ⎝ ⎭ ⎩ −ω0 Z3 0 1

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and we can show that ⎞ ⎛ Z1 + ωZ2− + ωZ3− ⎠, Z2+ λΛ = ⎝ Z3+

Zi+ = max{Zi , 0} and Zi− = min{Zi , 0}.

ˆ 2 = 0) with probability We can see that, asymptotically, we have α ˆ 1 = 0 (or α 1/2, and α ˆ1 = α ˆ 2 = 0 with probability 1/4. Consequently, for the test of ! the null hypothesis H0 : α1 = α2 = 0, the Wald statistic Wn = n α ˆ 21 + α ˆ22 has a discrete component, and thus cannot be the usual χ22 . More precisely, it is easy to see that under H0 d

Wn → W ∼

1 1 1 δ0 + χ21 + χ22 . 4 2 4

One can show that the LR test has the same nonstandard asymptotic distribution (in the Gaussian case), whereas the Lagrange Multiplier (LM) test conserves its usual χ2 asymptotic distribution, even when H0 puts θ0 on the boundary of the parameter space; see Demos and Sentana (1998), Francq and Zakoïan (2007). This is not very surprising, since the likelihood of the conˆ 2 = 0, strained model is equal to that of the unconstrained model when α ˆ1 = α ˆ2 = 0 (see Figure 2). but the score is not necessarily zero when α ˆ1 = α

α α ˆ = α01 = 0

α α01 = 0

α ˆ

ω , α) of an ARCH(1) model with Fig. 2 Projected Log-likelihood (full line) α → log Ln (ˆ ˆ > 0, ∂ log Ln (ˆ ω , α)/∂α = 0 and the score α01 = 0. In the right-hand graph we have a ω , 0)/∂α > 0. In the left-hand graph we have α ˆ = 0 and ∂ log Ln (ˆ ω , α)/∂α = ∂ log Ln (ˆ ω , 0)/∂α < 0. In both cases the score is almost surely non null. ∂ log Ln (ˆ

Another important consequence of the non standard asymptotic distribution (19), is that the Wald, LM and LR tests do not have the same local asymptotic power. The Wald test generally outperforms the LM test in terms of local asymptotic power. This is not surprising because the LM test do not take into account the one-sided nature of the alternatives. It is of course possible to derive one-sided versions of the LM test; see e.g. Demos and Sentana (1998).

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7 Conclusion Since many financial series exhibit heavy-tailed marginal distributions, it is particularly important to obtain estimation procedures which do not hinge on high-order moment assumptions. The QMLE is the most popular method for estimating GARCH models. In the general ARMA-GARCH case, the consistency is obtained without moment assumption on the observed process t , even when the parameter is on the boundary of the parameter space (a situation frequently encountered in test problems). In the pure GARCH case with ◦

θ0 ∈Θ the AN is also obtained without moment assumption on t , but addition assumptions are required in the general ARMA-GARCH case. When θ0 is on the boundary of the parameter space, the asymptotic distribution of the QMLE is no longer Gaussian, but is that of the projection of a Gaussian vector on a convex cone. The main drawbacks of the QMLE are that i) the estimator is not explicit and it requires a numerical optimization, ii) the AN requires the existence of fourth-order moments for the iid process ηt , iii) the estimator is in general inefficient, iv) the AN requires moments assumptions on t in the general ARMA-GARCH case, v) a fully parametric specification is required. Concerning the point iii) it is however interesting to note that the QMLE is not only efficient in the Gaussian case, but also when the distribution of ηt belongs to the class defined in Section 4. At least in the ARCH case, a two-step LSE should respond satisfactorily to the point i), but with a cost in terms of moment conditions. Weighted Lp and least absolute deviations estimators have been recently developed to alleviate the point ii). The MLE is a fully satisfactory response to the points ii) and iii), but requires a complete specification of the error distribution, unless adaptive estimators be employed. Also very recently, self-weighted LSE and self-weighted QMLE have been developed to respond to the point iv). Methods based on orthogonality conditions, such as the GMM, are simple and obviously more robust to model misspecifications, and are therefore worthwhile procedures for considering the points i) and v).

References Andrews, D.W.K. (1999): Estimation when a parameter is on a boundary. Econometrica 67,1341–1384. Berkes, I. and Horváth, L. (2003): The rate of consistency of the quasi-maximum likelihood estimator. Statistics and Probability Letters 61, 133–143. Berkes, I. and Horváth, L. (2004): The efficiency of the estimators of the parameters in GARCH processes Annals of Statistics 32, 633–655. Berkes, I., Horváth, L. and Kokoszka, P.S. (2003): GARCH processes: structure and estimation. Bernoulli 9, 201–227. Billingsley, P. (1961): The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.

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Bose, A. and Mukherjee, K. (2003): Estimating the ARCH parameters by solving linear equations. Journal of Time Series Analysis 24, 127–136. Bougerol, P. and Picard, N. (1992): Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127. Boussama, F. (1998): Ergodicité, mélange et estimation dans les modèles GARCH. PhD Thesis, Université Paris-7, Paris. Brockwell, P.J. and Davis, R.A. (1991): Time series: theory and methods. Springer, NewYork. Chen, M. and An, H.Z. (1998): A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica 8, 505–510. Demos, A. and Sentana, E. (1998): Testing for GARCH effects: a one-sided approach. Journal of Econometrics 86, 97–127. Drost, F.C. and Klaassen, C.A.J. (1997): Efficient estimation in semiparametric GARCH models. Journal of Econometrics 81, 193–221. Engle, R.F. (1982): Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 987–1007. Engle, R.F. and González-Rivera, G. (1991): Semiparametric ARCH models. Journal of Business and Econometric Statistics 9, 345–359. Escanciano, J.C. (2007): Quasi-maximum likelihood estimation of semi-strong GARCH models. Working documents. Francq, C. and Zakoïan, J.M. (2000): Estimating weak GARCH representations. Econometric Theory 16, 692–728. Francq, C. and Zakoïan, J.M. (2004): Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637. Francq, C. and Zakoïan, J.M. (2006): On efficient inference in GARCH processes. In: Bertail, P., Doukhan, P., Soulier, P. (Eds): Statistics for dependent data, 305–327. Springer, New-York. Francq, C. and Zakoïan, J.M. (2007): Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and their Applications 117, 1265–1284. Giraitis, L., Leipus, R. and Surgailis, D. (2008): ARCH(∞) models and long-memory properties. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 70–84 Springer, New York. González-Rivera, G. and Drost, F.C. (1999): Efficiency comparisons of maximum-likelihood based estimators in GARCH models. Journal of Econometrics 93, 93–111. Gouriéroux, C. (1997): ARCH models and financial applications. Springer, New York. Geweke, J. (1989): Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics 40, 63–86. Giraitis, L. and Robinson, P.M. (2001): Whittle estimation of ARCH models. Econometric Theory 17, 608–631. Hall, P. and Yao, Q. (2003): Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317. Hamilton, J.D. (1994): Time series analysis. Princeton University Press, Princeton. Horváth, L. and Liese, F. (2004): Lp -estimators in ARCH models. Journal of Statistical Planning and Inference 119, 277–309. Jensen, S.T. and Rahbek, A. (2004a): Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646. Jensen, S.T. and Rahbek, A. (2004b): Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226. Jordan, H. (2003): Asymptotic properties of ARCH(p) quasi maximum likelihood estimators under weak conditions. PhD Thesis, University of Vienna. Lee, S.W. and Hansen, B.E. (1994): Asymptotic theory for the GARCH(1,1) quasimaximum likelihood estimator. Econometric Theory 10, 29–52.

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Lindner, A. (2008): Stationarity, mixing, distributional properties and moments of GARCH(p,q)–processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 43–69. Springer, New York. Ling, S. (2003): Self-weighted LSE and MLE for ARMA-GARCH models. Unpublished working paper, HKUST. Ling, S. (2006): Self-weighted and local quasi-maximum likelihood estimators for ARMAGARCH/IGARCH models. Journal of Econometrics. To appear. Ling, S. and Li, W.K. (1997): On fractionally integreted autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194. Ling, S. and Li, W.K. (1998): Limiting distributions of maximum likelihood estimators for unstable ARMA models with GARCH errors. Annals of Statistics 26, 84–125. Ling, S. and McAleer, M. (2003): Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310. Ling, S. and McAleer, M. (2003): Adaptive estimation in nonstationry ARMA models with GARCH noises. Annals of Statistics 31, 642–674. Linton, O. (1993): Adaptive estimation in ARCH models. Econometric Theory 9, 539–564. Lumsdaine, R.L. (1996): Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575–596. Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R.J. (1995): Parameter estimation for ARMA models with infinite variance innovations. Annals of Statistics 23, 305–326. Mikosch, T. and Straumann, D. (2002): Whittle estimation in a heavy-tailed GARCH(1,1) model. Stochastic Processes and their Application 100, 187–222. Nelson, D.B. (1990): Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334. Peng, L. and Yao, Q. (2003): Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90, 967–975. Rich, R.W., Raymond, J. and Butler, J.S. (1991): Generalized instrumental variables estimation of autoregressive conditional heteroskedastic models. Economics Letter 35, 179–185. Straumann, D. (2005): Estimation in conditionally heteroscedastic time series models. Lecture Notes in Statistics. Springer, Berlin Heidelberg. Straumann, D. and Mikosch, T. (2006): Quasi-MLE in heteroscedastic time series: a stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495. Wald, A. (1949): Note on the consistency of the maximum likelihood estimate. The Annals of Mathematical Statistics 20, 595–601. Weiss, A.A. (1986): Asymptotic theory for ARCH models: estimation and testing. Econometric Theory 2, 107–131.

Practical Issues in the Analysis of Univariate GARCH Models Eric Zivot

Abstract This chapter gives a tour through the empirical analysis of univariate GARCH models for financial time series with stops along the way to discuss various practical issues associated with model specification, estimation, diagnostic evaluation and forecasting.

1 Introduction There are many very good surveys covering the mathematical and statistical properties of GARCH models. See, for example, Bera and Higgins (1995), Bollerslev et al. (1994), Pagan (1996), Palm (1996), Diebold and Lopez (1996) and Teräsvirta (2008). There are also several comprehensive surveys that focus on the forecasting performance of GARCH models including Poon and Granger (2003), Poon (2005), and Andersen et al. (2006). However, there are relatively few surveys that focus on the practical econometric issues associated with estimating GARCH models and forecasting volatility. This chapter, which draws heavily from Zivot and Wang (2005), gives a tour through the empirical analysis of univariate GARCH models for financial time series with stops along the way to discuss various practical issues. Multivariate GARCH models are discussed in the chapter by Silvennoinen and Teräsvirta (2008). The plan of this chapter is as follows. Section 2 reviews some stylized facts of asset returns using example data on Microsoft and S&P 500 index returns. Section 3 reviews the basic univariate GARCH model. Testing for GARCH effects and estimation of GARCH models are covered in Sections 4 and 5. Asymmetric and non-Gaussian GARCH models are discussed in Section 6,

Eric Zivot Department of Economics, University of Washington, Economics Box 353330, Seattle, WA 98195-3330, e-mail: [email protected]

T.G. Anderson et al., Handbook of Financial Time Series, DOI: 10.1007/978-3-540-71297-8_5, © Springer-Verlag Berlin Heidelberg 2009

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Mean Med

Min

Max Std. Dev Skew Kurt JB Daily Returns MSFT 0.0016 0.0000 -0.3012 0.1957 0.0253 -0.2457 11.66 13693 S&P 500 0.0004 0.0005 -0.2047 0.0909 0.0113 -1.486 32.59 160848 Monthly Returns MSFT 0.0336 0.0336 -0.3861 0.4384 0.1145 0.1845 4.004 9.922 S&P 500 0.0082 0.0122 -0.2066 0.1250 0.0459 -0.8377 5.186 65.75 Notes: Sample period is 03/14/86 - 06/30/03 giving 4365 daily observations. Table 1 Summary Statistics for Daily and Monthly Stock Returns.

and long memory GARCH models are briefly discussed in Section 7. Section 8 discusses volatility forecasting, and final remarks are given Section 9 1 .

2 Some Stylized Facts of Asset Returns Let Pt denote the price of an asset at the end of trading day t. The continuously compounded or log return is defined as rt = ln(Pt /Pt−1 ). Figure 1 plots the daily log returns, squared returns, and absolute value of returns of Microsoft stock and the S&P 500 index over the period March 14, 1986 through June 30, 2003. There is no clear discernible pattern of behavior in the log returns, but there is some persistence indicated in the plots of the squared and absolute returns which represent the volatility of returns. In particular, the plots show evidence of volatility clustering - low values of volatility followed by low values and high values of volatility followed by high values. This behavior is confirmed in Figure 2 which shows the sample autocorrelations of the six series. The log returns show no evidence of serial correlation, but the squared and absolute returns are positively autocorrelated. Also, the decay rates of the sample autocorrelations of rt2 and |rt | appear much slower, especially for the S&P 500 index, than the exponential rate of a covariance stationary autoregressive-moving average (ARMA) process suggesting possible long memory behavior. Monthly returns, defined as the sum of daily returns over the month, are illustrated in Figure 3. The monthly returns display much less volatility clustering than the daily returns. Table 1 gives some standard summary statistics along with the JarqueBera test for normality. The latter is computed as

* − 3)2 T  2 (kurt skew + , (1) JB = 6 4

1

All of the examples in the paper were constructed using S-PLUS 8.0 and S+FinMetrics 2.0. Script files for replicating the examples may be downloaded from http://faculty.washington.edu/ezivot

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Fig. 1 Daily returns, squared returns and absolute returns for Microsoft and the S&P 500 index.

* denotes the sample kur denotes the sample skewness and kurt where skew tosis. Under the null that the data are iid normal, JB is asymptotically distributed as chi-square with 2 degrees of freedom. The distribution of daily returns is clearly non-normal with negative skewness and pronounced excess kurtosis. Part of this non-normality is caused by some large outliers around the October 1987 stock market crash and during the bursting of the 2000 tech bubble. However, the distribution of the data still appears highly non-normal even after the removal of these outliers. Monthly returns have a distribution that is much closer to the normal than daily returns.

3 The ARCH and GARCH Model Engle (1982) showed that the serial correlation in squared returns, or conditional heteroskedasticity, can be modeled using an autoregressive conditional heteroskedasticity (ARCH) model of the form

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Fig. 2 Sample autocorrelations of rt , rt2 and |rt | for Microsoft and S&P 500 index.

yt = Et−1 [yt ] + t , t = zt σt ,

(2) (3)

σt2 = a0 + a1 2t−1 + · · · + ap 2t−p ,

(4)

where Et−1 [·] represents expectation conditional on information available at time t − 1, and zt is a sequence of iid random variables with mean zero and unit variance. In the basic ARCH model zt is assumed to be iid standard normal. The restrictions a0 > 0 and ai ≥ 0 (i = 1, . . . , p) are required for σt2 > 0. The representation (2) - (4) is convenient for deriving properties of the model as well as for specifying the likelihood function for estimation. The equation for σt2 can be rewritten as an AR(p) process for 2t 2t = a0 + a1 2t−1 + · · · + ap 2t−p + ut ,

(5)

where ut = 2t −σt2 is a martingale difference sequence (MDS) since Et−1 [ut ] = 0 and it is assumed that E(2t ) < ∞. If a1 + · · · + ap < 1 then t is covariance stationary, the persistence of 2t and σt2 is measured by a1 + · · · + ap and σ ¯ 2 = var(t ) = E(2t ) = a0 /(1 − a1 − · · · − ap ).

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Fig. 3 Monthly Returns, Squared Returns and Sample Autocorrelations of Squared Returns for Microsoft and the S&P 500.

An important extension of the ARCH model proposed by Bollerslev (1986) replaces the AR(p) representation in (4) with an ARMA(p, q) formulation σt2 = a0 +

p  i=1

ai 2t−i +

q 

2 bj σt−j ,

(6)

j=1

where the coefficients ai (i = 0, · · · , p) and bj (j = 1, · · · , q) are all assumed to be positive to ensure that the conditional variance σt2 is always positive.2 The model in (6) together with (2)-(3) is known as the generalized ARCH or GARCH(p, q) model. The GARCH(p, q) model can be shown to be equivalent to a particular ARCH(∞) model. When q = 0, the GARCH model reduces to the ARCH model. In order for the GARCH parameters, bj (j = 1, · · · , q), to be identified at least one of the ARCH coefficients ai (i > 0) must be nonzero. Usually a GARCH(1,1) model with only three parameters in the conditional variance equation is adequate to obtain a good model fit for financial time 2

Positive coefficients are sufficient but not necessary conditions for the positivity of conditional variance. See Nelson (1992) and Conrad and Haag (2006) for more general conditions.

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series. Indeed, Hansen and Lunde (2004) provided compelling evidence that is difficult to find a volatility model that outperforms the simple GARCH(1,1). Just as an ARCH model can be expressed as an AR model of squared residuals, a GARCH model can be expressed as an ARMA model of squared residuals. Consider the GARCH(1,1) model 2 σt2 = a0 + a1 2t−1 + b1 σt−1 .

(7)

Since Et−1 (2t ) = σt2 , (7) can be rewritten as 2t = a0 + (a1 + b1 )2t−1 + ut − b1 ut−1 ,

(8)

which is an ARMA(1,1) model with ut = 2t − Et−1 (2t ) being the MDS disturbance term. Given the ARMA(1,1) representation of the GARCH(1,1) model, many of its properties follow easily from those of the corresponding ARMA(1,1) process for 2t . For example, the persistence of σt2 is captured by a1 + b1 and covariance stationarity requires that a1 + b1 < 1. The covariance stationary GARCH(1,1) model has an ARCH(∞) representation with ai = a1 bi−1 1 , and the unconditional variance of t is σ ¯ 2 = a0 /(1 − a1 − b1 ). For the general GARCH(p, q) model (6), the squared residuals t behave p like an qARMA(max(p, q), q) process. Covariance stationarity requires a + i=1 i j=1 bi < 1 and the unconditional variance of t is σ ¯ 2 = var(t ) =

1−



a0

p i=1

ai +

q

.

(9)

j=1 bi

3.1 Conditional mean specification Depending on the frequency of the data and the type of asset, the conditional mean Et−1 [yt ] is typically specified as a constant or possibly a low order autoregressive-moving average (ARMA) process to capture autocorrelation caused by market microstructure effects (e.g., bid-ask bounce) or non-trading effects. If extreme or unusual market events have happened during sample period, then dummy variables associated with these events are often added to the conditional mean specification to remove these effects. Therefore, the typical conditional mean specification is of the form Et−1 [yt ] = c +

r  i=1

φi yt−i +

s  j=1

θj t−j +

L 

βl xt−l + t ,

l=0

where xt is a k × 1 vector of exogenous explanatory variables.

(10)

Practical Issues in the Analysis of Univariate GARCH Models

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In financial investment, high risk is often expected to lead to high returns. Although modern capital asset pricing theory does not imply such a simple relationship, it does suggest that there are some interactions between expected returns and risk as measured by volatility. Engle, Lilien and Robins (1987) proposed to extend the basic GARCH model so that the conditional volatility can generate a risk premium which is part of the expected returns. This extended GARCH model is often referred to as GARCH-in-the-mean or GARCH-M model . The GARCH-M model extends the conditional mean equation (10) to include the additional regressor g(σt ), which can be an arbitrary function of conditional volatility σt . The most common specifications are g(σt ) = σt2 , σt , or ln(σt2 ).

3.2 Explanatory variables in the conditional variance equation Just as exogenous variables may be added to the conditional mean equation, exogenous explanatory variables may also be added to the conditional variance formula (6) in a straightforward way giving σt2

= a0 +

p  i=1

ai 2t−i

+

q  j=1

2 bj σt−j +

K 

δk zt−k ,

k=1

where zt is a m × 1 vector of variables, and δ is a m × 1 vector of positive coefficients. Variables that have been shown to help predict volatility are trading volume, macroeconomic news announcements (Lamoureux and Lastrapes (1990a), Flannery and Protopapadakis (2002), Bomfin (2003)), implied volatility from option prices and realized volatility (Taylor and Xu (1997), Blair et al. (2001)), overnight returns (Gallo and Pacini (1997), Martens (2002)), and after hours realized volatility (Chen et al. (2008))

3.3 The GARCH model and stylized facts of asset returns Previously it was shown that the daily returns on Microsoft and the S&P 500 exhibited the “stylized facts” of volatility clustering as well as a non-normal empirical distribution. Researchers have documented these and many other stylized facts about the volatility of economic and financial time series. Bollerslev et al. (1994) gave a complete account of these facts. Using the ARMA representation of GARCH models shows that the GARCH model is capable

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of explaining many of those stylized facts. The four most important ones are: volatility clustering, fat tails, volatility mean reversion, and asymmetry. To understand volatility clustering, consider the GARCH(1, 1) model in (7). Usually the GARCH coefficient b1 is found to be around 0.9 for many daily or weekly financial time series. Given this value of b1 , it is obvious that 2 will be followed by large values of σt2 , and small values large values of σt−1 2 of σt−1 will be followed by small values of σt2 . The same reasoning can be obtained from the ARMA representation in (8), where large/small changes in 2t−1 will be followed by large/small changes in 2t . It is well known that the distribution of many high frequency financial time series usually have fatter tails than a normal distribution. That is, extreme values occur more often than implied by a normal distribution. Bollerslev (1986) gave the condition for the existence of the fourth order moment of a GARCH(1, 1) process. Assuming the fourth order moment exists, Bollerslev (1986) showed that the kurtosis implied by a GARCH(1, 1) process with normal errors is greater than 3, the kurtosis of a normal distribution. He and Teräsvirta (1999a) and He and Teräsvirta (1999b) extended these results to general GARCH(p, q) models. Thus a GARCH model with normal errors can replicate some of the fat-tailed behavior observed in financial time series. A more thorough discussion of extreme value theory for GARCH is given by Davis and Mikosch (2008). Most often a GARCH model with a non-normal error distribution is required to fully capture the observed fat-tailed behavior in returns. These models are reviewed in sub-Section 6.2. Although financial markets may experience excessive volatility from time to time, it appears that volatility will eventually settle down to a long run level. Recall, the unconditional variance of t for the stationary GARCH(1, 1) model is σ ¯ 2 = a0 /(1 − a1 − b1 ). To see that the volatility is always pulled toward this long run, the ARMA representation in (8) may be rewritten in mean-adjusted form as: ¯ 2 ) = (a1 + b1 )(2t−1 − σ ¯ 2 )