CONTENTS PIERRE DUBOIS, BRUNO JULLIEN, AND THIERRY MAGNAC: Formal and Informal Risk Shar-
ing in LDCs: Theory and Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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YACINE AÏT-SAHALIA AND JEAN JACOD: Fisher’s Information for Discretely Sampled Lévy
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ARIE BERESTEANU AND FRANCESCA MOLINARI: Asymptotic Properties for a Class of Partially Identified Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHARLES BELLEMARE, SABINE KRÖGER, AND ARTHUR VAN SOEST: Measuring Inequity Aversion in a Heterogeneous Population Using Experimental Decisions and Subjective Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROBERT M. ANDERSON AND ROBERTO C. RAIMONDO: Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets . . . . . . . . . . . . . . . MARTIN W. CRIPPS, JEFFREY C. ELY, GEORGE J. MAILATH, AND LARRY SAMUELSON: Common Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ANNOUNCEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FORTHCOMING PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES . . . . . . . . . .
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Econometrica, Vol. 76, No. 4 (July, 2008), 679–725
FORMAL AND INFORMAL RISK SHARING IN LDCS: THEORY AND EMPIRICAL EVIDENCE BY PIERRE DUBOIS, BRUNO JULLIEN, AND THIERRY MAGNAC1 We develop and estimate a model of dynamic interactions in which commitment is limited and contracts are incomplete to explain the patterns of income and consumption growth in village economies of less developed countries. Households can insure each other through both formal contracts and informal agreements, that is, selfenforcing agreements specifying voluntary transfers. This theoretical setting nests the case of complete markets and the case where only informal agreements are available. We derive a system of nonlinear equations for income and consumption growth. A key prediction of our model is that both variables are affected by lagged consumption as a consequence of the interplay of formal and informal contracting possibilities. In a semiparametric setting, we prove identification, derive testable restrictions, and estimate the model with the use of data from Pakistani villages. Empirical results are consistent with the economic arguments. Incentive constraints due to self-enforcement bind with positive probability and formal contracts are used to reduce this probability. KEYWORDS: Risk sharing, contracts, incomplete markets, informal transfers.
1. INTRODUCTION LIMITED COMMITMENT is an attractive way to accommodate consistent empirical evidence that full-risk sharing is rejected in village economies. In this paper, we nest complete markets and limited commitment in a unified setup. In the context of risk sharing, we show the empirical relevance of the joint consideration of incomplete markets and dynamic interactions using household data from villages in Pakistan. Limited commitment refers to situations in which there is no legal contract enforceability (Thomas and Worrall (1988)). Some level of insurance can nonetheless be achieved through informal agreements between agents engaged in a long-term relationship. These agreements take the form of voluntary transfers that are self-enforcing in a game theoretic sense. Parties abide by the informal agreement because they anticipate that defection would result in the breakdown of future profitable informal agreements. The existing literature has contrasted settings with a complete market structure and settings in which all agreements are informal, yet real institutions, and in particular village institutions, usually involve a mix of formal and informal agreements. Indeed, 1 We thank IFPRI for providing us with the data. We are indebted to Fabien Postel-Vinay for very detailed comments. We thank the co-editor and three referees for their comments, Orazio Attanasio, Richard Blundell, Andrew Foster, Patrick Gonzalez, Stéphane Grégoir, Ethan Ligon, Thomas Mariotti, Mark Rosenzweig, Bernard Salanié, Mike Whinston, and Tim Worall for discussions, and participants at seminars at Toulouse, CREST, Bristol, Southampton, University College of London, Montréal, Laval University, Cergy, Namur, CEMFI, IZA, Mannheim, Alicante, Munich, Marseille, and Yale and conferences (Jean Jacques Laffont’05, WCES’05 London, SBE’05 Natal, ASSET’06 Lisbon) for their helpful comments. The usual disclaimer applies.
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village institutions may be able to enforce specific types of formal contracts like sharecropping or other land-leasing contracts even if a complete market structure is beyond reach. In this paper we propose a model of risk sharing that allows for two types of transactions between agents—voluntary transfers and formal contracts. A distinguishing feature is the focus on the interaction between contracts and informal agreements. Informal agreements cannot only specify voluntary transfers in various states of nature, but also which contracts should be signed and when. As a consequence, all observed decisions in equilibrium are part of informal agreements including formal contracts. Formal contracting, instead of expanding the set of feasible transfers, serves to expand the set of incentive compatible allocations by allowing parties to use them as an enforcement tool for voluntary transfers. To some extent, formal contracts are used as collateral in informal lending–borrowing agreements in a risk sharing context. Individuals who receive a large transfer—because of negative income shocks, for instance—should, in exchange, concede more favorable terms on the formal contracts, raising the future prospects of the other party. The existence of formal contracting relaxes incentive constraints due to limited commitment and improves global efficiency. Another key contribution of this paper is the econometric and empirical implementation using data from village economies in Pakistan which show that our theoretical model is an empirically credible alternative. We do more than simply test the alternatives of complete markets, limited commitment, and the mix of formal and informal contracts. We exhibit the conditions under which the model is semiparametrically identified, derive its testable implications, and test them. Contract enforcement depends on specific institutions within each village and evaluating their degree of formality is a difficult task. This motivates us to work within the setup of Deaton (1997) in that we do not examine specific institutions in villages, but the consequences they have on household behavior in terms of the dynamics of income and consumption. To this end, we exploit the rich dynamics that our setting generates. We characterize the dynamics of household consumption and income by showing that these processes are functions of a state variable which is the relative marginal utility of consumption of the household with respect to the village. The process of this state variable retains a rich memory of the past in contrast to the limited commitment case in which once an incentive compatibility constraint is binding, all the past becomes irrelevant. Household relative marginal utility is a random walk with jumps (as in the limited commitment case of Ligon, Thomas, and Worrall (2002)) although the occurrence and size of jumps are not exogenous and depend on the relative marginal utility in the previous period. Our paper builds upon a rich empirical and theoretical literature on village economies in LDCs. Starting with the seminal paper of Townsend (1994), the empirical evaluation of the complete market hypothesis in village economies
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has led to a better understanding of market failures and to a better identification of the characteristics of the households that are most affected by these failures (Deaton (1997), Morduch (1999), Fafchamps (2003), Dercon (2004)). In developed countries, the question of interest is also how well agents are insured (Attanasio and Ríos-Rull (2003), Blundell, Pistaferri, and Preston (2003)). Regarding theoretical arguments, we specifically use the literature that highlights the issue of contract enforcement and limited commitment in risk sharing (see Thomas and Worrall (1988), Kocherlakota (1996)). Indeed, village economies lack the institutions needed to enforce contracts and in most villages there are no written records of transfers such as loans, nor are there proper institutions designed to enforce repayments, which can therefore be delayed or simply canceled. As a consequence, villagers use informal agreements that are Paretoimproving provided they can be self-enforced (Coate and Ravallion (1993), Fafchamps (1992)). This requirement restricts the set of informal agreements which in turn may not be rich enough to achieve complete risk sharing in the village. A few recent papers show that these arguments are empirically relevant and that self-enforcing transfers play a part in sharing risk within villages (Ligon, Thomas, and Worrall (2000, 2002)) and within networks or extended families (Foster and Rosenzweig (2001)). Full risk sharing and limited commitment are two polar cases, however. Some institutions might help enforce some formal contracts. Specifically, sharecropping and fixed-rent formal contracts are commonly observed in villages of LDCs and their role in allocating risk has been repeatedly emphasized (Stiglitz (1974), Newbery (1977)). The ability of landlords to enforce contracts on land is a critical issue and the design of such contracts is the subject of many empirical studies (Shaban (1987)). Dubois (2000) provided empirical evidence, using the same data as the present paper, that sharecropping or other land contracts allow risk sharing. Households use these instruments to smooth income as well as consumption (Morduch (1995)). Furthermore, Bell (1988) and Udry (1994) have emphasized that formal contracts are often accompanied by informal transfers that attenuate their effects in bad states of nature. Another branch of literature from which we depart focuses on the role of asymmetric information (see Thomas and Worrall (1990), Doepke and Townsend (2006), and references therein). The issue of formal and informal risk sharing is also of wider importance apart from this specific application, as it is relevant to many other domains, including international debt, finance, the internal labor market, and social networks. Outline and Main Findings In Section 2, we construct and analyze a theoretical model of long-term dynamic interactions between households in which several cases are nested: complete markets if all risks can be insured by formal contracts; a single noncontingent transfer as in Gauthier, Poitevin, and Gonzalez (1997); limited commitment and no formal contracts as in Ligon, Thomas, and Worrall (2002).
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A specific feature is that we embed both income and preference shocks into a two-agent model in which one agent, who ultimately represents the “village,” can commit himself while the other agent (the “household”) cannot. The market allocation (consumption, income, and contracts) is obtained as an equilibrium of an infinite horizon game. Using recursive methods, we characterize constrained Pareto efficient allocations where an allocation specifies the sequence of contracts and informal transfers to be implemented and which are possibly random to account for nonconcavity issues. We then derive optimality conditions and monotone comparative results. Any efficient equilibrium allocation is characterized by two monotone (stochastic) mappings: one maps current consumption into contracts and the other is an Euler equation in which consumption growth, instead of being a martingale, is affected by lagged consumption and current income. In Section 3, we specify the econometric model. It involves two equations of interest: the Euler equation of consumption dynamics and an income equation. We derive the testable predictions of our theoretical model using this specification. One such prediction is that income monotonically depends on lagged consumption. Other testable predictions include other monotonicity restrictions as well as parametric restrictions across the income and consumption equations. In Section 4, the model is estimated by semiparametric methods using a panel survey collected in Pakistan by the International Food Policy Research Institute (IFPRI) at the end of the 1980s. Because of nonlinearities, we estimate the model of income and consumption growth by penalized splines (Yu and Ruppert (2002)). Results are consistent with the predictions of the theoretical model and, in particular, lagged consumption affects income as predicted. The results hold true in the presence of measurement errors, of superior information that households might have on future income shocks, and of nutrition effects on productivity as the two last factors could generate a similar prediction. They are also robust to assumptions about labor market imperfections. Among other results, estimation of the consumption growth equation leads to an estimate of a lower bound on the probability that a self-enforcing constraint binds. This estimated bound is roughly equal to 15%. In the last section, we discuss the economic relevance of our empirical results, the limitations of our approach, and future research avenues. We conclude by discussing some policy implications. Proofs are in the Appendix. Data and programs can be found in the supplementary materials in Dubois, Jullien, and Magnac (2008). 2. A THEORETICAL MODEL OF RISK SHARING WITH FORMAL AND INFORMAL CONTRACTS
This section presents the theoretical model. We first define preferences, income processes, and the contractual environment as well as the timing of the game. Second, we characterize the constrained Pareto optimal allocations. To
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this end, we define two Bellman equations at the ex ante and interim stages and discuss their properties. Building on this, we propose a characterization of the dynamics of income and consumption processes. Last, we extend the model to account for the presence of observed heterogeneity. 2.1. The Model We consider a two-agent framework and assume full commitment by one agent and none by the other.2 Time is discrete and goes from 1 to infinity, t = 1 ∞. Let st denote a random state of the world at date t, identically and independently distributed (i.i.d.) on a finite support S . The extension to a general Markov process is undertaken at the end of the section. State s occurs with probability ps and agent i receives exogenous resources ωi (s) There is no storage possibility and no financial market. The utility function of agent 1 is written at date t ≥ 1 as ut (·) = ηt u(·) where (ηt )∞ t=1 is a random process. In the empirical part, ηt represents the new information affecting preferences that is revealed at the beginning of a period and that is never observed by the econometrician. In the theoretical part, its presence ensures enough variability in the allocation, implying in particular that the problem is strictly concave and the allocation is uniquely defined with probability 1. Setting η0 = 1 we denote χt = ηt /ηt−1 and we follow the literature by assuming that χt is i.i.d. on an interval of R+ , with mean 1 and positive density. The second agent has nonstochastic preferences, since what matters are ratios of marginal utility between the two agents. The second agent’s preferences are described by a von Neumann–Morgenstern utility function v(·) We assume that u and v are concave, twice continuously differentiable, u(0) = v(0) = 0 and u (0) = v (0) = +∞, and this ensures that consumption levels are positive. Both agents share the same discount rate ρ The ex ante expected in∞ t−1 tertemporal utility of agents 1 and 2 is, respectively, E[ ut (ct1 )] and t=1 ρ ∞ t−1 2 E[ t=1 ρ v(ct )]. Because of convexity issues, we allow agents to implement random allocations. To this end, we assume that there is a public random variable, εt , which is i.i.d. and continuous on a compact support. The precise timing of realization of the various events within period t, is the following: • Interim stage: At the beginning of period t the shocks χt and εt are publicly realized. A formal contract is signed. • Ex post stage: At the end of period t the income shock st is publicly realized. The formal contract is enforced. Then parties may complement it by voluntary transfers and consumption takes place. We shall also consider an ex ante stage posterior to date t − 1 consumption but anterior to the realization of date t shocks χt and εt 2 We analyzed the two-sided limited commitment case in a companion paper (Dubois, Jullien, and Magnac (2002)).
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Once signed, formal contracts are valid for period t and binding for both agents. They are restricted and incomplete as follows: 1. Contracts are short term, specifying how resources will be shared within the period only. 2. Contracts can be contingent only on the realization of exclusive events e ∈ E , where E is a finite partition of the set of states of nature S An event e ∈ E is a subset of S and occurs with probability pe = s∈e ps . 3. Contracts entail real costs, while voluntary transfers involve no real cost.3 A contract is represented by a vector (τ(e))e of net transfers received by agent 1. Transfers belong to some compact set ¯ where τ(e) < e [τ(e) τ(e)], 0 ≤ τ(e). ¯ Transferring τ in event e through a contract entails a cost ϕ(e τ) for agent 2 As a consequence, total resources in state s ∈ e are ω1 (s) + ω2 (s) − ϕ(e τ(e)) We assume that ϕ(e ·) is continuously differentiable, quasi-convex, with a minimum ϕ(e 0) = 0 at τ = 0. For conciseness, we also assume that the set of contracts is such that both agents enjoy positive consumption levels in all states.4 This could easily be relaxed by assuming that the transfers are set ex post to the minimum between the formal quantity and the agent’s resources (see our working paper, Dubois, Jullien, and Magnac, (2005)). When no contract is feasible (E = ∅), we obtain the model of informal risk sharing as in Thomas and Worrall (1988). We obtain the polar case of complete contracting when E = S and the cost of transfers is zero, ϕ ≡ 0, over the relevant range (and τ is sufficiently negative). Moreover, our setting nests the model of Gauthier, Poitevin, and Gonzalez (1997) since, among other things, they consider one ex ante transfer only (E = {S } i.e., card(E ) = 1). Denote Ht = [(sr−1 χr εr )r ]tr=1 to be the history of shocks up to the beginning of period t. An allocation is a random profile of formal transfers τt and consumptions cti that are measurable with respect to the relevant information, (i.e., τt = τ(Ht et ) and cti = c i (Ht st )), and that satisfy the resource constraint, for all Ht and st ,
×
(1)
ct1 ≥ 0
ct2 ≥ 0
and ct1 + ct2 = ωt
where ωt = ω1 (st ) + ω2 (st ) − ϕ(et τt ) denotes aggregate resources available at date t 2.2. Intertemporal Payoffs and Feasible Allocations In the benchmark case in which financial markets are complete, optimal insurance is achieved and consumption at date t only depends on the realization 3 This assumption is necessary to obtain an interior solution with one-sided commitment, it can be dispensed with in the case of bilateral limited commitment (see Dubois, Jullien, and Magnac (2002)). 4 More formally, for all e τ ∈ [τ(e) τ(e)] ¯ and s ∈ e and ω1 (s) + τ > 0 and ω2 (s) − τ − ϕ(e τ) > 0.
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of aggregate resources that are maximal at τ(et ) = 0. According to Borch’s rule, the ratio of marginal utility is the same in all states of nature so that consumption dynamics are characterized by ηt+1
1 u (ct+1 ) u (ct1 ) = η t 2 v (ct2 ) v (ct+1 )
In our setup there is no financial market, but agent 2 can fully commit to a set of future actions while agent 1 cannot commit at all and at any point in time maximizes his payoff. Any equilibrium allocation must be self-supporting in a game theoretic manner. Because of the presence of the random parameter ηt , our model is not strictly speaking a repeated game, yet it is stationary in the sense that the subgame starting at date t is identical to the game starting at date 1 provided that the utility levels are renormalized. To see that, define the expected utility at the beginning of date t normalized by ηt for agent 1 as ∞ 1 r ηt+r 1 (2) ρ u(ct+r ) Ht Ut = E u(ct ) + ηt r=1 ∞ r 2 Vt = E ρ v(ct+r ) Ht r=0
Now consider r the subgame starting at date t with Ht known. Given that ηt+r /ηt = l=1 χt+l the distribution of {ηt+r /ηt εt+r st+r }r≥1 is independent of the period t and the subgame starting at date t > 1 is identical to the initial game. The sets of equilibria of the two games coincide and thus the tools of repeated game theory apply. Specifically, there are minimum and maximum expected utility levels that can be supported in equilibrium for each agent. Since agent 2 can commit to transfer all her wealth, the minimum level is V = 0 On the other hand, agent 1 will reject any proposed agreement leading to a utility level Ut below the autarky level: U=
1 ps u(ω1 (s)) 1−ρ s
Since agent 2 can commit not to contract at all with agent 1 U is indeed the minimum utility that he can obtain. To prevent agent 1 from reneging on the agreement, it is optimal to coordinate in such a way that if he deviates, the equilibrium that follows is the worst equilibrium for him. In other words, one should apply an optimal penal code as defined by Abreu (1988): An allocation can be supported in equilibrium provided that, at any point in time, agent 1 prefers to abide by the informal
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agreement rather than to renege and be punished by receiving his minimum equilibrium expected utility. Thus at date t, agent 1 should be willing to sign the formal contract at the interim stage, (3)
Ut ≥ U
Once st is realized, at the ex post stage, agent 1 must also prefer to make the informal transfer rather than to enforce the formal contract only and continue in autarky next period: (4)
u(ct1 ) + ρE[χt+1 Ut+1 | Ht st ] ≥ u(ω1t + τt ) + ρU
In conclusion, the set F of allocations that satisfy the resource constraint (1), the participation constraint (3), and the incentive compatibility condition (4) at all dates and all histories is the set of feasible allocations. 2.3. Value Functions and Bellman Equations Our goal is to derive the set of constrained Pareto optimal allocations. An allocation is optimal if agent 1 receives the largest possible expected utility given agent 2’s expected utility among the feasible allocations. This set, or frontier, can be evaluated at various stages. It will be useful to consider two stages: before any shock is revealed, the ex ante definition, or after revelation of the shocks (χ and ε) at the beginning of the period, the interim definition. DEFINITION 1: At the interim stage, when H1 = (χ1 ε1 ) is known, the Pareto frontier is the set of feasible interim utility (P(V ) V ) characterized by the interim value function P(V ) =
max
(ct1 ct2 τt )∞ t=1 ∈F
U1
s.t.
V1 ≥ V
where U1 and V1 are date 1 normalized utility levels for agents 1 and 2 defined in equation (2). DEFINITION 2: At the ex ante stage, the Pareto frontier is the set of feasible ex ante utility (Q(V ) V ) characterized by the ex ante value function Q(V ) =
max
(ct1 ct2 τt )∞ t=1 ∈F
E[χ1 U1 ]
s.t.
E[V1 ] ≥ V
where the expectation operator refers to the joint distribution of (χ1 ε1 ). Optimality implies that at any date the allocation remains optimal conditional on history provided that agent 1 has not deviated. Thus, given stationarity, Ut = P(Vt ) with probability 1.
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Moreover, the interim and the ex ante frontier are linked by the relation Q(E[V1 ]) = Eχ1 P(V1 ) Denote F(· | χ1 ) to be the marginal distribution of date 1 utility V1 conditional on the realization of the random shock χ1 only. With the use of the random component ε1 it is possible to implement any distribution on support [V V¯ ] so that Q(V ) is the solution of the ex ante Bellman equation (5)
Q(V ) = max E
χ1 P(V1 ) dF(V1 | χ1 )
{F(·|χ1 )}
s.t.
E
V¯
V¯
V
V1 dF(V1 | χ1 ) ≥ V
V
where the expectation operator refers to the distribution of χ1 Again optimality implies that at any date, (6)
E(χt Ut | Ht−1 st−1 ) = Q(E(Vt | Ht−1 st−1 ))
with probability 1. Notice that, using ηt+1 = χt+1 ηt , Ut = E[u(ct1 ) + ρχt+1 Ut+1 | Ht ]
Vt = E[v(ct2 ) + ρVt+1 | Ht ]
Therefore, equation (6) implies that the solution of the interim program is such that Ut = E u(ct1 ) + ρQ(E(Vt+1 | Ht st )) | Ht (7) Combining this with Ut = P(Vt ) allows us to write the value function P(·) as the solution to the interim Bellman equation (8) ps u(c 1 (s)) + ρQ(V (s)) P(V ) = max (c 1 (s)c 2 (s)V (s))s (τ(e))e
(9)
s.t.
s∈S
ps v(c 2 (s)) + ρV (s) ≥ V
s∈S
(10)
u(c 1 (s)) + ρQ(V (s)) ≥ u(ω1 (s) + τ(e)) + ρU
(11)
c (s) + c (s) = ω (s) + ω (s) − ϕ(e τ(e))
(12)
0 = V ≤ V (s) ≤ V¯
1
2
1
2
∀e ∈ E ∀s ∈ e
∀e ∈ E ∀s ∈ e
∀s ∈ S
where the participation constraint (3) is embedded in constraint (12). The solution to the interim Bellman equation (8) maps utility Vt into a formal transfer τt consumption levels cti , and continuation expected utility at the end of period t contingent on the realization of the shock st Moreover, the solution of the ex ante Bellman equation (5) characterizes the distribution of period
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t + 1 utility Vt+1 as a function of the expected utility promised at the end of period t and of the shock χt+1 Using the argument of Thomas and Worrall (1988), the value functions Q(V ) and P(V ) are decreasing and continuous on the interval [V V¯ ]. We, however, need to address concavity and differentiability issues. 2.3.1. Properties of the Interim Value Function Consider the interim value function P Since Q is concave, the program defining P is concave for a fixed contract (τ(e))e However, when the contract ˆ be is optimally designed, P(·) may not be concave or differentiable. Let P(·) the convex hull of P(·), ˆ ) = max P(V
F∈ [V V¯ ]
V¯
P(w) dF(w) V
V¯
w dF(w) ≥ V
s.t. V
where F is a distribution on [V V¯ ] It is the distribution of Vt conditional on (Ht−1 st−1 ηt ) that is, after realization of the shock ηt but before randomization. ˆ ) is a concave and decreasing function. The argument of Benveniste P(V and Scheinkman (1979) can then be adapted along the line of Koeppl (2006) ˆ ) to show that because the incentive constraint cannot bind in all states, P(V ¯ is differentiable at V < V (Lemma 8 in the Appendix). Furthermore, as the ˆ t ) with probability 1 at an optimal agents can randomize over Vt P(Vt ) = P(V allocation. Moreover, (13)
Pˆ (E[Vt | Ht−1 st−1 χt ]) = P (Vt )
whenever Vt < V¯ (notice that 0 = V < Vt ). Thus P is differentiable with probability 1 over the support of any optimal allocation. ˆ t ) has another consequence. As Pˆ is concave, dualThe equality P(Vt ) = P(V ity theory yields a relationship between the slope of the interim Pareto frontier and the interim utility of agent 2. LEMMA 3: There exists a random variable μt = −Pˆ (Vt ) measurable with respect to history Ht such that with probability 1, (14)
Vt ∈ arg max {P(V ) + μt V } V ∈[V V¯ ]
ˆ ) − Pˆ (Vt )V } since P(·) ˆ is concave. UsPROOF: We have Vt ∈ arg maxV {P(V ˆ ˆ ing P(·) ≥ P(·) we obtain that when P(Vt ) = P(Vt ), then Vt ∈ arg maxV {P(V )− ˆ t )} = 1. Q.E.D. Pˆ (Vt )V } The result then follows from Pr{P(Vt ) = P(V
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The lemma implies that characterizing the set of optimal utility Vt amounts to characterizing the solutions of program (14) for various values of μt . In what follows, we describe the dynamics using the Pareto weight μt instead of Vt 2.3.2. Properties of the ex ante Value Function Consider now the ex ante value function Q Since the program is linear, the ˆ is differentiable. In addition, ex ante value function Q is concave, and, as P we show in Lemma 9 (Appendix) that function Q(·) is strictly concave thanks to the random effect χ1 . ˆ From above, we must have E(U1 | χ1 ) = P(E(V 1 | χ1 )) at date 1 Denoting the set of measurable functions from the support of χ1 into [V V¯ ] by B the solutions of the ex ante Bellman equation (5) can be written as ˆ Vˆ (χ1 )) s.t. E[Vˆ (χ1 )] ≥ V Q(V ) = max E χ1 P( (15) Vˆ (·)∈B
which determines the choice E[V1 | χ1 ] = Vˆ (χ1 ) for a given V and shock χ1 More generally, Vˆ (χt ) is the choice of E[Vt | Ht−1 st−1 χt ] as a function of V = E[Vt | Ht−1 st−1 ] and χt The ex ante and interim value functions are thus linked by optimality conditions implying that at χ1 such that Vˆ (χ1 ) < V¯ , it must be the case that Q (V ) = χ1 Pˆ (Vˆ (χ1 )) Thus the slopes of the two value functions are linked by the relationship Q (E[Vt | Ht−1 st−1 ]) = χt Pˆ (Vt ) Combined with (13), this provides a first relation between the ex ante and interim slopes of the Pareto frontiers at an optimal allocation: (16)
Q (E[Vt | Ht−1 st−1 ]) = χt P (Vt ) when
Vt < V¯
This expression is used below to describe how the (ex ante) value anticipated at the end of period t − 1 is transformed into an (interim) expected value at the beginning of period t A recursive characterization of an optimal allocation then consists of equation (16) and, for any interim value Vt a description of the contract, current period consumption levels, and the next period ex ante value as a function of realized shocks. 2.4. Dynamic Characterization The characterization of the dynamics is obtained in several stages. The starting point uses equation (16) above, relating the slopes of the ex ante and the interim value functions. Suppose that we know from past history the slope of the ex ante value function Qt Then we can infer the slope of the interim value function (17)
μt = P (Vt ) =
Qt χt
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The next step consists of characterizing date t allocation as a function of the slope μt This is done by solving the program (14) defining the interim frontier, that is, by maximizing P(V ) + μt V . This amounts to solving the program given by the interim Bellman equation (8). As part of the characterization we obtain the slope of the next period ex ante value Qt+1 . We can then start the process again to obtain a recursive characterization. The solution to the interim Bellman equation, max ps u(c 1 (s)) + ρQ(V (s)) (c 1 (s)c 2 (s)V (s))s (τ(e))e
s∈S
+ μt
ps v(c 2 (s)) + ρV (s)
s∈S
subject to the incentive compatibility condition (10), the resource constraint (11), and the participation constraint (12), is solved in two stages. In the first stage we fix the contract (τ(e))e and the weight μt of agent 2’s utility, and we characterize the solution (c 1 (s) c 2 (s) V (s))s The benefit of doing so is that the reduced program is concave. This determines date t promised consumption levels and continuation utility for a given contract (τ(e))e and μt A simple arbitrage condition shows that (18)
u (c 1 (s)) = −Q (V (s)) v (c 2 (s))
This corresponds to a risk sharing rule which determines the allocation in state s and which trades off consumption and continuation utility. Moreover, in all states in which agent 1’s incentive compatibility condition (10) is not binding, we have −Q (V (s)) = μt which equates the ratios of marginal utility across states, as required by Borch’s rule. However, when agent 1’s income is (too) large, maintaining the ratio of marginal utility at μt violates incentive compatibility. The future utility allocated to agent 1 must be increased to the level that he would obtain by reneging on the promise and enforcing the contract τ(e). In this case the allocation depends on the resources available and on the outside option, but not on the weight μt In the second stage, we consider the choice of the formal contract for a given weight μt . Although a full characterization is out of reach, a supermodularity argument shows that the contract is a monotonic function of the Pareto weight. As we move along the frontier P(V ) toward higher absolute slopes μ (and thus higher V ), the contract becomes uniformly less favorable to agent 1. The next proposition summarizes the results: PROPOSITION 4: Denote πt = ω1t + τt to be agent 1’s income at date t. There exist a function μ(ω ¯ π) with values in (−Pˆ (V¯ ) +∞) and a correspondence T¯ (μ) with values in e (τ(e) 0] such that an optimal allocation satisfies
×
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(i) the statements (19)
(τt (e))e ∈ T¯ (μt )
(20)
u (ct1 ) = min{μt μ(ω ¯ t πt )} v (ct2 )
(21)
μt+1 =
(ii) (iii) (iv)
1 u (ct1 ) χt+1 v (ct2 )
if
Vt+1 < V¯ ;
μ(ω ¯ π) is decreasing in π; T¯ (μ) is nonincreasing according to the strong order set5 ; T¯ (μt ) is single-valued with probability 1.
For a given formal contract, this proposition is a generalization of Thomas and Worrall (1988) in which one agent is risk neutral and no formal contract is allowed. When agent 1’s resources are large or the weight put on agent 2 is large, incentive compatibility and efficient risk sharing are conflicting. This is resolved in two ways. First, the consumption profile of agent 1 that follows such a state is shifted upward. Second, the problem can be alleviated ex ante by using the formal contract. As the utility V of agent 2 increases, it becomes more difficult to maintain the incentives of agent 1. Formal contracts are thus more extensively used. The dynamics can be described by means of the evolution of the weight μt associated with the utility Vt chosen after history Ht . At this stage, agents sign a contract (τt (e))e ∈ T¯ (μt ). Consumption is given as a function of μt , the transfer τt , and the state of nature st . This also defines the slope Q (E(Vt+1 | Ht st )) at the end of period t. Then, at date t + 1, Ht+1 is realized and condition (16) gives the new value of the weight μt+1 . Whenever the Pareto frontier is locally strictly concave, T¯ (μ) is single valued so that the proposition completely characterizes the dynamics. If P(·) is not strictly concave, T¯ (μ) can be multivalued. Notice that it is single valued for all values μt for which program (14) has a unique solution, that is, for which −Pˆ (Vt ) = μt has a unique solution Vt . It is shown in Lemma 9 (Appendix) that this occurs with probability 1 due to the effect of the shock ηt . Thus in equilibrium T¯ (μt ) is single valued with probability 1. When there is no formal contract, the variable μ(ω ¯ t πt ) is exogenous. The dynamics of consumption after a date t when the incentive constraint binds is independent of past history and depends only on state st Introducing the possibility of formal contracts dramatically changes this conclusion. Indeed the 5
The transfer is strictly decreasing in all events such that the incentive constraint binds with a positive probability and income is positive in some state. If the incentive constraint does not bind in all states of event e then the transfer is τ(e) = 0.
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resources of the agent depend on past history of shocks through the contract Tt , even after a period in which the incentive constraint binds. Notice that the proposition implies that income πt is monotonic in the parameter μt COROLLARY 5: Income πt is nonincreasing with μt in all states st PROOF: Proposition 4(iii) implies that transfer τt does not increase in all Q.E.D. events when μt increases. Thus income is uniformly nonincreasing. The same conclusion holds for the allocation of consumption levels and future utility (ct1 Ut+1 ).6 Notice also that if there were no shock ηt the weight μt would be decreasing over time. It would converge to some fixed value and the contract should eventually be backloaded in favor of the agent who cannot commit. But due to the shock ηt , there is a positive probability that μt increases. Thus starting from any point, there is a positive probability that the incentive compatibility constraint binds in finite time, and the asymptotic distribution of allocations is not degenerate. For our argument to make empirical sense there must be enough variability in the shocks ηt .7 2.5. Additional Sources of Observed Heterogeneity We have so far assumed time-invariant utility functions (except ηt ) and a stationary resource process. Suppose now that the utility is u(c 1 ; x1t ) and v(c 2 ; x2t ). Suppose also that resources depend on state st and that a state variable qt describes the information set. Letting xt = (x1t x2t ) suppose that (xt qt ) follows a Markov process whose realization is revealed to the agents at the beginning of period t. Then the value functions at date t are functions P(V ; xt qt ) and Q(V ; xt−1 qt−1 ). The value function Q is the maximum of ˆ t ; xt qt ) | xt−1 qt−1 } E{χt P(V subject to E{Vt | xt−1 qt−1 } ≥ V . The value function P(V ; xt qt ) is equal to the maximum of E u(c 1 (xt qt st ); x1t ) + ρQ(V (xt qt st ); xt qt ) | xt qt subject to incentive and participation constraints. All proofs generalize to this setup. The function μ¯ depends only on ωt , πt , and xt qt , but not on χt , while the ratio of marginal utility has to be conditioned on xit only. We thus obtain u (ct1 ; x1t ) = min{μt μ(ω ¯ t πt ; xt qt )} v (ct2 ; x2t ) 6 This follows from the fact that at the solution of the subprogram with a given contract, both cs2 and V (s) are nondecreasing with μ and nonincreasing with τ(e). 7 See Thomas and Worrall (1988), Beaudry and DiNardo (1991), and Ray (2002). We thank a referee for this point and the references.
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The contract T¯ (μt ; xt qt ) depends on μt and xt qt , and the dynamics of the multiplier μt are unchanged. 3. THE STRUCTURAL ECONOMETRIC MODEL The main purpose of constructing an empirical model from these theoretical underpinnings is to test whether it leads to an empirically credible alternative to the perfect insurance and the limited commitment cases, and to assess the degree of discrepancy with those. The new insight that our theoretical setup brings about is that we should analyze the income process in addition to consumption growth. If income were exogenous, the hypothesis of complete markets would be tested by looking at the significance of the correlation between the residuals of the consumption growth equation and income. This procedure is correct since income is excluded from the marginal utility of consumption if leisure and consumption are separable. In our case, the choice of formal contracts makes income endogenous and there are indeed three endogenous variables (consumption, formal contracts, and income). A test procedure that markets are complete should now be looking for exogenous or predetermined variables that affect formal contracts, thus income, and are independent of random preference shocks.8 Moreover, the absence of formal contracts, the case of pure limited commitment, is evaluated by testing for income exogeneity. Nevertheless, we want to achieve more than simply testing null hypotheses, and this section focuses on formalizing an econometric model suited to the theory developed in the previous section and that can be identified using household panel data on income and consumption. Alternatives have been proposed in the literature to understand the empirical evidence that we obtain, but we defer the discussion of their relevance until the empirical section. Some discussion of the most striking omissions is nonetheless in order. The first issue concerns the adaptation of the two-agent framework to a village, that is, a multiagent framework. We follow Ligon, Thomas, and Worrall (2002) by assuming that each household plays a two-person game with the rest of the village or with a pivotal person in the village. The characteristics and the equilibrium realizations of endogenous variables of the other agent are written as village-and-period indicators. The second issue is that we do not model individual savings or “investment” technologies that would enable households to transfer resources from one period to the next in an idiosyncratic way. As demonstrated by Ligon, Thomas, and Worrall (2000), the manipulation of incentive constraints becomes intractable in empirical applications. The level of aggregate savings is assumed to be determined by some unmodelled mechanism and captured by village-andperiod dummy variables in the empirical model. 8 The issue of income endogeneity is dealt with in some papers in a reduced-form setting (e.g., Jacoby and Skoufias (1998), Jalan and Ravallion (1999)).
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Third, we use information on consumption and income only. It is difficult to tell in the data whether contracts are formal so that we only capture their impact through the income process. Furthermore, we refrain from using information on informal transfers as Foster and Rosenzweig (2001) did because it seems difficult to us to draw a precise line between truly exogenous transfers and those that result from the very endogenous informal contracts that we model here. Consumption growth would eventually include those informal transfers (Deaton (1997)). For the sake of clarity and for lack of a better device, we shall adopt the convention that “observable” refers from now on to variables which we, the econometricians, observe and we keep the expression “revealed at time t” to designate what households observe at time t. Moreover, we shall adopt the following notation: NOTATION 3.1: (i) All parameters indexed by vt control for village-and-period effects. (ii) Household i’s consumption at time t is denoted cit . (iii) Household i’s agricultural profits at time t net of input costs including nonfamily labor costs, are denoted πita . Household i’s income at time t including income from formal contracts, is denoted πit = πita + πite , where πite is other income. The latter stands for other nonlabor income or exogenous transfers such as remittances from abroad (excluding informal transfers). (iv) Observed preference shifters (household demographics for instance) and unobserved preference shocks, respectively denoted xit and ηit , are revealed to agents at the beginning of period t. (v) Besides taste shifters, the information set at the beginning of the period contains observed variables qit (such as owned land) that are revealed to agents at the beginning of period t. (vi) Income shocks are revealed at mid-period. Observed shocks9 are denoted zit (days of illness and craft income for instance); unobservable shocks are denoted ξit . The set of all observables at mid-period is denoted mit = (xit qit zit ) We start below with the specification of utility functions, of formal contracts, and of income processes. We continue by stating the identifying restrictions. All specification assumptions and restrictions are collected in the different items of Assumption 3.2. A proposition stating semi-parametric identification follows and the section closes by explaining the estimation method by penalized splines. 9
In the theoretical model, we should assume that zit and (xit qit ) are independent. If they are not, we would replace zit by the innovation in zit independent of (xit qit ), that is, by the unanticipated income shock
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3.1. The Specification of Consumption and Income Equations 3.1.1. Consumption Dynamics Following most papers in this literature (exceptions are Ogaki and Zhang (2001) for HARA estimation and Dubois (2000) for heterogenous CRRA functional forms), we assume that households have constant relative risk aversion and that other goods, such as leisure, are excluded. ASSUMPTION 3.2: (i) The ratio of marginal utility of consumption for household i at time t, relative to the marginal utility of the village, v, is written as (22)
ηit
uit (cit ) = ηit exp(δvt ) exp(σxit β) · cit−σ uvt (cvt )
where σ > 0 and where the log of the marginal utility of the village is captured by the village-and-period effect δvt . We denote λit (β) = ln(cit ) − xit β to be the adjusted consumption index. Write the multiplier μit defined in equation (21) and specified by the marginal utility (22) as (23)
ln μit = − ln χit − σλit−1 (β) + δvt−1
where the first difference shock is given by ln χit = ln ηit − ln ηit−1 . Consumption dynamics is derived from the Euler equation (20) and the multiplier definition (23). Whether the incentive constraint facing household i is binding or not defines two regimes, −σλit (β) + δvt ⎧ ln μ¯ vt (πit xit qit ) ⎪ ⎪ ⎨ if − ln χ − σλ (β) + δ ¯ vt (πit xit qit ) it it−1 vt−1 > ln μ = ⎪ ⎪ ⎩ − ln χit − σλit−1 (β) + δvt−1 if − ln χit − σλit−1 (β) + δvt−1 < ln μ¯ vt (πit xit qit ) where the notation μ¯ vt replaces μ¯ in the Euler equation (20) because aggregate resources are now captured by the village-and-period index (Notation 3.1). The functional form of the bound, ln μ¯ vt (πit xit qit ), will be specified later, yet it should be noted immediately that this bound is deterministic. It could implicitly be made random if preference shifters and information variables (xit qit ) were allowed to include some unobserved heterogeneity components. The structure of stochastic shocks, however, is already sufficiently rich because of the income variable πit . As we allow for measurement errors in income, the
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absence of unobserved heterogeneity in (xit qit ) does not seem to be such a tight assumption in this model.10 Because we cannot observe whether the incentive constraint is binding or not, the two regimes cannot be distinguished. As a consequence, the system of equations above is observationally equivalent to a single equation describing the dynamics of consumption: (24)
σ λit (β) = δvt + φit · 1{φit ≥ 0} + ln χit
where (25)
φit = δvt−1 − σλit−1 (β) − ln χit − ln μ¯ vt (πit xit qit )
Two remarks are in order. First, complete markets is an interesting special case in which the incentive constraints never bind. We have φit < 0 so that the usual Euler equation holds: ln cit = xit β + ( δvt + ln χit )/σ More generally, the existence of self-enforcing constraints implies that dynamics are sometimes consistent with the predictions derived from the hypothesis of complete markets and sometimes not, depending on the sign of φit . Second, when the incentive constraint is binding, the second term in the right-hand side of the Euler equation (24) is positive. It means that consumption growth is more than what would be expected under full insurance. It is the “rent” to pay to keep the household in the self-enforced informal arrangement when the income shock in the period is “too” favorable for the household. Variable φit depends on the income process πit that we now detail. 3.1.2. Formal Contracts and the Income Process A formal contract is described by equation (19) in Proposition 4. The vector of formal transfers (i.e., in any state e) is a function of the multiplier μit and of variables belonging to the information set (xit qit ), τit = τvt (− ln χit − σλit−1 (β) + δvt−1 xit qit ) when the multiplier ln μit is replaced by its specification using equation (23). We have in mind that formal transfers τit can be supported by land-leasing contracts (in or out) such as sharecropping contracts and their sharing rules or 10 A more difficult issue that we do not deal with here is the presence of unobserved household effects in xit qit . Household effects are notoriously difficult to handle in nonlinear dynamic settings, although some advances could be made along the lines proposed by Cunha, Heckman, and Navarro (2005).
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fixed-rent contracts for which the rent is fixed at the village level. Reducedform estimation indeed showed that such contracts do help sharing risks (Dubois (2000)). By contrast, it seems dubious that labor contracts can be formally enforced since permanent labor contracts are infrequently observed in village economies (see Bliss and Stern (1982)). As a consequence, agricultural profit is a function of the characteristics of these contracts and, in particular, of the marginal utility of consumption μit and of variables in the information set (xit qit ). It is also a function of observed and unobserved income shocks zit and ξit that are revealed after the signature of the contracts so that the reduced form of agricultural income is (using mit = (xit qit zit ); see Notation 3.1) πita = π a (− ln χit − σλit−1 (β) + δvt−1 mit ξit ) We could have specified a structural production function or the structural relationships between the quantities of land under sharecropping and fixed-rent contracts and marginal utility. Given the complicated endogenous structure of land exploitation, this would be fraught with severe robustness issues. Household net income is not only composed of agricultural profits, but also of nonagricultural profits or other income, πite . For more generality, we also include πite among variables zit (Notation 3.1(vi)), on top of adding it to πita By adding some measurement errors, ςit , and adopting the semiparametric assumption that agricultural profits are a linear function, we finally obtain measured income π˜ it ≡ πita + πite + ςit as follows: ASSUMPTION 3.2—Continued: (ii) Measured income is (26)
π˜ it = πvt + π0 σλit−1 (β) + mit πm + π0 ln χit + ξit + ςit
where the village-and-period effect, πvt , absorbs parameters δvt−1 Corollary 5 implies that parameter π0 is positive, which provides the first restriction of our model. The equations describing consumption growth and income, (24) and (26), form our structural model. Assumptions about shocks and incentive constraints can now be spelled out. 3.2. Identifying Restrictions and Reduced Forms Notation 3.1 and the consumption growth and income equations (24) and (26) lead to the list of covariates wit = (ln cit−1 , xit−1 , xit , qit , zit ), which respectively, stand for lagged consumption, lagged and current preference shifters, information variables, and other income shocks. Identifying restrictions on heterogeneity terms, measurement error, and predetermined variables are stated as follows:
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ASSUMPTION 3.2—Continued: (iii) The vector of unobserved shocks in preferences and income (ln χit ξit ) is independent of covariates wit and of measurement errors ςit , and is identically distributed and independent across households and periods. It has an absolutely continuous distribution and its support is a compact set of R2 . (iv) Measurement error ςit is mean-independent of covariates wit and is independent across households and periods. (v) Variables in wit = (ln cit−1 xit−1 xit qit zit ) have a compact support in Rr . If wvt = E(wit | v t) denotes their expectations within a village-and-period, there is full within-variation of the covariates, that is, rank E[(wit − wvt )(wit − wvt ) ] | v t = r The independence assumption, Assumption 3.2(iii), is slightly stronger than the ones generally used in linear dynamic models. Nonlinearities, due here to the presence of a bound, require more than mean-independence assumptions. We also used the assumption of compact support to facilitate the proof of identification. Assumption 3.2(iv) is weaker as it takes advantage of linearity. Assumption 3.2(iii) and (iv) both make the covariate process wit weakly exogenous. Assumption 3.2(v) implies that the distribution function of predetermined variables wit is not degenerate. This assumption is substantive since it would be violated if the information set does not include at least one variable qit which is excluded from preferences (an asset variable for instance). Such variables provide exogenous variability in income. The final step to complete the specification of consumption dynamics is to specify the bound φit appearing in equation (25). As income πit , without measurement errors as described in Notation 3.1(iii), depends on marginal utility ln χit + σλit−1 (β) − δvt−1 and on the index πvt + mit πm + ξit , the unobserved variable φit defined in equation (25) can be written as a function of the arguments φit = φ(ln χit + σλit−1 (β) − δvt−1 πvt + mit πm + ξit xit qit φvt ) where φvt is a village-and-period effect. Using the independence assumption, Assumption 3.2(iii), we can evaluate the conditional expectation of the nonlinear term φit 1{φit ≥ 0} of the consumption growth equation (24): E(φit 1{φit ≥ 0} | ln cit−1 xit−1 xit qit zit ) ≡ H(λit−1 (β) − δvt−1 /σ πvt + mit πm xit qit φvt ) where the positive function H is derived from the joint distribution function of (ln χit ξit ).
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We did not follow this nonparametric route11 and chose to specify the bound as a linear index: ASSUMPTION 3.2—Continued: (vi) ln μ¯ vt (πit xit qit ) is linear in all its arguments. We then have the following lemma: LEMMA 6: Under Assumption 3.2(vi), φit is linear in all its arguments: ∂ ln μ¯ vt (λit−1 (β) + mit φm + φvt ) + εit0 (χit ξit ) φit = −σ 1 + π0 ∂πit Furthermore, (27)
E(φit 1{φit ≥ 0} | ln cit−1 xit−1 xit qit zit ) = H(λit−1 (β) + mit φm + φvt )
where H(·) is an unknown positive, decreasing, convex, and twice differentiable function, where ∂ ln μ¯ vt H (λit−1 (β) + mit φm + φvt ) = − Pr(φit > 0) · σ 1 + π0 (28) < 0 ∂πit and if there is more than one z variable, (29)
φz(j) /φz(1) = πz(j) /πz(1)
for any j > 1
See Appendix B for the proof. Summarizing these results, the structural model consists of the following equations: First the consumption growth moment condition derived from equation (24) and equation (27) is (30)
E(σ · λit (β) − δvt | ln cit−1 xit−1 xit qit zit ) = H(λit−1 (β) + mit φm + φvt )
Second, using equation (26), the income moment condition is (31)
E(π˜ it | ln cit−1 xit−1 xit qit zit ) = πvt + π0 σλit−1 (β) + mit πm
We allow for correlation between these equations. This correlation is not informative about the parameters of interest since it depends on the unrestricted joint distribution of random shocks. We now study the identification of the system of structural equations (30) and (31). 11 The proof of identification remains, however, very similar in this more general case. It suffices to condition on mit and village-and-period effects in the following discussion.
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3.3. Semiparametric Identification The model depends on the population parameters θ = (σ β π0 πm φm ) ∈ Θ
= ({δvt πvt φvt }vt ) ∈ Ξ
H(·) ∈ H
where θ is the parameter of interest. Village-and-period effects and the functional parameter H(·) are treated as nuisance parameters. The identification analysis proceeds as follows. First, parameter σ > 0 is not identified since the intertemporal elasticity of substitution is not identified in a Euler equation when the interest rate is unknown. The two equations of interest are invariant to transforming the vector of parameters (σ, π0 , δvt , H(·)) into (1, π0 /σ, δvt /σ, H(·)/σ) (holding other parameters constant). We shall therefore normalize σ = 1 and change the interpretation of other parameters accordingly. Second, the parameters πvt , π0 , −βπ0 , and πm of the reduced form of the income equation (31) are identified because of the full rank assumption, Assumption 3.2(v). This is the most one can get from the income equation. As π0 and −βπ0 are identified, β is identified, which shows that, in the consumption growth equation (30), the dependent variable λit (β) = ln cit − xit β is identified. The consumption equation (32)
E( λit (β) | λit−1 (β) mit ) = δvt + H(λit−1 (β) + mit φm + φvt )
is an index model which identification was analyzed by Ichimura and Lee (1991). We adapt their assumptions to our special case as H is a positive, decreasing, and convex function (Lemma 6) and as all regressors are bounded. The main problem we have to face is that the intercept term is identified at infinity only (Heckman (1990)). We restrict function H and its argument as follows: ASSUMPTION 3.2—Continued: (vii) For any (v t) and at the true parameter value, the support of λ−1 + mφm is a connected interval of R. Denote a1vt to be the value such that Median(λ−1 + mφm + a1vt ) = 0 The support of λ−1 + mφm + a1vt is denoted SI(vt) bounded, decreasing, convex, and twice differentiable func(viii) H is the set of tions taking values on vt SI(vt) and such that H(0) = 0. As shown in the proof of the next proposition, Assumption 3.2(vii) and (viii) are normalizations if we observe one village at one period. The identification of H follows from the normalization of its location and the location of its argument. Imposing these conditions for any village at any period is testable although the assumption is weak. In its absence, identification would not be robust to small departures from the other assumptions that H(·) does not directly
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depend on v t which would be generated by allowing for heteroskedasticity in ξit for instance. Finally, we cannot impose that function H(·) is positive. All these conditions are sufficient to obtain the following proposition: PROPOSITION 7: Under Assumption 3.2(i)–(viii), (θ H) is identified in Θ× H. See Appendix B for the proof. 3.4. Estimation Using Penalized Splines To estimate the parameters of the consumption growth equation (32), we use penalized spline regression following Yu and Ruppert (2002). Letting u = λit−1 (β) + mit φm , properly normalized using Assumption 3.2(vii), we choose function H(·) in the set H(u; γ) = γ1 u + · · · + γp up +
K−1
γp+k Sp (u − κk )
k=1
where
Sp (u − κk ) =
(u − κk )p 1{u > κk } (κk − u)p 1{u < κk }
for k > p/2, for k ≤ p/2,
where 1{A} is the indicator function of A. Parameters to estimate are (β φm { δvt } γ), where { δvt } is the collection of village-and-period effects and where γ is the vector of parameters of the spline. The number of knots K − 1 is considered fixed (see Ruppert (2002), for a procedure of selection) and the locations of the knots κk are supposed to be given by the 1/K quantiles of u. By construction, H(0) = 0. The estimation method consists of finding the global minimum of 2 Q(β φm {δvt } γ) = λit (β) − δvt − H(λit−1 (β) + mit φm ; γ) it
+ ν · γ Gγ where ν is a penalty weight. We chose to penalize equally the elements γp+1 to γp+K−1 , writing γ Gγ =
K−1
2 γp+k
k=1
so that we do not impose that function H(·) is decreasing and convex. The reader is referred to Yu and Ruppert (2002) for all proofs of asymptotic prop-
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P. DUBOIS, B. JULLIEN, AND T. MAGNAC
erties including the sandwich formula for the variance–covariance matrix of estimates.12 4. THE EMPIRICAL ANALYSIS 4.1. Data Description and Economic Analysis The data come from a survey conducted by IFPRI (International Food Policy Research Institute) in Pakistan between 1986 and 1989 (see Alderman and Garcia (1993), for a thorough presentation) whose information is roughly equivalent to the famous ICRISAT data set. The survey consists of a random sample of around 900 households interviewed 12 times and coming from four districts in three regions. This rich data set conveys information on household demographics, incomes from various sources, individual labor supply, endowments and owned assets, agrarian structure, crops and productions, and land contracts such as sharecropping and fixed rent. Our endogenous variables are income and consumption growth. Sources of income are wages, agricultural profits, rents from property rights, and pensions. Agricultural profits are computed at each survey round using the information since the previous round and consist of cash income from staples, milk products, and animal poultry and livestock production, net of total input expenditures including nonfamily wage costs, feeding costs of productive animals, fertilizers, and pesticides. Whether or not family labor costs should be substracted from agricultural profits is evaluated below. Other sources of income such as household’s handicraft income, asset income (including property rents, fixed pensions regularly received from the government and rentals of other assets) and nonagricultural labor income are considered as “other income.” Household wage income in off-farm agricultural tasks is considered as one of the support of informal transfers and is not included in income. Concerning the consumption variable, we chose food expenditures as our consumption variable because they represent the largest part of nondurable expenditures and are well measured. More details on the consumption variables are given in Appendix C and descriptive statistics are presented in Table I. In our model, observed heterogeneity between households is of different types: Preferences, x, other variables in the information set, q or unanticipated shocks, z. First, Dubois (2000) showed, using the same data, that household size is the main preference shifter. Other variables such as the number of children never proved to be significant. As for variables in the information set, various empirical analyses (see Jalan and Ravallion (1999), for instance) report evidence that contrasting rich and poor households is the main relevant 12
Other details concerning the algorithm, the choice of the smoothing parameter ν by generalized cross-validation, and the choice of other regularization parameters are presented in the working paper version (Dubois, Jullien, and Magnac (2005)).
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RISK SHARING IN LDCS TABLE I DESCRIPTIVE STATISTICSa (ALL PERIODS, 8053 OBSERVATIONS) Variable
Food consumption Other nondurable expenditures (heating, . . . ) Total owned land area (acres) Land owner (0, landless; 1, owner) Total land owned in village (acres) Irrigated land (acres) Nonirrigated land (acres) Household size Number of children (≤15 years old) Number of days of illness per week (male) Number of days of illness per week (female) Pensions Agricultural profits Transfers Other income Total income (without transfers) Sharecropping dummy variable (renting in) Fixed rent dummy variable (renting in) Total area under sharecropping (renting in; acres) Total area under fixed rent (renting in; acres) Total area under sharecropping (renting out; acres) Total area under fixed rent (renting out; acres)
Average
Std. Err.
17629 4366 942 060 666 332 179 773 456 052 025 1876 −1690 8703 1541 15119 035 008 265 047 255 035
10321 18898 2181 048 132 894 476 273 221 175 090 199 6567 4082 6283 8407 047 026 496 339 984 380
a All income and expenditure variables are in rupees per week.
differentiation when the complete markets hypothesis is evaluated. As income is endogenous in our model, the quantity of owned land is a good indicator of a household’s productive assets and a predetermined predictor of income. The quantity of owned rain-fed land gives additional information about the quality of productive land. These two variables constitute the information set, denoted qit above. Third, craft or trade profits or spells of illness of household members are unlikely to be contractible and are regarded as a component of unexpected income shocks zt . Decomposing the variance of agricultural income into the village-and-period variance, the household variance, and a residual variance confirms that rural households are mostly affected by large idiosyncratic risks. This is also true for other sources of income like asset income, wages, and other income. In contrast, 52.7% of the variance of the logarithm of food expenditures comes from village-and-period effects and 23.7% comes from household effects. The first component refers to aggregate shocks; the second refers to the relative economic status of the household in the village, that is, its Pareto weight. We find the same pattern when looking at expenditures corrected by household size or at total expenditures. Turning to first differences in the logarithm of
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food expenditures, we find substantial, though quite imperfect, risk sharing since village–time effects explain 24.1% of these changes, and household effects explain 2.2%.13 Of course, the variance decomposition of marginal utility also depends on preference shifters, and first differencing magnifies measurement errors. Nevertheless, testing whether income shocks affect consumption growth, controlling for preference shifters, allowed one of the authors to reject perfect risk sharing using this data set (Dubois (2000)). Finally, the land rental market is very active in these villages since we observe in the data that the per period rate of change of area under contract such as sharecropping and fixed rent is 20% on average. This statistic provides a lower bound for the frequency of contract changes because we are not able to observe with whom the household contracted or contractual changes that did not involve a change in the land area under contract. This confirms that these contracts, which are natural candidate empirical counterparts for our formal contracts, are subject to frequent renegotiation. 4.2. Structural Estimates When interpreting the results of the estimation of the income and consumption growth equations that are presented below, we shall examine the empirical relevance of the following testable structural restrictions: C1: π0 ≥ 0 denotes monotonicity of the contract with respect to lagged marginal utility as derived from Corollary 5. C2: The coefficients of preference shifters xit−1 in the income equation and in both linear indices of the consumption growth equation are all equal to β. C3: H is decreasing. There are also other restrictions derived from auxiliary econometric assumptions in Assumption 3.2(i) and (viii) that we directly describe in the text. 4.2.1. The Income Equation We present several sets of estimation results of the income equation (26) in Table II. As predicted by our model, the significance of the coefficient of lagged consumption in the income equation provides some evidence that households smooth income. In column 1, we report two stage least squares (2SLS) estimates. This procedure controls for the presence of measurement error in lagged consumption by introducing residuals of the instrumental regression of lagged consumption on other variables of the regression and a further lag of consumption. The estimated coefficient of those residuals is indeed significant and is three times larger than the ordinary least squares (OLS) estimate of the coefficient of lagged consumption. We also tested for the presence of 13 At the district level, risk sharing is significantly less effective and it is why we decided to pursue the analysis at the village level.
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RISK SHARING IN LDCS TABLE II INCOME EQUATIONa 2SLS Dependent Variable: π˜ it
π0 −π0 β: Lagged (log) household size πq : Land owned Total land in village Rain-fed land in village πx (log) household size πz Income shocks Female illness days Male illness days Other income Consumption residuals (for ln ct−1 equation in lags) Information controls (from rented-in land equation) Lagged log-calorie consumption
1
2
3
4
0194 (0060) 0032 (0079) 0149 (0016) −0195 (0031) −0129 (0079) −0004 (0011) −0008 (0005) 0159 (0038) −0135 (0064)
0177 (0062) 0035 (0079) 0150 (0016) −0193 (0031) −0125 (0080) −0005 (0011) −0008 (0005) 0157 (0036) −0116 (0065) −0005 (0002)
0169 (0061) 0013 (0092) 0149 (0016) −0195 (0032) −0133 (0081) −0005 (0011) −0009 (0005) 0157 (0036) −0112 (0064) −0005 (0002) 0040 (0078)
0166 (0061) 0016 (0092) 0148 (0016) −0195 (0032) −0134 (0081) −0005 (0011) −0008 (0005) 0157 (0036) −0109 (0064) −0005 (0002) 004 (0078) 0002 (0003)
7133
7133
7131
7131
Lagged adults body mass index Observations
a Robust standard errors in parentheses. Village-and-period effects (46 villages, 11 rounds) not shown. ln c it−1 is instrumented by ln cit−2 and all other variables.
measurement errors in the other variables using lags and we did not reject their exogeneity. A robustness analysis to econometric assumptions is detailed in Section 4.2.3 below. In conformity with restriction C1, the coefficient of lagged consumption π0 is positive and significant. We also tried to use total expenditures with quite similar results. Competing explanations that we now detail could be consistent with this result. Household Superior Information: If all relevant information is not included in the income equation, the significantly positive coefficient of lagged consumption in the income equation could also reflect that lagged consumption is a good indicator of the information that households have about future in-
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come shocks that we, as analysts, do not observe. If households have superior information, income and lagged consumption are indeed positively correlated. To investigate the importance of this effect, we use, as proposed by Campbell (1987), another decision variable that might reveal the superior information that households have on future income shocks under the assumption that this information is single dimensional. To be on par in terms of information with contractual decisions, this decision should be coincident with the signature of contracts. Net rented-in area14 is such a candidate variable since it is a productive factor. We regress net rented-in area on variables that are known at the beginning of the period and we interpret residuals as a measure of the superior information that households have. Table A.I reports results of this estimation. The regression is very well determined and lagged consumption, which is instrumented as before, negatively affects rented-in area. Note that it is because lagged consumption is instrumented that residuals indeed contain new superior information if any. In the second column of Table II, the residual of the rented-in land equation used as information controls is found to be significant. Households have superior information and, as the coefficient is negative, tend to rent-in more land in anticipation of an adverse shock in income either in off-farm or onfarm activities. Our prediction C1 of a positive effect of lagged consumption is nevertheless robust to the presence of superior information. The estimated coefficient of lagged consumption remains significant and positive although it decreases marginally. Nutrition Effects on Productivity: The positive effect of lagged consumption on income could also be due to nutrition effects on future productivity, since more food consumption can help improve the productivity of agricultural laborers as analyzed by Behrman, Foster, and Rosenzweig (1997) and Fafchamps and Quisumbing (1999). Both articles use working samples extracted from the same survey we used. When planting labor effort is incompletely rewarded in the labor market and the productivity of labor is assumed to depend on current consumption, Behrman, Foster, and Rosenzweig (1997) argued that planting stage calories positively affect harvest stage profits. Using a subsample of wheat producers at two pairs of planting and harvest periods, they reported empirical evidence about this positive relationship. The availability of data on calorie intake at the household level actually allows us to distinguish a nutrition effect from the mechanism of risk sharing through formal contracting. As expected, calorie consumption and food expenditures are correlated, although we found no effect of lagged calorie intake on income when we use this variable along with or instead of food expenditures.15 14
Equivalently, cultivated land, as owned land is controlled for. As lagged food expenditures, lagged calorie intake is instrumented by the second lag of calorie intake because of measurement errors. 15
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For instance, column 3 of Table II shows that when lagged food expenditures and lagged calorie intake are both included in the estimation, the coefficient of lagged expenditures only is significantly different from zero. When using only lagged calorie intake without food expenditures, the former estimated coefficient is not significant. The discrepancy with the results cited above might come from differences in the sample selection and construction of the data. In particular, by using all periods, we show that harvest stage consumption has an impact on the next planting stage profits as well. This is difficult to reconcile with a pure nutrition effect on productivity. As an aside, independent variation of food expenditures and calories reveals that preferences over food matter more for households than the productive role of food through calorie intake. As nutrition could have longer term effects, we also use information about the body mass index (weight over height squared) of household members. This variable is positively correlated with food calorie intake. Introducing the average or total body mass index of household adults, or of male adults only, as additional variables, does not change the main results and shows a small positive but insignificant effect on productivity (column 4 of Table II). This is in line with Fafchamps and Quisumbing (1999), who found that such impacts are small. Labor Market Imperfections: Our construction of agricultural profits in the income equation assumes that the labor market is imperfect. We do not substract family labor costs from agricultural profits, so we implicitly assume that there are no substitutes for family labor on the market. A polar assumption would be that labor markets are perfect. As a consequence, the construction of agricultural profits should be modified and they should be computed net of family labor costs. This is what we did by using village-level information on male and female wages at each period as well as male and female hours of work to impute family labor costs. The income equation results hardly change and if there is any effect, the coefficient of lagged consumption in the income equation is larger by a nonsignificant factor of 10%. We also repeated the analysis substracting either male or female labor costs with no change. Other Determinants: Regarding the estimated coefficients of other variables in Table II, household demographics should partly compensate for the effect of lagged consumption, and lagged household size indeed does, though insignificantly. The quantity of owned land affects household income, although rain-fed land compensates for it so that only their difference—irrigated land— seems to matter. A larger number of days of illness for males and females decreases income as expected, though insignificantly. Informal transfers could take the form of additional labor from the village that compensates for the sickness of household members since it might be less costly to intervene at that stage than afterward (Fafchamps (1992)). Sickness would not affect household income in that case, but the way we construct income would be contaminated
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P. DUBOIS, B. JULLIEN, AND T. MAGNAC
by informal transfers that are implicitly already smoothing it. Errors in selfreporting wealth is another possible explanation (Strauss and Thomas (1998)), but our data limitations preclude any further analysis of these arguments. Finally, other income is positive and very significant. We tested and did not reject the exogeneity of this variable, using the various individual income components as instruments. 4.2.2. Consumption Growth Equation In Table III, we report the estimates of the consumption growth equation using penalized spline regressions. The number of knots of the spline is small and equal to 8, and the penalization parameter is optimally chosen by generalized cross validation for the specification corresponding to column 1. We estimate the system of income and consumption growth equations by allowing for some correlation between equations. This correlation is insignificant in magnitude (0.001) and statistically. The likelihood ratio statistic that contrasts estimations allowing or not allowing for correlated equations is equal to 04 and has 1 degree of freedom. It shows that even if residual shocks in income are correlated with the residuals of consumption growth through incentive constraints, the influence of these shocks is small relative to measurement errors in both equations and preference shocks in consumption dynamics. We do not report the estimates of the income equation because they are very similar to what is reported in Table II. As before, we follow a control function approach in nonlinear models (see Blundell and Powell (2003)) and we introduce, as an explanatory variable, the residual of the regression of lagged consumption on its second lag and all other variables to take into account the measurement errors in lagged consumption. Consumption exogeneity is rejected very strongly indeed and the magnitude of the estimated coefficient denotes the presence of large measurement errors in consumption. In column 2 we also introduce the residual of the net rentedin area equation (Table A.I) as a control for superior information. It is very significant (the likelihood ratio statistic for the system of equations is around 60 and has 2 degrees of freedom) though it does not affect the estimation of the other coefficients. We now consider testing the restrictions C2 and C3. First, the hypothesis that the coefficients of preference variables in the consumption and income equations are equal (C2) cannot be rejected. The likelihood ratio statistic is equal to 1.4 in column 2 of Table III. It has 2 degrees of freedom although we should account for the fact that we were unable to precisely estimate the coefficients of lagged household size in both linear indices because function H(·) is close to a linear function (see below). Nevertheless, even if H(·) is linear and the statistic has 1 degree of freedom only, we cannot reject restriction C2 at any reasonable level of confidence. Figure 1 reports estimates of the function H(·) using optimal smoothing. Although it should not be taken as a formal test, its shape is not in contra-
709
RISK SHARING IN LDCS TABLE III CONSUMPTION GROWTHa System Estimation
Consumption Growth
Linear index ln(hh_size)t : β Residual (ln cit−1 ) Linear argument of H(·) ln cit−1 − ln(hh_size)t−1 : β ln(hh_size)t Owned land Rain-fed land Female illness Male illness Other income
1
0248 (0075) −0571 (0024) 1
0249 (0035) −0565 (0024) 1
Round ≤ 7 3
Round ≥ 8 4
0212 (060) −0652 (0075)
0267 (034) −0505 (0040)
(—) 0248 (—) −0466 (0083) −0020 (0006) −0041 (0014) −0231 (0066) 0008 (00011) −0181 (01367)
(—) 0249 (—) −0458 (015) −0015 (0003) −0052 (00264) −0220 (0062) 0004 (00085) −0197 (01426) 0011 (00016)
(—) 0212 (—) −0332 (046) −0032 (00157) 0448 (038) −0193 (041) 0013 (00057) −0311 (06563) 0034 (00166)
1 (—) 0267 (—) −0435 (017) 0005 (0007) −0173 (030) −0203 (0070) −0038 (00480) −0168 (00249) 0007 (00031)
−850.0 7133
−829.6 7133
204.8 3704
−809.8 3429
Information control Likelihood Observations
2
1
a Quadratic splines. 8 knots. Penalization parameter =1.69. Robust standard errors below estimates in parenthesis. Village-and-period effects (46 villages, 11 rounds) not shown. ln cit−1 is instrumented by ln cit−2 and all other variables. Coefficients whose standard errors are (—) are normalized or set by a structural restriction. The likelihood value is for the full model including the income equation not shown here.
diction with hypothesis C3 as it is decreasing. Incentive constraints bind with some strictly positive probability and the model is statistically able to capture the type of market incompleteness at work in this economy. Moreover, recall from equation (28) that the slope of H(·) is the product of the probability that an incentive constraint is binding multiplied by 1 + π0 (∂ ln μ¯ vt /∂πit ) > 0. As (∂ ln μ¯ vt /∂πit ) < 0 (see Appendix B), the slope is a lower bound on this probability and a rough estimate of this probability is equal to 0.15. We analyze the most complete results of Table III, column 2, by starting with parameters in the linear index of the equation of consumption growth. As ex-
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P. DUBOIS, B. JULLIEN, AND T. MAGNAC
FIGURE 1.—Quadratic splines: optimal smoothing.
pected, household size has a positive and significant effect on the marginal utility of consumption in a way that is comparable to what was obtained in a more usual setting (Dubois (2000)). It is more difficult to analyze the effects of variables in the linear index, which is the argument of function H(·). Remember that the larger this index is, the smaller is the probability that the incentive constraint binds. Positive coefficients indicate that we are moving away from the constraint. It is only the case for variables such as male days of illness (though the effect is insignificant) and for information controls that we have seen to be negatively correlated to income (Table II). It is negative for household size, land assets, female days of illness, and other income. Intuitively, we should expect that beneficial income shocks should increase the probability that the incentive constraint binds. Nevertheless, theory does not completely comform to these insights because there could be compensating effects, due to the information correlated with these variables, that affect the contracting choices of the household. An intriguing result is what is observed, however, for all variables except for female days of illness. As already mentioned, the issue that days of illness can be immediately compensated for by informal transfers of labor seems to us to be an interesting path to explore. Moreover days of illness could affect function H(·) in a more complicated way if the assumption of independence (As-
RISK SHARING IN LDCS
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sumption 3.2(iii)) in this regressor is violated. As a matter of fact, we did test another consequence of independence (Assumption 3.2(iii)) in unexpected income shocks (equation (29)) and we were able to reject this restriction. More complicated dependence of H on zt stemming from heteroskedasticity of ξit+1 , for instance, should thus be allowed for. As residual variances in income are larger after the first half of the sample period (see below), we split the sample into two periods and reestimated the consumption growth equation. Results are reported in the two last columns of Table III and are very stable across subsamples although many estimates lose their significance. In Table A.II, we report tests of the separability of consumption and leisure obtained by regressing the residuals of the consumption growth equation on male and female labor supply variables. None is significant and we do not reject separability. 4.2.3. Robustness Analysis A thorough analysis of serial correlation and sensitivity to auxiliary parameters used in penalized spline regression was presented by Dubois, Jullien, and Magnac (2005) and we briefly summarize it now. Serial Correlation in Incomes: Analyzing the autocorrelation of unobservables affecting income, we found that during the sample period, two subperiods can be distinguished: the first one with small variances (until round 7); the second one with a lot more variance. Alderman and Garcia (1993) reported that the sample period was a quite uncertain and troubled period for agricultural activities, yet it seems to be true above all for the second half of the sample period. This overall view does not indicate the presence of strong household specific effects. We also tried to first-difference the income equation, but we lost all significance. The main question that remains is whether households are aware of the structure of autocorrelation, because we use econometric restrictions that they are not. More prosaically, the presence of serial correlation might also modify standard errors of the estimates of the income equation. We did try to correct for serial correlation by assuming that it is unrestricted over time but constant across individuals. This indeed modified standard errors of coefficients, though by a maximal factor of 50%, without affecting any diagnostic of significance that was used, employing the robust to heteroskedasticity White correction (Table II). Serial Correlation in Consumption Growth: We computed the residual variances of the consumption growth equation along with their autocorrelations. In contrast to income residuals, there is no clear time pattern for consumption residual variance, only two peaks at rounds 6 and 9. They might be attributed to measurement errors since the length of the period is somewhat variable across rounds even though we considered weekly income and consumption levels. The
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pattern of autocorrelation is more surprising since first-order correlation varies between −0.2 and 0.05, second-order correlation between −0.17 and 0, thirdorder correlation between 0 and 0.10, and fourth-order correlation between 0.05 and 0.15. In the case in which the function H(·) is linear, we developed (Dubois, Jullien, and Magnac (2005)) a model with i.i.d. preference shocks and classical measurement errors, and we calibrated it on our data. Autocorrelation coefficients are broadly in line with the empirical ones. The calibration also shows that the magnitude of preference shocks and measurement errors is roughly equal in the residuals of the consumption growth equation. Other Auxiliary Parameters: The auxiliary parameters in penalized spline regression are the penalization parameter, the number of knots, and the order of the spline. The income equation is not affected by the variations in these parameters as correlation between equations is virtually equal to zero. Making the penalization parameter vary between one-tenth of its optimal value and ten times its optimal value produces no noticeable difference on estimated coefficient values, but only differences in estimated standard errors for some coefficients. We tested structural restrictions by using likelihood ratio statistics which are more robust than Wald tests. We also ran the same kind of experiment by varying the number of knots and the order of the splines. In all cases, the central results of this paper remain the same, though standard errors seem affected by those auxiliary parameters. Regarding the estimates of function H(·) when the order of the spline varies, it is much smoother using quadratic and cubic splines, although the estimation using linear splines does not contradict our main empirical conclusions. Spatial Correlation: We also estimated auxiliary regressions of cross-products of income residuals and of cross-products of consumption growth residuals within a village on the absolute difference between households of characteristics such as land assets or household size. These regressions are intended to investigate whether unexplained components of consumption growth and income are associated across households of similar characteristics in each village. Results are reported in Table A.III. Interestingly the cross-product of residuals is always a negative function of the distance of characteristics between households. Furthermore, while the distance in land assets has a strong influence on the cross-product between income residuals, its influence is much attenuated in the consumption growth equation. This is not the case for the distance in household size. In the absence of household fixed effects, it is tempting to interpret these results as consistent with partial risk sharing of productivity shocks attached to asset levels but no risk sharing of demographic shocks. 5. DISCUSSION AND EXTENSIONS We developed a theoretical setting which nests the case of complete markets when all risk can be insured by formal contracts and the case in which only
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informal agreements are available. Its empirical implementation builds upon the estimation of a Euler equation of consumption dynamics and an income equation using data from village economies in Pakistan. Overall, our results indicate that those villages are far from the case of perfect risk sharing in which lagged consumption would explain neither income nor consumption growth or from the case of limited commitment in which lagged consumption would not explain income. Both cases are rejected in a significant way. The importance of formal contracts and limited commitment can be evaluated from the empirical estimates of the two structural equations. First, from results in Table II, we can derive that 12% of the within-village variance of income is explained by lagged consumption and lagged preference shifters, and we have been arguing that this comes from formal contracting. Specifically, while information variables including land assets explain four times as much as “other income,” formal contracts are on a par with “other income” in terms of explaining the total within-village variance of agricultural profits. Income smoothing due to formal contracting is thus economically significant. Second, we can also evaluate the explanatory power provided by limited commitment and formal contracts in the consumption growth equation, although it is low, as is usual with consumption growth. Whereas 24% of the variance of consumption growth is explained by village-and-period effects, only 1.6% is explained by the nonlinear function H(·) and its arguments, and 0.3% is explained by preference shifters. Nonetheless, as incentive constraints do not bind at all periods, one in seven being the estimated lower bound of the frequency of its occurrence, the economic magnitude of such a result is still significant. Policy evaluation by simulation would require full structural estimates, as discussed in the limitations below. Nonetheless, we can already draw interesting policy implications. First, by providing evidence on the impact of informal agreements, our work points to important but often neglected mechanisms interlinking formal and informal agreements. This is in line with evidence on contract interlinkages (Bell (1988)). One consequence is that short-term land agreements react to future anticipated shocks in a nontrivial way, since anticipations affect the incentives to renege on informal agreements. Our analysis also corroborates evidence explaining in part the success of microcredit that relies on similar informal mechanisms. Moreover, the dimension of informal risk sharing should be incorporated both into the evaluation and into the design of development programs. Specifically, the policy designer should evaluate whether an intervention complements informal agreements or weakens the ability of agents to cooperate. In the latter case, gains in terms of access to new instruments or markets should outweigh the cost in terms of lost commitment. As Albarran and Attanasio (2003) showed, policies that aim at improving formal risk sharing could crowd out informal mechanisms.
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P. DUBOIS, B. JULLIEN, AND T. MAGNAC
Furthermore, the focus on commitment issues also suggests designing policies that improve the enforceability of contracts within communities prior to extending the set of contractual parties (e.g., governments, banks, international institutions. . . ). It may also be desirable to adjust interventions at the local level, with field studies aiming at identifying and exploiting channels of informal risk sharing. Localized programs may in particular aim at identifying the credible parties (able to commit) and associating them to policy. Much remains to be done before one can draw a complete picture of the role of informal agreements in village economies. In a companion paper (Dubois, Jullien, and Magnac (2002)), we analyze a model in which no agent can commit. Conclusions about contract monotonicity are preserved although the dynamics of the ratio of marginal utility is affected by a second threshold which corresponds to the incentive constraint of agent 2. Besides, although here we focused attention on short-term contracts, our theoretical and empirical conclusions should extend to the case of longer but finite horizon contracts, provided that these contracts can be renegotiated every period at no cost. What really matters is the ability to use contracts to affect the distribution of the outside options of the agents. Long-term contracts can be used to affect the continuation utility of deviating agents, and then can be renegotiated (including voluntary cancellation of contracts) to improve the efficiency of the future relationship. It was beyond the scope of this paper to try to identify structural correspondence functions between the space of contracts and the marginal rate of substitution of consumption between two successive periods. It is nevertheless high on the research agenda since it would lead to full structural modelling and allow simulation of the impact of economic policies. Whereas we posited a reduced form for modelling the incentive compatibility constraint, the precise determination of which contracts can be enforced would lead to the determination of the full structure. This line of research would certainly involve simplifying the description of risks that households face and contracts that they can use. Our analysis provides support for the relevance of our setting, thereby encouraging attempts at such a construction. The issue of saving and, more generally, of any device linking one period to the next (nutrition, investment in agricultural technology, . . . ) is difficult to address. Some recent progress has been made by Gobert and Poitevin (2006) in the case of public savings. They assumed that breaching an informal agreement triggers the loss of the saving assets. It would make sense in our context if the asset were borne by the agent who is able to commit. Beyond consumption smoothing, savings provide resources that can be used as collateral for informal promises and that help relax incentive constraints. When savings are publicly monitored but privately owned, it would affect the outside options of a deviant agent and would also raise convexity issues (see Ligon, Thomas, and Worrall (2000)). Another critical issue is asymmetric information when savings are not publicly observed. As pointed out by Cole and Kocherlakota (2001), hidden savings drastically limit the ability to use even formal contracts.
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Toulouse School of Economics (Groupe de Recherche en Economie Mathématique et Quantitative, Institut d’Economie Industrielle, Institut National de la Recherche Agronomique), Manufacture des Tabacs, 21 alleé de Brienne, 31000 Toulouse, France;
[email protected], Toulouse School of Economics (Groupe de Recherche en Economie Mathématique et Quantitative, Institut d’Economie Industrielle), Manufacture des Tabacs, 21 alleé de Brienne, 31000 Toulouse, France;
[email protected], and Toulouse School of Economics (Groupe de Recherche en Economie Mathématique et Quantitative, Institut d’Economie Industrielle), Manufacture des Tabacs, 21 alleé de Brienne, 31000 Toulouse, France;
[email protected]. Manuscript received May, 2005; final revision received December, 2007.
APPENDIX A: APPENDIX TO SECTION 2 ˆ ) is differentiable on ]V V¯ [ LEMMA 8: P(V PROOF: From Carathéodory’s theorem (see Rockafellar (1970, Corollary 17.1.5)), the optimum obtains with a two-point support distribution, say Vχ ∈ {Vχ1 Vχ2 } with probability p and 1 − p. Thus an optimal allocation is such that conditional on χ, the distribution of the future utility solves the program ˆ χ ) = max {pP(V 1 ) + (1 − p)P(V 2 )} P(V χ χ p∈]01]
s.t.
pV + (1 − p)Vχ2 = Vχ 1 χ
ˆ ) so that there is For conciseness, we now focus on a point at which P(V ) = P(V no randomization and p = 1 and consider the solution of program P. Notice that it must be the case that ∀e ∈ E τ(e) ≤ 0 Otherwise a reduction in τ(e) would raise the utility of agent 2 without affecting the other constraints. Notice also that given that P is decreasing on ]V V¯ [ agent 1 receives a utility strictly larger than the autarky level for V < V¯ As a consequence, the constraint (10) is not binding in at least one state sˆ , since otherwise agent 1 could not receive more than the autarky level. Then, for x small, consider the allocation obtained by simply adjusting c 2 (ˆs) by γ(x) such that psˆ (v(c 2 (ˆs) + γ(x)) − v(c 2 (ˆs))) = x and c 1 (ˆs) to c 1 (ˆs) − γ(x) The new utility of agent 1 is ˆ + x) ˆ ) + psˆ u(c 1 (ˆs) − γ(x)) − u(c 1 (ˆs)) ≤ P(V H(V + x) = P(V as the new allocation is feasible and yields V + x to agent 2. Moreover, H is ˆ ) From Lemma 1 in Benveniste concave and differentiable, and H(V ) = P(V ˆ and Scheinkman (1979), the function P(·) is differentiable. Q.E.D.
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P. DUBOIS, B. JULLIEN, AND T. MAGNAC
LEMMA 9: Q(·) is strictly concave. PROOF: First notice that for V = 0 the solution is to set τt = 0 and ct1 = ωt 1 1 2 ˆ at all dates so that P(0) = P(0) = 1−ρ s ps u(ω (s) + ω (s)) and no incentive constraint is binding. Given that v (0) = +∞, we obtain Pˆ (0) = 0 > Pˆ (V¯ ) Consider the set B of slopes b such that b = −Pˆ (V ) occurs for more than one value V . The solution of Q(V ) is a function Vˆ (χ) with χPˆ (Vˆ (χ)) = Q (V ) if
χ>
Q (V ) Pˆ (V¯ )
Vˆ (χ) = V¯
if
χ≤
Q (V ) Pˆ (V¯ )
This defines Vˆ (χ) uniquely except when Q χ(V ) ∈ B which occurs with zero probability since B is a countable set and χ is a continuous random variable. Consider now V > V with a solution V (χ). It is impossible that Q (V ) = Q (V ) because this would imply V (χ) = V (χ) with probability 1 and thus conQ.E.D. tradicts E{V (χ)} = V > V Thus Q (·) must be decreasing.
A.1. Proof of Proposition 4 ˆ T ), where We solve the program Φ(μ) = maxT Φ(μ ˆ Φ(μ T) = max ps u(c 1 (s)) + ρQ(V (s)) (c 1 (s)c 2 (s)V (s))s
s∈S
+ μ[v(c 2 (s)) + ρV (s)]
s.t. (10) , (11), (12). ¯ π) with values in [−Pˆ (V ) −Pˆ (V¯ )[, LEMMA 10: There exists a function μ(ω decreasing in π whenever interior, such that the solution satisfies u (c 1 (s)) = −Q (V (s)) = min μ μ(ω(s) ¯ π(s)) v (c 2 (s)) PROOF: Given the separability in τ(e), we can optimize events by events: ˆ Φ(μ T ) = e pe Φe (μ τ(e)), where max E u(c 1 (s)) + ρQ(V (s)) | e Φe (μ τ(e)) = (c 1 (s)c 2 (s)V (s))
s.t. (A.1)
ps 1 λ pe s
+ μE v(ω(s) − c 1 (s)) + ρV (s) | e
u(c 1 (s)) + ρQ(V (s)) ≥ u(π(s)) + ρv1
∀s ∈ e
RISK SHARING IN LDCS
(A.2) (A.3)
ps ργ¯ s pe ps ργ pe s
717
V (s) ≤ V¯
V ≤ V (s)
where the terms in parentheses are Lagrange multipliers. The Lagrangian of the program is ps s∈e
pe
u(c 1 (s)) + ρQ(V (s)) + μ v(ω(s) − c 1 (s)) + ρV (s) + λ1s u(c 1 (s)) + ρQ(V (s)) + (γ s − γ¯ s )ρV (s)
As the program is strictly concave, the first-order conditions are necessary and sufficient for optimality. After elimination of γ s , γ¯ s , they reduce to μ u (c 1 (s)) = 2 v (c (s)) 1 + λ1s μ (= if V (s) < V¯ ) −Q (V¯ ) ≤ 1 + λ1s μv (c 2 (s)) = u (c 1 (s))(1 + λ1s ) Let φ(·) be the (increasing) inverse function of −Q (·) and let ψ1 (ω μ) be the solutions of u (ψ1 ) = μ v (ω − ψ1 ) The solution coincides with c 1 (s) = ψ1 (ω(s) μ) and V (s) = φ(μ) in all states where the incentive constraint is not binding: u ψ1 (ω(s) μ) + ρQ(φ(μ)) ≥ u(π(s)) + ρv1 (A.4) The left-hand side of the condition decreases with μ Define μ(ω ¯ π) as the solution of equation (A.4) holding with equality. Then μ(ω ¯ π) decreases with π and the incentive constraint is not binding whenever μ ≤ μ(ω(s) ¯ π(s)). Now suppose that μ ≥ μ(ω(s) ¯ ω1 (s)+τ(e)). Then (u (c 1 (s)))/(v (c 2 (s))) = ¯ π(s)) = μ/(1 + λ1s ) satisfies the first-order conditions −Q (V (s)) = μ(ω(s) and thus is the solution. Q.E.D. ˆ LEMMA 11: The mapping T¯ (μ) : μ → arg maxT Φ(μ T ) is a monotone nonincreasing correspondence in μ (according to the strong order set).
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PROOF: We can restrict attention to the subset τ(e) ≤ 0, ϕ decreases with z. The result follows from Milgrom and Shannon (1994, Theorem 4). Given the ˆ separability in τ(e), Φ(μ T ) is quasi-supermodular in T The following lemma shows that it also satisfies the single crossing condition in (T ; μ). Q.E.D. ˆ
) LEMMA 12: For a.e. τ(e) μ1 ∂Φ(μT is nonincreasing with μ decreasing if (10) ∂τ(e) is binding in some state of event e
PROOF: From the envelope theorem, at any point where Pr{ω1 (s) + τ(e) = 0 | e} = 0 and thus for all except a finite number of values of τ(e), we have ∂Φe (μ τ(e)) ∂τ(e) ∂ϕ(e τ(e)) + λ1s u (ω1 (s) + τ(e)) e = −E μv (c 2 (s)) ∂τ(e) ˆ Denote by ψ(μ ω) the solution of ψˆ = u (x) = μv (ω − x) ˆ μ When μ > μ ¯ then λ1s = μ/(μ(ω(s) ¯ π(s))) − 1 and μv (c 2 (s)) = μμ¯ ψ( ¯ ω(s)) 1 2 ˆ When μ < μ ¯ then λs = 0 and μv (c (s)) = ψ(μ ω(s)) Thus ∂Φe (μ τ(e)) ∂τ(e) ∂ϕ(e τ(e)) ˆ ¯ e prob(μ < μ¯ | e) = −E ψ(μ ω(s)) μ < μ ∂τ(e) μˆ ∂ϕ(e τ(e)) μ −E ψ(μ ¯ ω(s)) + − 1 u (ω1 (s) + τ(e)) μ¯ ∂τ(e) μ¯ μ > μ ¯ e prob(μ > μ¯ | e) This function is differentiable in μ for all μ such that μ¯ is not equal to μ with probability 1 Otherwise the argument which follows applies to the right and left derivatives in μ. We have ∂ ∂Φe (μ τ(e)) μ ∂μ ∂τ(e) ˆ ∂ψ(μ ω(s)) ∂ϕ(e τ(e)) ¯ e prob(μ < μ¯ | e) = −E μ μ < μ ∂μ ∂τ(e)
719 ∂ϕ(e τ(e)) μ 1 μˆ ψ(μ ¯ ω(s)) + u (ω (s) + τ(e)) μ > μ −E ¯ e μ¯ ∂τ(e) μ¯ RISK SHARING IN LDCS
× prob(μ > μ¯ | e) =
∂Φe (μ τ(e)) ∂τ(e) ˆ ∂ψ(μ ω(s)) ∂ϕ(e τ(e)) ˆ ¯ e + E ψ(μ ω(s)) − μ μ < μ ∂μ ∂τ(e) × prob(μ < μ¯ | e) ¯ e prob(μ > μ¯ | e) − E u (ω1 (s) + τ(e)) | μ > μ
As u ∂ψˆ >0 = μv 1 − ψˆ − μ ∂μ u + μv and ∂ϕe /(∂τ(e)) ≤ 0 we have ∂Φe (μ τ(e)) ∂ ∂Φe (μ τ(e)) ≤ μ ∂μ ∂τ(e) ∂τ(e) Equality arises only if prob(μ¯ > μ | e) = 1 and τ(e) = 0, which means that the incentive constraints is not binding. This shows that for a.e. τ(e), ∂ 1 ∂Φe ∂ 1 ∂Φˆ pe = ∂μ μ ∂τ(e) ∂μ μ ∂τ(e) pe ∂Φe ∂Φe ∂ = 2 μ − ≤0 μ ∂μ ∂τ(e) ∂τ(e) with equality only if the incentive constraint is never binding in event e Q.E.D. PROOF OF PROPOSITION 4: Condition (19) follows from Lemma 11, condition (20) follows from Lemma 10, and condition (21) follows from Lemma 10 and condition (16). The set of values where T¯ (μ) is not singled valued is countable since T¯ (μ) is increasing with a compact support. From the argument of Lemma 9, the probability that a μt belongs to this set is zero. Q.E.D.
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APPENDIX B: APPENDIX TO SECTION 3 A.1. Lemma 6 Using equation (25) and the linearity of ln μ¯ vt by Assumption 3.2(vi), simple algebra delivers the first statement: ∂ ln μ¯ vt ∂ ln μ¯ vt ∂ ln μ¯ vt φit = −σ 1 + π0 λit−1 (β) − xit + πx π0 ∂πit ∂xit ∂πit ∂ ln μ¯ vt ∂ ln μ¯ vt − qit + πq π0 ∂qit ∂πit ∂ ln μ¯ vt − zit πz π0 + φ˜ vt + εit0 (χit ξit ) ∂πit ∂ ln μ¯ vt (λit−1 (β) + mit φm + φvt ) + εit0 = −σ 1 + π0 ∂πit
(B.1)
(B.2)
where φvt is a linear combination of village-and-period effects, and where εit0 is a linear combination of ηit and ξit . εit0 is independent of any covariates wit and is absolutely continuous by Assumption 3.2(iii). As a consequence, the second statement follows: E(φit 1{φit ≥ 0} | wit ) = E(φit 1{φit ≥ 0} | φ0it ) = H(φ0it ) where φ0it = λit−1 (β) + mit φm + φvt and function H(x) is positive. Furthermore (1+π0 (∂ ln μ¯ vt /∂πit )) > 0 (although ∂ ln μ¯ vt /∂πit < 0 by Proposition 4(ii)) because the ratio of marginal utility tomorrow, λit+1 (β) is an increasing function of marginal utility today, λit (β), because of the strict concavity of function Q(·) Moreover, function H(x) is equal (up to a positive scale) to H(x) ∝
(−x + u)f (u) du u>x
and thus H (x) ∝ − Pr(u > x) which proves that H(·) is decreasing, convex, and twice differentiable. The scale parameter is given by the coefficient of λit−1 (β) in equation (B.2) which proves the third statement of the lemma. Finally, equation (B.1) implies the fourth statement that the coefficients of zit are proportional to πz Q.E.D.
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A.2. Proposition 7 The only remaining parameters of interest to identify φm and H(·) are in the consumption equation. We first show that Assumption 3.2(vii) and (viii) are normalizations at the village-and-period level. Then we show how φm and H(·) are identified on the support SI(vt) and thus are globally identified. Fix v and t, and drop those indices. Write the consumption equation as (B.3)
E( λ | λ−1 m) = δ + H(λ−1 + mφm + φ)
As function H is positive, decreasing, and convex, and as its argument u = λ−1 + mφm + φ is bounded, this moment condition is invariant to transformations of ( δ φ H(u)) into ( δ + a0 φ + a1 −a0 + H(u − a1 )), where (a0 a1 ) are any pair of scalars such that a0 < minu H(u − a1 ). Hence, ( δ φ H) is not identified and we can adopt any normalization (a0 a1 H0 ) for the true ( δ φ H). More precisely, we can normalize the location of H(·) and the location of its argument. Recall that by Assumption 3.2(iii) and (v), the variable λ−1 is absolutely continuous and (λ−1 m) are bounded so that λ−1 + mφm + a1 is a bounded index whose distribution is continuous on the support SI The support is connected by Assumption 3.2(vii). Any quantile normalization such as the median in Assumption 3.2(vii) implies that 0 is an interior point of SI . Assumption 3.2(viii) then posits that, at that value 0, H takes value 0. We now return to the proof of the proposition. As function H is bounded, decreasing, and twice differentiable, and as the index is continuous on SI , E( λ | λ−1 m) necessarily continuously decreases from cH to cL (say) where c L < cH Conversely, for any c ∈ [cL cH ], the set S0 = {(λ−1 m) such that E( λ | λ−1 m) = c} is defined and equal to a linear manifold λ−1 + mφm = b. Therefore φm is identified because of Assumption 3.2(v). By Assumption 3.2(vii), a1 = − Median(λ−1 + mφm ). Denote by S0 the set of values of (λ−1 m) such that λ−1 + mφm + a1 = 0. Then a0 = E( λ | (λ−1 m) ∈ S0 ) is identified. For any value b = (λ−1 + mφm + a1 ) in support SI , we can then construct function H as the function of b such as H(b) = E( λ | λ−1 + mφm + a1 = b) − a0 and H is identified in SI . Finally, as (φm H) are identified at the village-and-period level, they are identified globally, H(·) on the union of the previous sets which all contain 0. APPENDIX C: DATA The data provided by IFPRI consist of a sample of 927 households (first round) interviewed 12 times between 1986 and 1989. The survey was realized in four districts in three regions (Attock and Faisalabad in Punjab, Badin in the Sind, and Dir in the North West Frontier Province). In each of the four dis-
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tricts, villages were randomly chosen from a comprehensive list of villages classified in three sets according to their distances to two markets (mandis), and households were randomly drawn from a comprehensive list of village households. Some attrition is observed in the data (887 households in the final survey) that seems to stem from administrative and political problems rather than from self-selection of households (Alderman and Garcia (1993)). We shall assume that attrition is exogenous. In the survey, household demographics are directly available. Household food consumption is initially reported by food item, in quantity and value, or quantity and price. It comprises meals at home, including home-produced goods, and meals taken outside for all household members except meals that were the result of invitation or rewards in kind as the information on those items was not available. The nondurable nonfood expenditures correspond mainly to heating expenditures. Other expenditures are related to travel, education, entertainment (very few), health, hygiene, clothes and tobacco, electricity, and gas (some of which were missing in the sample for several periods). Transfers correspond to transfers received from relatives and friends, and from solidarity funds of local mosques (zakat).
C.1. Tables TABLE A.I RENTED -IN LAND AREAa
Net Rented-In Area
ln ct−1 qit : Land owned Total land in village Rain-fed land in village xit : (log) household size xit−1 : Lag (log) household size Constant Observations R-squared
1 OLS
2 2SLS
−1807 (0275)
−5323 (0851)
−5192 (0237) −5152 (0785) 1318 (1241) 1735 (1256) 5815 (1598)
−5054 (0249) −4875 (0837) 1915 (1340) 2523 (1327) 19666 (3754)
8053 053
7133
a Robust standard errors in parentheses. All village-and-period effects (46 villages, 11 rounds) not shown. In the case of 2SLS estimates, ln cit−1 is instrumented by ln cit−2 and all other variables.
723
RISK SHARING IN LDCS TABLE A.II CONSUMPTION AND LEISURE SEPARABILITY Dependent Variable Consumption-Growth Residuals Explanatory Variables
Male labor (days) Female labor (days)
Coeff.
Std. Error
−0.0009 −0.0045
(0.0010) (0.0024)
Observations R-squared
7133 0.0007
TABLE A.III WITHIN-VILLAGE CORRELATIONS OF INCOME AND CONSUMPTION GROWTHa Dependent Variable Cross-Product of Residuals (i j) Within-Village Income Correlations Explanatory Variables
|xit − xjt | Total land in village Rain-fed land in village (log) household size Observations R-squared
Within-Village Consumption Growth Correlations
Coeff.
Std. Error
Coeff.
Std. Error
−0.140 −0.061 −0.19
(0.0035) (0.010) (0.012)
−0.030 −0.023 −0.21
(0.0030) (0.0084) (0.0095)
137561 0.024
137561 0.0056
a This is the regression of all cross-products of residuals of the income equation (or consumption growth equation) within each village and each period on the distance between explanatory variables. No correction for first stage estimation. Village-and-period effects are included and not shown.
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Econometrica, Vol. 76, No. 4 (July, 2008), 727–761
FISHER’S INFORMATION FOR DISCRETELY SAMPLED LÉVY PROCESSES BY YACINE AÏT-SAHALIA AND JEAN JACOD1 This paper studies the asymptotic behavior of Fisher’s information for a Lévy process discretely sampled at an increasing frequency. As a result, we derive the optimal rates of convergence of efficient estimators of the different parameters of the process and show that the rates are often nonstandard and differ across parameters. We also show that it is possible to distinguish the continuous part of the process from its jumps part, and even different types of jumps from one another. KEYWORDS: Lévy process, jumps, rate of convergence, optimal estimation.
1. INTRODUCTION CONTINUOUS TIME MODELS in finance often take the form of a semimartingale, because this is essentially the class of processes that precludes arbitrage. Among semimartingales, models allowing for jumps are becoming increasingly common. There is a relative consensus that, empirically, jumps are prevalent in financial data, especially if one incorporates not just large and infrequent Poisson-type jumps, but also infinite activity jumps that can generate smaller, more frequent, jumps. This evidence comes from direct observation of the time series of various asset prices, studying the distribution of primary asset and that of derivative returns, especially as high frequency data becomes more commonly available for a growing range of assets. On the theory side, the presence of jumps changes many important properties of financial models. This is the case for option and derivative pricing, where most types of jumps will render markets incomplete, so the standard arbitragebased delta hedging construction inherited from Brownian-driven models fails, and pricing must often resort to utility-based arguments or perhaps bounds. The implications of jumps are also salient for risk management, where jumps have the potential to alter significantly the tails of the distribution of asset returns. In the context of optimal portfolio allocation, jumps can generate the types of correlation patterns across markets, or contagion, that is often documented in times of financial crises. Even when jumps are not easily observable or even present in a given time series, the mere possibility that they could occur will translate into amended risk premia and asset prices: this is one aspect of the peso problem. These theoretical differences between models driven by Brownian motions and those driven by processes that can jump are fairly well understood. How1 Financial support from the NSF under Grants SBR-0350772 and DMS-0532370 is gratefully acknowledged. We are also grateful for comments from the editor, three anonymous referees, George Tauchen, and seminar and conference participants at Princeton University, The University of Chicago, MIT, Harvard, and the 2006 Econometric Society Winter Meetings.
727
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ever, comparatively little is known about the estimation of models driven by jumps, even in the special case where the jump process has independent and identically distributed increments, that is, is a Lévy process (see, e.g., Woerner (2004) and the references cited therein), and among different inference procedures for parametrized Lévy processes, even less is known about their relative efficiency, especially for high frequency data. In this paper, we take a few steps toward a better understanding of the econometric situation, in a parametric model that is far from being fully realistic, but has the advantage of being tractable and yet is rich enough to deliver some surprising results. In particular, we will see that jumps give rise to a nonstandard econometric situation where the rates of convergence of estima√ tors of the various parameters of the model can depart from the usual n in a number of unexpected ways: another comparable, although unrelated, situation happens as one either stays on or crosses the unit root barrier in classical time series analysis. We study a class of Lévy processes for a log-asset price X, indexed on three parameters. We split X into the sum of two independent Lévy processes as (1.1)
Xt = σWt + θYt
Here, we have σ > 0 and θ ∈ R, and W is a standard symmetric stable process with index β ∈ (0 2]. We are often interested in the classical situation where β = 2 and so W is a Wiener process, hence continuous; σ and θ are scale parameters. In the case where W is a Wiener process, σ is the (Brownian or continuous) volatility of the process. As for Y , it is another Lévy process, viewed as a perturbation of W and “dominated” by W in a sense we will make precise below. In some applications, Y may represent frictions that are due to the mechanics of the trading process or, in the case of compound Poisson jumps, it may represent the infrequent arrival of relevant information related to the asset. In the latter case, W is then the driving process for the ordinary fluctuations of the asset value. Y is independent of W , and its law is either known or a nuisance parameter. When β = 2, W has, as already mentioned, continuous paths, and we will then be working with a model that has both continuous and discontinuous components. But when β < 2, W is itself a jump process, and we will be able to study the effect on each other’s parameters of including two separate jump components in the model, with different relative levels of jump activity. Our results cover both cases and we will describe the relevant differences. While there is a large literature devoted to stable processes in finance (starting with the work of Mandelbrot in the 1960s and see, for example, Rachev and Mittnik (2000) for new developments), a model based on stable processes is arguably quite special. Rather than an attempt at a fully realistic model intended to be fitted to the data as is, we have chosen to capture only some of the salient features of financial data, such as the presence of jumps, in a reasonably tractable framework. Inevitably, we leave out some other important
SAMPLED LEVY PROCESSES
729
features such as stochastic volatility2 or market microstructure noise, but in exchange we are able to derive explicitly some new and surprising econometric results in a fully parametric framework. The log-asset price X is observed at n discrete instants separated by n units of time. In financial applications, we are often interested in the high frequency limit where n → 0. The length Tn = nn of the overall observation period may be a constant T or may go to infinity. With Y viewed as a perturbation of W , we studied in an earlier paper (Aït-Sahalia and Jacod (2007)) the estimation of the single parameter σ, showing in particular that it can be estimated with the same degree of accuracy as when the process Y is absent, at least asymptotically. We proposed estimators for σ designed to achieve the efficient rate despite the presence of Y .3 We now study the more general problem where the parameter vector of interest is either (σ θ) or the full (σ β θ), viewing the law of Y nonparametrically. Our objective is to determine the optimal rate at which these parameters can be estimated. So we are led to consider the behavior of Fisher’s information. Through the Cramer–Rao lower bound, this yields the rate at which an optimal sequence of estimators should converge and the lowest possible asymptotic variance. Unlike the situation where n is fixed or the estimation of σ we previously studied, we are now in a nonstandard situation where the rates √ of convergence of optimal estimators of (β θ) can depart from the usual n in a number of different and often unexpected ways. Even for such a simple class of models as (1.1), there is a wide range of different asymptotic behaviors for Fisher’s information if we wish to estimate the three parameters σ, θ, and β or even σ and θ only when β is known, for example, when β = 2. We will show that different rates of convergence are achieved for the different parameters and for different types of Lévy processes. √ and Jacod (2007). Now, The optimal rate for σ is n as seen in Aït-Sahalia √ for β, the optimal rate will be the faster n log(1/n ), and is unaffected by the presence of Y (as was the case for σ). To estimate θ, the optimal rate is √ not n and, unlike the one for β, depends heavily on the specific nature of the Y process. Furthermore, unlike what happens for σ or β, which are immune to the presence of Y , the presence of the other process (W in this case) strongly affects the rate for θ. We study different examples of that situation; one in particular consists of the case where Y is also a stable process with index α < β; another is the jump-diffusion situation where Y is a compound Poisson process. 2 There has been much activity devoted to estimating the integrated volatility of the process in that context, with or without jumps: see, for example, Andersen, Bollerslev, and Diebold (2003), Barndorff-Nielsen and Shephard (2004), and Mykland and Zhang (2006). 3 Another string of the recent literature focuses on disentangling the volatility from the jumps: see, for example, Aït-Sahalia (2004) and Mancini (2001). In our case, this is related to the estimation of σ when β = 2.
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FIGURE 1.—Convergence as → 0 of Fisher’s information for the parameter σ from the model X = σW + θY , where W is Brownian motion (β = 2) and Y is a Cauchy process (α = 1), to Fisher’s information from the model X = σW which is 2/σ 2 as we will show in Theorem 2. Fisher’s information for the model X = σW + θY (solid line) is computed numerically using the exact expressions for the densities of the two processes and numerical integration of their convolution. The figure is in log–log scale. The parameter values are σ = 30%/year and θ = 03.
To illustrate some of these effects numerically, consider the example where W is a Brownian motion (β = 2) and Y is a Cauchy process (α = 1). We plot in Figures 1 and 2 the convergence, as the sampling interval decreases, of the (numerically computed) Fisher’s information for σ and θ, respectively, to their limits derived in Theorems 2 and 7, respectively. While, as illustrated in Figure 1, we can estimate σ as well as if Y were absent when n → 0, the symmetric statement is not true for θ. The limiting behavior in Figure 2 is quite distinct from a convergence to Fisher’s information for θ in the model without W , that is, X = θY . The latter is simply 1/(2θ2 ). In fact, Fisher’s information about θ in the presence of W tends to zero when n → 0.√Consequently, any optimal estimator of θ will converge at a rate slower than n. The paper is organized as follows. In Section 2, we set up the model. In Section 3, we derive the asymptotic behavior of Fisher’s information about the parameters (σ β θ). Then we show in Section 4 that the optimal rate for θ varies substantially according to the structure of the process Y , and we illustrate the versatility of the situation through several examples that display the variety of convergence rates that arise. Section 5 concludes. Proofs are in the Appendix.
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731
FIGURE 2.—Convergence as → 0 of Fisher’s information for the parameter θ from the model X = σW + θY , where W is a Brownian motion (β = 2) and Y is a Cauchy process (α = 1), to the theoretical limiting curve → (πθσ)−1 1/2 [log(1/)]−1/2 derived in Theorem 7. Fisher’s information for the model X = σW + θY (solid line) is computed numerically using the exact expressions for the densities of the two processes, and numerical integration of their convolution. The x-axis is in log scale. The parameter values are identical to those of Figure 1.
2. THE MODEL Because the components of X in (1.1) are all Lévy processes, they have an explicit characteristic function. Consider first the component W . The characteristic function of Wt is (2.1)
E(eiuWt ) = e−t|u|
β /2
The factor 2 above is perhaps unusual for stable processes when β < 2, but we put it here to ensure continuity between the stable and the Gaussian cases. The essential difficulty in this class of problems is the fact that the density of W is not known in closed form except in special cases (β = 1 or 2).4 Without the density of Y in closed form, it is of course hopeless to obtain the density of X , the convolution of the densities of σW and θY , in closed form. But when goes to 0, the density of X and its derivatives degenerate in a way 4 For fixed , there exist representations in terms of special functions (see Zolotarev (1995) and Hoffmann-Jørgensen (1993)), whereas numerical approximations based on approximations of the densities may be found in DuMouchel (1971, 1973b, 1975) or Nolan (1997, 2001), or also Brockwell and Brown (1980).
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which is controlled by the characteristics of the Lévy process, thus allowing us to explicitly describe, in closed form, the limiting behavior of Fisher’s information. To understand this behavior, we will need to be able to bound various functionals of the density of X . As is well known, when β < 2, we have E(|Wt |ρ ) < ∞ if and only if 0 < ρ < β. The density hβ of W1 is C ∞ (by repeated integration of the characteristic function) for all β ≤ 2, even, and its nth derivative h(n) β behaves, as |w| → ∞, as
(2.2)
(n) hβ (w) ∼
⎧ ⎪ ⎪ ⎨
cβ (β + i + 1) if β < 2, n+1+β |w| i=0 ⎪ ⎪ ⎩ n −w2 /2 √ |w| e / 2π if β = 2 n−1
(in the case n = 0, an empty product is defined to be 1), where cβ is the constant given by ⎧ β(1 − β) ⎪ ⎪ if β = 1, β < 2, ⎨ 4Γ (2 − β) cos(βπ/2) (2.3) cβ = ⎪ ⎪ ⎩ 1 if β = 1. 2π This follows from the series expansion of the density due to Bergstrøm (1952), the duality property of the stable densities of order β and 1/β (see, e.g., Chapter 2 in Zolotarev (1986)), with an adjustment factor in cβ to reflect our definition of the characteristic function in (2.1). In addition, ∞the cumulative distribution function of W1 for β < 2 behaves according to w hβ (x) dx ∼ cβ /(βwβ ) when w → ∞ (and symmetrically as w → −∞). The second component of the model is Y . Its law as a process is entirely specified by the law G of the variable Y for any given > 0. We write G = G1 , and we recall that the characteristic function of G is given by the Lévy– Khintchine formula
cv2 + F(dx) eivx − 1 − ivx1{|x|≤1} (2.4) E(eivY ) = exp ivb − 2 where (b c F) is the “characteristic triple” of G (or, of Y ): b ∈ R is the drift of Y , and c ≥ 0 is the local variance ofthe continuous part of Y , and F is the Lévy jump measure of Y , which satisfies (1 ∧ x2 )F(dx) < ∞ (see, e.g., Chapter II.2 in Jacod and Shiryaev (2003)). We need to put some additional structure on the model. We define the “domination” of Y by W by the property that G belongs to one of the classes defined below, for some α ≤ β. Let first Φ be the class of all nonnegative continuous
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functions on [0 1] with φ(0) = 0. If φ ∈ Φ, we set (2.5)
G (φ α) = the set of all infinitely divisible distributions with c = 0
and
(2.6)
∀x ∈ (0 1]
Gα =
⎧ α x F([−x x]c ) ≤ φ(x) ⎪ ⎪ ⎪ ⎨ 2 x F([−x x]c ) ≤ φ(x) and
⎪ ⎪ ⎪ |y|2 F(dy) ≤ φ(x) ⎩ {|y|≤x}
if α < 2,
if α = 2,
G (φ α)
φ∈Φ
We then have ⎧ α ∈ (0 2] ⇒ ⎪ ⎪ ⎨ α c G = G is infinitely divisible c = 0 lim x F([−x x] ) = 0 α (2.7) x↓0 ⎪ ⎪ ⎩ α = 2 ⇒ G2 = {G is infinitely divisible c = 0} For example, if Y is a compound Poisson process or a gamma process, then G is in ∈(02] Gα . If Y is an α-stable process, then G is in α >α Gα , but not in Gα . Recall also that the Blumenthal–Getoor index γ(G) of Y (or G or F ) is the infimum of all γ such that {|x|≤1} |x|γ F(dx) < ∞, equivalently, s≤t |Ys − Ys− |γ is almost surely finite for all t. If Hα denotes the set of all G such that c = 0 and γ(G) ≤ α, one may show that Gα ⊂ Hα ⊂ α >α Gα . Hence saying that G ∈ Gα is a little bit more restrictive than saying that G ∈ Hα . Our purpose is to understand the properties of the best possible estimators of the parameters of the model. The variables under consideration in the model (1.1) have densities which depend smoothly on the parameters, so in light of the Cramer–Rao lower bound, Fisher’s information is an appropriate tool for studying the optimality of estimators. X is observed at n times n 2n nn . Recalling that X0 = 0, this amounts to observing the n increments Xin − X(i−1)n . So when n = is fixed, we observe n independent and identically distributed variables distributed as X . If, further, these variables have a density depending smoothly on a parameter η, a very weak assumption, we are on known grounds: the Fisher’s information at stage n has the form In (η) = nI (η), where I (η) > 0 is the Fisher’s information (an invertible matrix if η is multidimensional) of the model based on the observation of propthe single variable X − X0 ; we have the locally asymptotically normal √ erty; the asymptotically efficient estimators η n are those for which n( ηn − η) converges in law to the normal distribution N (0 I (η)−1 ), and the maximum likelihood estimator (MLE) does the job (see, e.g., DuMouchel (1973a)).
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For high frequency data, though, things become more complicated since the time lag n becomes small. The corresponding asymptotics require n → 0 as the number n of observations goes to infinity. Fisher’s information at stage n still has the form Inn (η) = nIn (η), but the asymptotic behavior of the information In (η), not to speak about the behavior of the MLE for example, is now far from obvious: the essential difficulty comes from the fact that the law of Xn becomes degenerate as n → ∞. In the model (1.1), the law of the observed process X depends on the three parameters (σ β θ) to be estimated, plus on the law of Y which is summarized by G. The law of the variable X has a density which depends smoothly on σ and θ, so that the 2 × 2 Fisher’s information matrix (relative to σ and θ) of our experiment exists; it also depends smoothly on β when β < 2, so in this case the 3 × 3 Fisher’s information matrix exists. In all cases we denote the information by Inn (σ β θ G), and it has the form Inn (σ β θ G) = nIn (σ β θ G), where I (σ β θ G) is the Fisher’s information matrix associated with the observation of a single variable X . We denote the elements of the matrix I (σ β θ G) as Iσσ (σ β θ G), Iσβ (σ β θ G), etcetera. G may appear as a nuisance parameter in the model, in which case we may wish to have estimates for the Fisher’s information that are uniform in G, at least on some reasonable class of G’s. Let us also mention that in many cases the parameter β is indeed known: this is particularly true when W is a Brownian motion, β = 2. 3. THE GENERAL CASE We are first interested in determining the optimal rates (and constants) at which one can estimate (σ β), while leaving the distribution G ∈ Gβ as unspecified as possible. As we will see, the optimal rate for the estimation of θ is heavily dependent on the precise nature of G. 3.1. Inference on (σ β) It turns out that the limiting behavior of Fisher’s information for (σ β) is given by the corresponding limits in the situation where Y = 0, that is, we directly observe X = σW . In our general framework, this corresponds to setting G = δ0 , a Dirac mass at 0, and we set the (now unidentified) parameter θ to 0, or for that matter to any arbitrary value. We start by studying whether the limiting behavior of Iσσ (σ β θ G) when β = 2 and of the (σ β) block of the matrix I (σ β θ G) when β < 2 is affected by the presence of Y . We have the intuitively obvious majoration of Fisher’s information in the presence of Y by the one for which Y = 0. Note that in this result no assumption whatsoever is made on Y (except of course that it is independent of W ): THEOREM 1: For any > 0, we have (3.1)
Iσσ (σ 2 θ G) ≤ Iσσ (σ 2 0 δ0 )
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and, when β < 2, the difference
σσ I (σ β 0 δ0 ) Iσβ (σ β 0 δ0 ) Iσβ (σ β 0 δ0 ) Iββ (σ β 0 δ0 ) σσ
I (σ β θ G) Iσβ (σ β θ G) − σβ I (σ β θ G) Iββ (σ β θ G) is a positive semidefinite matrix; in particular we have (3.2)
Iββ (σ β θ G) ≤ Iββ (σ β 0 δ0 )
Next, how does the limit as → 0 of I (σ β θ G) compare to that of I (σ β 0 δ0 )? To answer this question, we need some further notation. For the first two derivatives of the density hβ of W1 , we write hβ and hβ , and we also introduce the functions (3.3)
h˘ β (w) = hβ (w) + whβ (w)
h˘ β (w)2 hβ (w) = hβ (w)
Then h˜ β is positive, even, continuous, and h˜ β (w) = O(1/|w|1+β ) as |w| → ∞; hence h˜ β is Lebesgue-integrable. Let
(3.4) I (β) = hβ (w) dw This is a positive number, which takes the value I (β) = 2 when β = 2. When Y = 0, we have p (x|σ β 0 δ0 ) = (1/σ1/β )hβ (x/σ1/β ) and, therefore, (3.5)
Iσσ (σ β 0 δ0 ) =
1 I (β) σ2
which does not depend on . I (β) is simply the Fisher’s information at point σ = 1 for the statistical model in which we observe σW1 . The limiting behavior of the (σ β) block of Fisher’s information is given by the following theorem: THEOREM 2: (a) If G ∈ Gβ , we have, as → 0, (3.6)
Iσσ (σ β θ G) →
1 I (β) σ2
and also, when β < 2, (3.7)
1 Iββ (σ β θ G) → 4 I (β) 2 log(1/) β
Iσβ (σ β θ G) 1 I (β) → log(1/) σβ2
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(b) For any φ ∈ Φ and α ∈ (0 β], and K > 0, we have as → 0, σσ I (β) I (σ β θ G) − → 0 (3.8) sup 2 σ G∈G (φα)|θ|≤K ⎧ ββ I (σ β θ G) I (β) ⎪ ⎪ ⎪ sup ⎪ (log(1/))2 − β4 → 0 ⎨ G∈G (φα)|θ|≤K β<2 ⇒ σβ ⎪ I (σ β θ G) I (β) ⎪ ⎪ → 0 ⎪ − sup ⎩ log(1/) σβ2 G∈G (φα)|θ|≤K (c) For each n, let Gn be the standard symmetric stable law of index αn , with αn a sequence strictly increasing to β. Then for any sequence n → 0 such that (β − αn ) log n → 0 (i.e., the rate at which n → 0 is slow enough), the sequence of numbers Iσσn (σ β θ Gn ) (resp. Iββn (σ β θ Gn )/(log(1/n ))2 when further β < 2) converges to a limit which is strictly less than I (β)/σ 2 (resp. I (β)/β4 ). Since the result (a) is valid for all G ∈ Gβ , it is valid in particular when G = δ0 , that is, when Y = 0. In other words, at their respective leading orders in , the presence of Y has no impact on the information terms Iσσ , Iββ , and Iσβ , as soon as Y is dominated by W : so, in the limit where n → 0, the parameters σ and β can be estimated with the exact same degree of precision whether Y is present or not. Moreover, part (b) states the convergence of Fisher’s information is uniform on the set G (φ α) and |θ| ≤ K for all α ∈ [0 β]; this settles the case where G and θ are considered as nuisance parameters when we estimate σ and β. But as α tends to β, the convergence disappears, as stated in part (c). The part of the theorem related to the σσ term alone is discussed in Aït√ Sahalia and Jacod (2007), where we use it to construct explicit n-converging estimators of the parameter σ√that are strongly efficient in the sense that not only do they converge at rate n, but they also √ have the same asymptotic variance when Y is present as when it is not: n( σn − σ) converges in law to N (0 σ 2 /I (β)) as soon as n → 0. This is the case even in the semiparametric case where nothing is known about Y , other than the fact that Y is sufficiently away from W , meaning that G ∈ Gα for α small enough: α < β is not enough and the precise condition on α depends on the behavior of the pair (n n ); see Theorem 5 of that paper. When it comes to estimating β, with σ known, things are different. When n → 0 and the true value is β < 2, we can hope for estimators converging to β √ at the faster rate n log(1/n ), in view of the limit for Iββ given in (3.7). In fact, for a single observation, say X , there exists of course no consistent estimator = − log(1/)/ log |X |, for σ, but there is one for β when → 0: namely β which converges in probability to β as , and the rate of convergence is log(1/); all these follow from the scaling property of W .
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Estimators for β could be constructed using the jumps of the process of a size greater than some threshold (see Höpfner and Jacod (1994)). Estimating β was studied by DuMouchel (1973a), who computed numerically the term Iββ (σ β 0 δ0 ), including also an asymmetry parameter. Note that if one suspects that β = 2, then one should perform a test, as advised by DuMouchel (1973a), and in that case the behavior of the Fisher’s information about β does not provide much insight. The introduction of β as a parameter to be estimated makes the joint estimation of the pair (σ β) a singular problem. The asymptotic (normalized) 2 × 2 Fisher’s information matrix given by (3.6) and (3.7) is not invertible. In particular, there is no guarantee that the joint MLE for the pair (σ β), for example, behaves well. However, the estimation of either parameter √ knowing the other remains a regular problem, albeit with a rate faster than n for β. 3.2. Inference on θ For the entries of Fisher’s information matrix involving the parameter θ, things are more complicated. First, observe that Iθθ (0 β θ G) (that is, the Fisher’s information for the model X = θY ) does not necessarily exist, but of course if it does we have an inequality similar to (3.1) for all σ: (3.9)
Iθθ (σ β θ G) ≤ Iθθ (0 β θ G)
Contrary to (3.1), however, this is a very rough estimate, which does not take into account the properties of W . The (θ θ) Fisher’s information is usually much smaller than what the right side above suggests, and we give below a more accurate estimate when Y has second moments, but without the domination assumption that G ∈ Gβ . For this, we see another information appear, namely the Fisher’s information associated with the estimation of the real number a for the model where one observes the single variable W1 + a. This Fisher’s information is
hβ (w)2 (3.10) J (β) = dw hβ (w) Observe in particular that J (β) = 1 when β = 2. THEOREM 3: If Y1 has finite variance v and mean m, we have (3.11)
Iθθ (σ β θ G) ≤
J (β) 2 2−2/β 1−2/β + v m σ2
This estimate holds for all > 0. The asymptotic variant, which says that (3.12)
lim sup 2/β−1 Iθθ (σ β θ G) ≤ →0
vJ (β) σ2
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is sharp in some cases and not in others, as we will see in the examples below. These examples will also illustrate how the “translation” Fisher’s information J (β) comes into the picture here. Theorem 3 does not solve the problem in full generality: for a generic Y , we only obtain a bound. For some specific Y ’s, we will be able to give a full result in Section 4. It is indeed a feature of this problem that the estimation of θ depends heavily on the specific nature of the Y process, including the fact that the optimal rate at which θ can be estimated depends on the process Y . Hence it is not surprising that any general result regarding θ, such as Theorem 3, can at best state a bound: there is no single common rate that applies to all Y ’s. 4. SPECIFIC EXAMPLES OF LÉVY PROCESSES The calculations of the previous section involving the parameter θ can be made fully explicit if we specify the distribution of the process Y . We can then illustrate the surprisingly wide range of situations that can occur to the rates of convergence. For simplicity, let us consider β as known in these examples and focus on the joint estimation of (σ θ). 4.1. Stable Process Plus Drift Here we assume that Yt = t, so G = δ1 and θ is a drift parameter: THEOREM 4: The 2 × 2 Fisher’s information matrix for estimating (σ θ) is σσ
I (σ β θ δ1 ) Iσθ (σ β θ δ1 ) (4.1) Iσθ (σ β θ δ1 ) Iθθ (σ β θ δ1 )
1 I (β) 0 = 2 0 2−2/β J (β) σ This has several interesting consequences (we denote by Tn = nn the length of the observation window): 1. If θ is known, (4.1) suggests the existence of estimators σn satisfying √ d 2 n( σn − σ) −→ N (0 σ /I (β)). This is the case and, in fact, since θ is known, observing Xin and observing Win are equivalent. √ 2. If σ is known, one may hope for estimators θn satisfying nn1−1/β ( θn − √ d 2 θ) −→ N (0 σ /J (β)). If β = 2, the rate is thus Tn : this is in accordance with the classical fact that for a diffusion the rate for estimating the √drift coefficient is the square root of the total observation window, that is, Tn here. Moreover, in this case, the variable XTn /Tn is N (θ σ 2 /Tn ), so θn = XTn /Tn is an asymptotically efficient estimator for θ (recall that J (β) = 1 when √ β = 2). When β < 2, we have 1 − 1/β < 1/2, so the rate is bigger than T√ n and it increases when β decreases; when β < 1, this rate is even bigger than n.
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3. Observe that here Y1 has mean m = 1 and variance v = 0, so the bound (3.11) in Theorem 3 is indeed an equality. The fact that the translation Fisher’s information J (β) appears here is transparent. θn such 4. If both σ and one may hope for estimators σn and √ √ θ are unknown, θn − θ)) converge in law to the product σn − σ), nn1−1/β ( that the pairs ( n( N (0 σ 2 /I (β)) ⊗N (0 σ 2 /J (β)). 4.2. Stable Process Plus Poisson Process Here we assume that Y is a standard Poisson process (for simplicity jumps of size 1, intensity 1) whose law we write as G = P. We can describe the limiting behavior of the (σ θ) block of the matrix I (σ β θ P) as → 0. THEOREM 5: If Y is a standard Poisson process, we have, as → 0, 1 I (β) σ2
(4.2)
Iσσ (σ β θ P) →
(4.3)
1/β−1/2 Iσθ (σ β θ P) → 0
(4.4)
2/β−1 Iθθ (σ β θ P) →
1 J (β) σ2
Since P ∈ Gβ , (4.2) is nothing else than the first part of (3.6). One could prove more than (4.3), namely that sup 1/β−1 |Iσθ (σ β θ P)| ≤ ∞. Since Y1 has mean m = 1 and variance v = 1, the asymptotic estimate (3.12) in Theorem 3 is sharp in view of (4.4). Here again, we deduce some interesting consequences for the estimation: θn − 1. If σ is known, one may hope for estimators θn satisfying nn1−2/β ( √ d 2 θ) −→ N (0 σ /J (β)). So the rate is faster than n, except when β = 2. More generally, if both σ and θ are unknown, σn and one may hope for estimators √ 1−2/β σn − σ) nn (θn − θ)) converge in law to the θn such that the pairs ( n( product N (0 σ 2 /I (β)) ⊗ N (0 σ 2 /J (β)). 2. However, the above-described behavior of any estimator θn cannot be true when Tn = nn does not go to infinity, because in this case there is a positive probability that Y has no jump on the biggest observed interval, and so no information about θ can be drawn from the observations in that case. It is true, though, when Tn → ∞, because Y will eventually have infinitely many jumps on the observed intervals. There is a large literature on this subject, starting with the base case where the Poisson process is directly observed, for which the same problem occurs. 4.3. Stable Process Plus Compound Poisson Process Here we assume that Y is a compound Poisson process with arrival rate λ and law of jumps μ: that is, the characteristics of G are b = λ {|x|≤1} xμ(dx)
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and c = 0 and F = λμ. We then write G = Pλμ , which belongs to Gβ . If β = 2, we are then in the classical context of jump diffusions in finance. We will further assume that μ has a density f satisfying lim uf (u) = 0 sup |f (u)|(1 + |u|) < ∞ (4.5) |u|→∞
u
We also suppose that the “multiplicative” Fisher’s information associated with μ (that is, the Fisher’s information for estimating θ in the model when one observes a single variable θU with U distributed according to μ) exists. It then has the form
(xf (x) + f (x))2 (4.6) dx L= f (x) We can describe the limiting behavior of the (σ θ) block of the matrix I (σ β θ Pλμ ) as → 0. THEOREM 6: If Y is a compound Poisson process satisfying (4.5) and such that L in (4.6) is finite, we have, as → 0, (4.7) (4.8) (4.9)
Iσσ (σ β θ Pλμ ) →
1 I (β) σ2
1 √ Iσθ (σ β θ Pλμ ) → 0 2
(xf (x) + f (x))2 1 1 λ dx ≤ lim inf Iθθ (σ β θ Pλμ ) ≤ 2 L cβ σ β θ2 θ λf (x) + β 1+β θ |x|
and also, when β = 2, (4.10)
1 1 θθ I (σ 2 θ Pλμ ) → 2 L θ
As for the previous theorem, (4.7) is nothing else than the first part of (3.6). We could prove more than (4.8), namely that sup 1 |Iσθ (σ β θ Pλμ )| < ∞. Here again, we draw some interesting consequences: 1.√ Equations (4.9) and (4.10) suggest that there are estimators θn such that Tn ( θn − θ) is tight (the rate is the same as for the case Yt = t) and is even asymptotically normal when β = 2. But this is of course wrong when Tn does not go to infinity, for the same reason as in the previous theorem. 2. When the measure μ has a second order moment, the right side of (3.11) is larger than the result of the previous theorem, so the estimate in Theorem 3 is not sharp.
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The rates for estimating θ in the two previous theorems, and the limiting Fisher’s information as well, can be explained as follows (supposing that σ is known and that we have n observations and that Tn → ∞): 1. For Theorem 5: θ comes into the picture whenever the Poisson process has a jump. On the interval [0 Tn ] we have an average of Tn jumps, most of them being isolated in an interval (in (i + 1)n ]. So it essentially amounts to observing Tn (or rather the integer part [Tn ]) independent variables, all distributed as σ1/β n W1 + θ. The Fisher’s information for each of those (for estimating θ) is J (β)/σ 2 2/β , and the “global” Fisher’s information, namely nIθθn , is approximately Tn J (β)/σ 2 2/β ∼ J (β)/σ 2 n2/β−1 . n 2. For Theorem 6: Again θ comes into the picture whenever the compound Poisson process has a jump. We have an average of λTn jumps, so it essentially amounts to observing λTn independent variables, all distributed as σ1/β n W1 + θV , where V has the distribution μ. The Fisher’s information for each of those (for estimating θ) is approximately L/θ2 (because the variable θθ σ1/β n W1 is negligible), and the “global” Fisher’s information nIn is approxi2 2 mately λTn L/θ ∼ nn L/θ . This explains the rate in (4.9), and is an indication that (4.10) may be true even when β < 2, although we have been unable to prove it thus far. 4.4. Two Stable Processes Our last example is about the case where Y is also a symmetric stable process with index α, α < β. We write G = Sα . Surprisingly, the results are quite involved, in the sense that for estimating θ we have different situations according to the relative values of α and β. We obviously still have (4.7), so we concentrate on the term Iθθ and ignore the cross term in the statement of the following theorem. The constant cβ below is the one occurring in (2.2). THEOREM 7: If Y is a standard symmetric stable process with index α < β, we have the following situations as → 0: (a) If β = 2, (4.11)
(log(1/))α/2 θθ 2αcα βα/2 I (σ β θ S ) → α (β−α)/β θ2−α σ α (2(β − α))α/2
(b) If β < 2 and α > β2 , (4.12)
(2(β−α))/β Iθθ (σ β θ Sα ) 1
( R |y|1−α dy 0 (1 − v)hβ (x − yv) dv)2 α2 cα2 θ2α−2 dx → σ 2α hβ (y)
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(c) If β < 2 and α = β2 , (4.13)
(2(β−α))/β
1 θθ 2α(β − α)cα2 θ2α−2 I (σ β θ Sα ) → log βcβ σ 2α
(d) If β < 2 and α < β2 , (4.14)
1 θθ α2 cα2 θ2α−2 I (σ β θ Sα ) → σ 2α
cβ |z|
1+2α−β
dz + cα θα |z|1+α /σ α
Then if σ is known, one may hope to find estimators θn with un ( θn − θ) −→ N(0 V ), with ⎧√ nn(2−α)/4 /(log(1/n ))α/4 if β = 2, ⎪ ⎪ ⎪ √ ⎪ ⎨ n(β−α)/β if β < 2 α > β/2, n un = √ 1/2 ⎪ nn log(1/n ) if β < 2 α = β/2, ⎪ ⎪ ⎪ ⎩√ nn if β < 2 α < β/2, d
and of course the asymptotic variance V should be the inverse of the right-hand sides in (4.11)–(4.14). 5. CONCLUSIONS We studied in this paper a tractable parametric model with jumps, also including a continuous component, and examined the impact of the presence of the different components of the model (continuous part vs. jump part and/or jumps with different relative levels of activity) on the various parameters of the model. We determined how optimal estimators should behave where the parameter vector of interest is (σ β θ), viewing the law of Y√ nonparametrically. While optimal estimators for σ should converge at rate n and have the same asymptotic √ variance with or without Y , the optimal rate for β is faster and given by n log(1/n ), and is also unaffected by the presence of Y . But the rate of convergence of optimal estimators of θ is very dependent on the law of Y and is affected by the presence of W . Given that we now know what can be achieved by efficient estimators, the natural next step is to exhibit actual, and tractable, estimators with those optimal properties. We did not attempt here to construct estimators of these parameters, and leave that question to future work. As a partial step in that direction, we studied in Aït-Sahalia and Jacod (2007) estimation procedures, for the parameter σ only, that satisfy the property that those estimators are asymptotically as efficient as when the process Y is absent.
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Dept. of Economics, Bendheim Center for Finance, Princeton University, 26 Prospect Avenue, Princeton, NJ 08540-5296, U.S.A. and NBER; yacine@ princeton.edu and Institut de Mathématiques de Jussieu, CNRS–UMR 7586, 175 Rue due Chevaleret, 75013 Paris, France and Université P. et M. Curie, Paris-6, France;
[email protected]. Manuscript received May, 2006; final revision received October, 2007.
APPENDIX: PROOFS A. Fisher’s Information When X = σW + θY By independence of W and Y the density of X in (1.1) is the convolution (recall that G is the law of Y )
x − θy 1 p (x|σ β θ G) = (A.1) G (dy)hβ σ1/β σ1/β We now seek to characterize the entries of the full Fisher’s information matrix. Since hβ , h˘ β , and h˙ β are continuous and bounded, we can differentiate under the integral in (A.1) to get
x − θy 1 ˘ ∂σ p (x|σ β θ G) = − 2 1/β G (dy)hβ (A.2) σ σ1/β σ log ∂σ p (x|σ β θ G) β2
x − θy G (dy)yhβ σ1/β
(A.3)
∂β p (x|σ β θ G) = v (x|σ β θ G) −
(A.4)
∂θ p (x|σ β θ G) = −
1 σ 2/β 2
where (A.5)
1 v (x|σ β θ G) = σ1/β
x − θy ˙ G (dy)hβ σ1/β
The entries of the (σ θ) block of the Fisher’s information matrix are (leaving implicit the dependence on (σ β θ G))
(A.6)
σσ
I
=
θθ
I
=
∂σ p (x)2 dx p (x) ∂θ p (x)2 dx p (x)
σθ
I
=
∂σ p (x)∂θ p (x) dx p (x)
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When β < 2, the other entries are (A.7)
Iσβ = Jσβ −
σ log σσ I β2
(A.8)
Iββ = Jββ −
2σ log σβ σ 2 (log )2 σσ J + I β2 β4
where (A.9)
Jσβ =
βθ
J
=
Iβθ = Jβθ −
∂σ p (x)v (x) dx p (x)
Jββ =
σ log σθ I β2
v (x)2 dx p (x)
v (x)∂θ p (x) dx p (x)
B. Proof of Theorem 1 The proof is standard and given for completeness, and given only in the case where β < 2 (when β = 2 take v = 0 below). What we need to prove is that, for any u v ∈ R, we have
(u∂σ p (x|σ β θ G) + v∂β p (x|σ β θ G))2 (B.1) dx p (x|σ β θ G)
(u∂σ p (x|σ β 0 δ0 ) + v∂β p (x|σ β 0 δ0 ))2 dx ≤ p (x|σ β 0 δ0 ) We set q(x) = p (x|σ β θ G)
q0 (x) = p (x|σ β 0 δ0 )
r(x) = u∂σ p (x|σ β θ G) + v∂β p (x|σ β θ G) r0 (x) = u∂σ p (x|σ β 0 δ0 ) + v∂β p (x|σ β 0 δ0 ) Observe that by (A.1), q(x) = G (dy)q0 (x − θy) hence r(x) = G (dy) × r0 (x − θy) as well. Apply the Cauchy–Schwarz inequality to G with r0 = √ √ q0 (r0 / q0 ) to get
r0 (x − θy)2 2 r(x) ≤ q(x) G (dy) q0 (x − θy) Then
r(x)2 dx ≤ q(x)
dx
r0 (x − θy)2 G (dy) = q0 (x − θy)
r0 (z)2 dz q0 (z)
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by Fubini’s theorem and a change of variable: this is exactly (B.1). C. Proof of Theorem 2 It is easily seen from the characteristic function that β → hβ (w) is differentiable on (0 2], and we denote by h˙ β (w) its derivative. However, instead of (2.2), one has ⎧ cβ log |w| ⎪ ⎪ ⎨ |w|1+β if β < 2, |h˙ β (w)| ∼ (C.1) ⎪ 1 ⎪ ⎩ if β = 2 |w|3 as |w| → ∞, by differentiation of the series expansion for the stable density. Therefore the quantity
˙ hβ (w)2 (C.2) dw K(β) = hβ (w) is finite when β < 2 and infinite for β = 2. This is the Fisher’s information for estimating β, upon observing the single variable W1 . First, part (b) of the theorem implies (a). Second, the claim about the σσ term (C.3)
Iσσn (σ β θn Gn ) →
I (β) σ2
and its uniformity are proved in Aït-Sahalia and Jacod (2007), so we only need to prove the claims about I ββ and I σβ in (b) and (c). We assume β < 2. If (b) fails, one can find φ ∈ Φ, α ∈ (0 β], and ε > 0, and also a sequence n → 0 and a sequence Gn of measures in G(φ α) and a sequence of numbers θn converging to a limit θ, such that ββ In (σ β θn Gn ) I (β) Iσσn (σ β θn Gn ) I (β) + ≥ε − − (log(1/ ))2 β4 log(1/n ) σβ2 n for all n. In other words, we will get a contradiction and (b) will be proved as soon as we show that (C.4)
Iββn (σ β θn Gn ) I (β) → 2 (log(1/)) β4
Iσβn (σ β θn Gn ) I (β) → log(1/) σβ2
Recall (A.7) and (A.8): with the notation Jσβ (σ β θ G) of (A.9), we see that (C.5) Jσβ (σ β θ G) ≤ Iσσ (σ β θ G)Jββ (σ β θ G)
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by a first application of the Cauchy–Schwarz inequality. A second application of the same plus (A.1) and (A.5) yields v (x|σ β θ G)2 ≤
p (x|σ β θ G) σ1/β
G (dy)
h˙ 2β x − θy hβ
σ1/β
Then Fubini’s theorem and the change of variable x ↔ (x − θy)/σ1/β in (A.9) leads to (C.6)
Jββ (σ β θ G) ≤ K(β)
Next, (C.4) readily follows from (C.5), (C.6), and (C.3), and also (A.7) and (A.8). Finally, it remains to prove the part of (c) concerning I ββ . We already know that the sequence Iσσn (σ β θ Gn ) converges to a limit which is strictly less than I (β)/σ 2 . Then by (A.8), (C.5), (C.6), and (C.3), it is clear that Iββn (σ β θ Gn )/(log(1/n ))2 also converges to a limit which is strictly less than I (β)/β4 . D. Proof of Theorem 3 The Cauchy–Schwarz inequality gives us, by (A.1) and (A.4), |∂θ p (x|σ β θ G)|2 1 ≤ 3 3/β p (x|σ β θ G) σ
G (dy)y 2
[hβ ((x − θy)/σ1/β )]2 hβ ((x − θy)/σ1/β )
Plugging this into (A.6), applying Fubini’s theorem, and doing the change of variable x ↔ (x − θy)/σ1/β leads to (D.1)
θθ
I
1 ≤ 2 2/β σ
G (dy)y
2
hβ (x)2 hβ (x)
dx
Since E(Y2 ) = m2 + v, we readily deduce (3.11). E. Proof of Theorem 4 When Yt = t we gave G = δ . Then (4.1) follows directly from applying the formulae (A.2) and (A.4), and from the change of variable x ↔ (x − θ)/σ1/β in (A.6), after observing that the function h˘ β hβ / hβ is integrable and odd, hence has a vanishing Lebesgue integral.
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F. Proofs of Theorems 5 and 6 F.1. Preliminaries We suppose that Y is a compound Poisson process with arrival rate λ and law of jumps μ, and that μk is the kth fold convolution of μ. So we have G = e−λ
∞ (λ)k
k!
k=0
Set
x − θu μk (du)hβ σ1/β
x − θu ˘ μk (du)hβ σ1/β
x − θu μk (du)uhβ σ1/β
(F.1)
1 γ (k x) = σ1/β
(F.2)
γ(2) (k x) =
1 σ 1/β
(F.3)
γ(3) (k x) =
1 σ 2/β
(1)
2
2
μk
We have (recall that μ0 = δ0 ) (F.4)
p (x|σ β θ G) = e−λ
∞ (λ)k k=0
(F.5)
∂σ p (x|σ β θ G) = −e−λ
k!
∞ (λ)k k=0
(F.6)
∂θ p (x|σ β θ G) = −e−λ
γ(1) (k x)
k!
∞ (λ)k k=1
k!
γ(2) (k x) γ(3) (k x)
Omitting the mention of (σ β θ G), we also set
(F.7)
i = 2 3
Γ(i) (k k ) =
(F.8)
Γ
(4)
(k k ) =
γ(i) (k x)γ(i) (k x) dx p (x) γ(2) (k x)γ(3) (k x) dx p (x)
By the Cauchy–Schwarz inequality, we have (i) (i) i = 2 3 Γ (k k ) ≤ Γ (k k)Γ(i) (k k ) (F.9) (4) Γ (k k ) ≤ Γ (2) (k k)Γ (3) (k k ) (F.10)
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For any k ≥ 0 we have p (x) ≥ e−λ ((λ)k )/k!γ(1) (k x). Therefore,
(i) γ (k x)2 eλ k! (i) (F.11) i = 2 3 Γ (k k) ≤ dx (λ)k γ(1) (k x) Finally, if we plug (F.5) and (F.6) into (A.6), we get (F.12)
Iσθ = e−2λ
∞ ∞ (λ)k+l k=0 l=1
(F.13)
θθ
I
−2λ
=e
k!l!
∞ ∞ (λ)k+l k=1 l=1
k!l!
Γ(4) (k l) Γ(3) (k l)
F.2. Proof of Theorem 5 When Y is a standard Poisson process, we have λ = 1 and μk = εk . Therefore, we get
x − θk 1 (F.14) γ(1) (k x) = h β σ1/β σ1/β
x − θk 1 (F.15) γ(2) (k x) = 2 1/β h˘ β σ σ1/β
x − θk k (F.16) γ(3) (k x) = 2 2/β hβ σ σ1/β Plugging this into (F.11) yields (F.17)
Γ(2) (k k) ≤
e k! I (β) σ 2 k
Γ(3) (k k) ≤
e k2 k! J (β) σ 2 k+2/β
Recall that (4.2) follows from Theorem 2, so we need to prove (4.3) and (4.4). In view of (F.12) and (F.13), this amounts to proving the two properties ∞ ∞ k+l+1/β−1/2 k=0 l=1
k!l!
∞ ∞ k+l+2/β−1 k=1 l=1
k!l!
Γ(4) (k l) → 0
Γ(3) (k l) →
1 J (β) σ2
If we use (F.9) and (F.17), it is easily seen that the sum of all summands in the first (resp. second) left side above, except the one for k = 0 and l = 1 (resp. k = l = 1), goes to 0. So we are left to prove (F.18)
1/2+1/β Γ(4) (0 1) → 0
1+2/β Γ(3) (1 1) →
1 J (β) σ2
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1 , so (1 + β1 )(1 − ω) = 1 + 1/2β. Assume first that β < 2. Then Let ω = 2(1+β) 1/β ω if |x| ≤ |θ/σ | , we have for some constant C ∈ (0 ∞), possibly depending on θ, σ, and β, that changes from one occurrence to the other and provided ≤ (2θ/σ)β , hβ (x) ≥ Cω(1+1/β) hβ x + rθ/σ1/β ≤ C1+1/β
when r ∈ Z\{0}, and thus ω |x| ≤ θ/σ1/β ⇒ hβ x + rθ/σ1/β ≤ Chβ (x)(1+1/β)(1−ω) = Chβ (x)1+1/2β When β = 2, a simple computation on the normal density shows that the above property also holds. Therefore, in view of (F.4) and (F.14) we deduce that in all cases
ω x−θ θ ≤ σ1/β σ1/β implies
∞ k x−θ e− + C1+β/2 1 + hβ p (x) ≤ σ1/β σ1/β k! k=2
x−θ e− 1 + Cβ/2 hβ ≤ 1/β−1 1/β σ σ
By (F.7) it follows that Γ
(3)
e (1 1) ≥ 3 1+3/β σ ≥
e σ 2 1+2/β
hβ ((x − θ)/σ1/β )2 {|(x−θ)/σ1/β |≤(θ/σ1/β )ω }
hβ (x)2 {|x|≤(θ/σ1/β )ω }
hβ (x)
hβ ((x − θ)/σ1/β )
dx
dx
We readily deduce that lim inf→0 1+2/β Γ(3) (1 1) ≥ J (β)/σ 2 . On the other hand, (F.17) yields lim sup→0 1+2/β Γ(3) (1 1) ≤ J (β)/σ 2 , and thus the second part of (F.18) is proved. Finally h˘ β / hβ is bounded, so (F.15), (F.16), and the fact that p (x) ≥ e− hβ x/σ1/β /σ1/β yield (4) Γ (0 1) ≤
e σ 3 2/β
x−θ dx ≤ e h |hβ (x)| dx β σ1/β σ 2 1/β
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Y. AÏT-SAHALIA AND J. JACOD
and the first part of (F.18) readily follows. F.3. Proof of Theorem 6 We use the same notation as in the previous proof, but here the measure μk has a density fk for all k ≥ 1, which further is differentiable and satisfies (4.5) uniformly in k, while we still have μ0 = δ0 . Exactly as in (3.3), we set f˘k (u) = ufk (u) + fk (u) Recall the Fisher’s information L defined in (4.6), which corresponds to estimating θ in a model where one observes a variable θU, with U having the law μ. Now if we have n independent variables Ui with the same law μ, the Fisher’s information associated with the observation of θUi for i = 1 n is of course nL; if instead one observes only θ(U1 + · · · + Un ), one gets a smaller Fisher’s information Ln ≤ nL. In other words, we have
˘ fn (u)2 du ≤ nL (F.19) Ln := fn (u) Taking advantage of the fact that μk has a density, for all k ≥ 1, we can rewrite γ(i) (k x) as (using further an integration by parts when i = 3 and the fact that each fk satisfies (4.5))
x − yσ1/β 1 (1) (F.20) γ (k x) = dy hβ (y)fk θ θ
x − yσ1/β 1 (2) ˘ (F.21) γ (k x) = hβ (y)fk dy σθ θ
x − yσ1/β 1 (3) ˘ (F.22) γ (k x) = 2 hβ (y)fk dy θ θ Since the fk ’s satisfy (4.5) uniformly in k, we readily deduce that (F.23) k ≥ 1 i = 1 2 3 ⇒ γ(i) (k x) ≤ C x 1 1 ˘ x (1) (3) (F.24) γ (1 x) → f γ (1 x) → 2 f θ θ θ θ Let us start with the lower bound. Since γ(1) (0 x) = (1/σ1/β )hβ (x/σ1/β ) we deduce from (2.2), (F.23), and (F.4) that 1 cβ σ β λ x (F.25) p (x) → 1+β + f |x| θ θ as soon as x = 0, and with the convention c2 = 0. In a similar way, we deduce
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from (F.23) and (F.6) that (F.26)
1 λ x ∂θ p (x) → − 2 f˘ θ θ
Then plugging (F.25) and (F.26) into the last equation in (A.6), we conclude by Fatou’s lemma and after a change of variable that
f˘(x)2 I θθ λ2 lim inf ≥ 2 dx →0 θ λf (x) + cβ σ β /θβ |x|1+β It remains to prove (4.8) and the upper bound in (4.9) (including when β = 2). By the Cauchy–Schwarz inequality, we get (using successively the two equivalent versions for γ(i) (k x))
1 γ(2) (k x)2 ≤ 3 γ(1) (k x) μk (du) hβ (x − θu)/σ1/β σ
f˘k ((x − yσ1/β )/θ)2 1 (1) (3) 2 dy γ (k x) ≤ 3 γ (k x) hβ (y) θ fk ((x − yσ1/β )/θ) Then it follows from (F.11) and (F.19) that (F.27)
Γ(2) (k k) ≤
eλ k! I (β) σ 2 (λ)k
Γ(3) (k k) ≤
eλ kk! L θ2 (λ)k
We also need an estimate for Γ(4) (0 1). By (F.1), (F.2), and (F.4) we obtain p (x) ≥ e−λ hβ (x/σ1/β )/σ1/β and γ(2) (0 x) = h˘ β (x/σ1/β )/σ 2 1/β . Then use (F.22) and the definition (F.7) to get
˘ (4) hβ x − yσ1/β x 1 ˘ dy (F.28) Γ (0 1) ≤ dx hβ (y)f σθ2 hβ σ1/β θ
˘ hβ 1 x x − yσ1/β ˘ = hβ (y) dy f dx σθ2 hβ σ1/β θ ≤ C where the last inequality comes from the facts that h˘ β / hβ is bounded and that f˘ is integrable (due to (4.5)). At this stage we use (F.9) together with (F.12) and (F.13), and the fact that 2|xy| ≤ ax2 + y 2 /a for all a > 0. Taking arbitrary constants akl > 0, we deduce from (F.27) that ∞
1/2 ∞ ∞ L −λ (λ)k k (λ)k+l kk! ll! θθ +2 I ≤ 2 e θ k! k!l! (λ)k (λ)l k=1 k=1 l=k+1
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Y. AÏT-SAHALIA AND J. JACOD
∞ ∞ ∞ ∞ L (λ)l k (λ)k l 1 akl + ≤ 2 λ + θ l! k! akl k=1 l=k+1 k=1 l=k+1 Then if we take akl = (λ)(k−l)/2 for l > k, a simple computation shows that indeed Iθθ ≤
L λ + C3/2 2 θ
for some constant C, and thus we get the upper bound in (4.9). In a similar way, replacing L/θ2 above by the supremum between L/θ2 and I (β)/σ 2 , we see that in (F.12) the sum of the absolute values of all summands except the one for k = 0 and l = 1 is smaller than a constant times Λ. Finally, the same holds for the term for k = 0 and l = 1, because of (F.28), and this proves (4.8). G. Proof of Theorem 7 G.1. Preliminaries In the setting of Theorem 7, the measure G admits the density y → hα (y/1/α )/1/α . For simplicity, we set (G.1)
u = 1/α−1/β
(so u → 0 as → 0), and a change of variable allows to write (A.1) as
x σy 1 p (x|σ β θ G) = dy h − y hα β θ1/α σ1/β θu Therefore, 1 ∂θ p (x|σ β θ G) = − 2 1/α θ
hβ
x σy dy − y h˘ α σ1/β θu
and another change of variable in (A.6) leads to (G.2)
Iθθ =
θ2α−2 u2α Ju σ 2α
where
(G.3)
Ru (x)2 dx Su (x) 1+α
σy σ ˘ Ru (x) = dy hβ (x − y)hα θu θu
Ju =
753
yθu ˘ σ hα (y) dy hβ x − = θu σ
σy yθu σ hβ (x − y)hα dy = hβ x − hα (y) dy Su (x) = θu θu σ SAMPLED LEVY PROCESSES
α
Below, we denote by K a constant and by φ a continuous function on R+ with φ(0) = 0, both of them changing from line to line and possibly depending on the parameters α β σ θ; if they depend on another parameter η, we write them as Kη or φη . Recalling (2.2), we have, as |x| → ∞,
cα 2cα hα (x) ∼ 1+α (G.4) hα (y) dy ∼ |x| α|x|α {|y|>|x|}
αcα 2cα h˘ α (y) dy ∼ − α h˘ α (x) ∼ − 1+α |x| |x| {|y|>|x|} Another application of (2.2) when β < 2 and of the explicit form of h2 gives if β < 2 Khβ (x) (G.5) |y| ≤ 1 ⇒ |hβ (x − y)| ≤ hβ (x) := 2 −x2 /3 if β = 2 K(1 + x )e We will now introduce an auxiliary function
1 Dη (x) = (G.6) hβ (x − y) 1+α dy |y| {|y|>η} which will help us control the behavior of Ru and Su as stated in the following lemma. LEMMA 1: We have
(θu)α (G.7) Su (x) − hβ (x) + cα σ α D1 (x) ≤ (hβ (x) + uα D1 (x))φ(u) + Kuα hβ (x)
Ru (x) − cα 2hβ (x) − αDη (x) ηα
hβ (x) ≤ + Dη (x) φη (u) + Khβ (x)η2−α ηα with (G.8)
Dη (x) ∼
1 |x|1+α
as |x| → ∞
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Y. AÏT-SAHALIA AND J. JACOD
PROOF: We split the first two integrals in Ru and Su into sums of integrals on the two domains {|y| ≤ η} and {|y| > η} for some η ∈ (0 1] to be chosen later. We have |hβ (x − y) − hβ (x) + hβ (x)y| ≤ hβ (x)y 2 as soon as |y| ≤ 1, so the fact that both f = h˘ α and f = hα are even functions gives
σy σy dy − hβ (x) dy hβ (x − y)f f θu θu {|y|≤η} {|y|≤η}
σy dy ≤ hβ (x) y 2 f θu {|y|≤η} On the one hand we have with f = h˘ α or f = hα , and in view of (G.4),
3 3
σy 2 dy = θ u y f z 2 |f (z)| dz ≤ Ku1+α η2−α 3 θu σ {|y|≤η} {|z|≤ση/θu} On the other hand, the integrability of hα and h˘ α , and the fact that h˘ α (y) dy = 0 yield
σy θu θu dy = (1 + φ(u)) hα hα (z) dz = θu σ {|z|≤σ/θu} σ {|y|≤1}
θu σy ˘ hα h˘ α (z) dz dy = θu σ {|z|≤ση/θu} {|y|≤η}
θu h˘ α (z) dz =− σ {|z|>ση/θu} = 2cα
(θu)1+α (1 + φη (u)) σ 1+α ηα
Putting all these facts together yields
θu σy (G.9) dy − h h (x − y)h (x) β α β θu σ {|y|≤1} ≤ uφ(u)hβ (x) + Ku1+α hβ (x) and (G.10)
σy (θu)1+α ˘ dy − 2cα 1+α α hβ (x) hβ (x − y)hα θu σ η {|y|≤η}
φη (u) 1+α 2−α ≤u hβ (x) + Khβ (x)η ηα
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For the integrals on {|y| > η} we observe that by (G.4) we have hα (σy/θu) − cα (θu/σ|y|)1+α ≤ (θu/σy)1+α φ(u) and the same for h˘ α except that cα is substituted with −αcα . Given the definition of Dη we readily get
σy (θu)1+α (G.11) dy − cα hβ (x − y)hα D1 (x) 1+α θu σ {|y|>1}
≤ D1 (x)u1+α φ(u) σy (θu)1+α ˘ dy + αcα hβ (x − y)hα Dη (x) 1+α θu σ {|y|>η} ≤ Dη (x)u1+α φ(u)
and if we put together (G.9) and (G.11), we obtain (G.7). Finally, we prove (G.8). We split the integral in (G.6) into the sum of the integrals, say Dη(1) and Dη(2) , over the two domains {|y| > η |y − x| ≤ |x|γ } and {|y| > η |y − x| > |x|γ }, where γ = 4/5 if β = 2 and γ ∈ ((1 + α)/(1 + β) 1) if β < 2. On the one hand, Dη(2) (x) ≤ Kη hβ (|x|γ ), so with our choice of γ we obviously have |x|1+α Dη(2) (x) → 0. On the other hand, |x|1+α Dη(1) (x) is clearly equivalent, as |x| → ∞, to {|y−x|≤|x|γ } hβ (x − y) dy, which equals {|z|≤|x|γ } hβ (z) dz, which in turn goes to 1. Hence we get (G.8) for all η > 0. This concludes the proof of the lemma. Q.E.D. Using the lemma, we can now obtain the behavior of Ru and Su as u → 0. First, an application of (G.4) to the last formula in (G.3) and Lebesgue’s theorem readily give (G.12)
lim Su (x) = hβ (x) u→0
Also, by (G.4), (G.5), (G.7), and (G.8), we get ⎧ C ⎪ ⎪ if β < 2, ⎨ 1 + |x|1+β
(G.13) Su (x) ≥ ⎪ uα ⎪ ⎩ C e−x2 /2 + if β = 2, 1 + |x|1+α for some C > 0 depending on the parameters and all u small enough. In the same way, we see that uα Ru (x) → 0, but this not enough. However, if rη = 2hβ /ηα − αDη , we deduce from (G.7) that for all η ∈ (0 1], (G.14)
lim sup |Ru (x) − cα rη (x)| ≤ Khβ (x)η2−α u→0
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A simple computation and the second order Taylor expansion with integral remainder for hβ yield
rη (x) = α
hβ (x) − hβ (x − y) dy |y|1+α
1 |y|1−α dy (1 − v)hβ (x − yv) dv
{|y|>η}
= −α
{|y|>η}
0
By Lebesgue’s theorem, rη converges pointwise as η → 0 to the function r given by
r(x) = −α
|y|
1−α
R
1
dy
(1 − v)hβ (x − yv) dv
0
Then, taking into account (G.14) and using once more (G.7) together with (G.4), (G.5), and (G.8), and also α < β, we get (G.15)
lim Ru (x) = cα r(x) u→0
|Ru (x)| ≤
K 1 + |x|1+α
G.2. The Case (b) in Theorem 7 We are now in a position to prove (4.12), where β < 2 and α > β/2 (and of course α < β). Since α > β/2, we see that (G.12), (G.13), and (G.15) allow us to apply Lebesgue’s theorem in the definition (G.3) to get that Ju → dx (cα r(x))2 / hβ (x): this is (4.12) (obviously |r(x)| ≤ K/(1 + |x|1+α ), while hβ (x) > C/(1 + |x|1+β ) for some C > 0; hence the integral in (4.12) converges). G.3. The Other Cases The other cases are a bit more involved, because Lebesgue’s theorem does not apply and we will see that Ju goes to infinity. First, we introduce the functions ⎧ cβ cα θα uα ⎪ ⎪ + ⎪ ⎨ |x|1+β σ α |x|1+α if β < 2, αcα R (x) = 1+α Su (x) = 2 ⎪ |x| e−x /2 cα θα uα ⎪ ⎪ + α 1+α if β = 2. √ ⎩ σ |x| 2π Below we denote by ψ(u Γ ) for u ∈ (0 1] and Γ ≥ 1 the sum φ (u) + φ (1/Γ ) for any two functions φ and φ changing from line to line, that are continuous
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on R+ with φ (0) = φ (0) = 0. We deduce from (G.4), (G.5), (G.7) for η = 1, and (G.8) that (G.16)
|x| > Γ
⇒
|Ru (x) + R (x)| ≤ ψ(u Γ )R (x) +
⎧ ⎨ ⎩
K |x|1+β
if β < 2, 2
Kx2 e−x /3
if β = 2
In a similar way, we get |x| > Γ ⇒
u
|Su (x) − S (x)| ≤
ψ(u Γ )Su (x)
if β < 2, −x2 /3
ψ(u Γ )Su (x) + K0 uα x2 e
if β = 2
where K0 is some constant. Then for any ϕ > 0, we denote by Γϕ the smallest α 2 αθ number bigger than 1, such that K0 x2 e−x /3 ≤ ϕ σ αc|x| 1+α for all |x| > Γϕ . The last estimate above for β = 2 reads as |Su − Su | ≤ Su (ψ(u Γ ) + ϕ), so in all cases we have, for some fixed function ψ0 as above, (G.17)
Su (x) = Su (x)(1 + ρu (x))
where
|ρu (x)| ≤
ψ0 (u Γ )
if β < 2 |x| > Γ ,
ψ0 (u Γ ) + ϕ if β = 2 |x| > Γ > Γ0
At this stage, we set
R (x)2 (G.18) JuΓ = dx {|x|>Γ } Su (x) Observe that Ju = JuΓ +
(1) = JuΓ
J
(4) JuΓ
(i) JuΓ , where
{|x|≤Γ }
{|x|>Γ }
(Ru (x) + R (x))2 dx Su (x)
(3) uΓ
i=1
Ru (x)2 dx Su (x)
(2) = JuΓ
4
R (x)(Ru (x) + R (x)) dx Su (x) {|x|>Γ }
R (x)2 R (x)2 dx − dx = {|x|>Γ } Su (x) {|x|>Γ } Su (x) = −2
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Y. AÏT-SAHALIA AND J. JACOD
In light of (G.13), (G.15), and (G.16), we get, for some u0 > 0, (1) < ∞ sup JuΓ
u∈(0u0 ]
⇒
u ∈ (0 u0 ]
(4) (2) JuΓ ≤ K + ψ(u Γ ) JuΓ + JuΓ
The Cauchy–Schwarz inequality yields (3) J ≤ 2 J (2) J (4) + JuΓ 1/2 uΓ uΓ uΓ Finally, (G.13) and (G.17), and the definition of R yield (with ψ0 as in (G.17)) (4) J uΓ if β < 2 ψ0 (u Γ ) < 1/2, 2ψ0 (u Γ )JuΓ ≤ 2(ψ0 (u Γ ) + ϕ)JuΓ if β = 2 Γ > Γϕ ψ0 (u Γ ) + ϕ < 1/2. Therefore, we get Ju ≤ K + ψ(u Γ ) + 2(ψ0 (u Γ ) + ϕ) − 1 J J uΓ uΓ as soon as ψ0 (u Γ ) + ϕ < 1/2 and Γ > Γϕ , and with the convention that ϕ = 0 and Γ0 = 1 when β < 2. Then, remembering that limu→0 limΓ →∞ ψ(u Γ ) = 0, and the same for ψ0 , and that ϕ = 0 when β < 2 and ϕ is arbitrarily small when β = 2, we readily deduce the following fact: suppose that for some function u → γ(u) going to +∞ as u → 0 and independent of Γ we have proved that (G.19)
JuΓ ∼ γ(u) as
u → 0 ∀Γ > 1
Then Ju ∼ γ(u) and, therefore, by (G.2) we get (G.20)
Iθθ ∼
θ2α−2 u2α γ(u) σ 2α
G.3.1. The Case (d) in Theorem 7: Now let β < 2 and α < β/2. The change of variables z = xuα/(β−α) in (G.18) yields
1 2 2 (G.21) JuΓ = α cα dx 1+2α−β + c θα uα |x|1+α /σ α c |x| α {|x|>Γ } β
1 α2 cα2 = (α(β−2α))/(β−α) dz 1+2α−β u + cα θα |z|1+α /σ α {|z|>Γ uα/(β−α) } cβ |z|
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and we have (G.19) with (G.22)
γ(u) =
α2 cα2 (α(β−2α))/(β−α)
u
cβ |z|
1+2α−β
1 dz + cα θα |z|1+α /σ α
So if we combine this with (G.20), we get (4.14). G.3.2. The Case (c) in Theorem 7: Suppose now that 2α = β < 2. Then
∞ 1 2 2 (G.23) JuΓ = 2α cα dx x(cβ + cα θα uα xα /σ α ) Γ For v > 0 we let Hv be the unique point x > 0 such that cα θα vα xα /σ α = cβ , so in fact Hv = ρ/v for some ρ > 0. We have
K ∞ 1 2α2 cα2 Hu 1 2α2 cα2 1 dx + α + K dx ≤ log JuΓ ≤ 1+α cβ x u Hu x cβ u Γ On the other hand, if μ > 0 and Γ < x < Huμ , we have cβ + cα θα uα xα /σ α < cβ (1 + 1/μα ), hence
2α2 cα2 1 1 2α2 cα2 Huϕ 1 JuΓ > dx ≥ − Kϕ log α α cβ x(1 + 1/μ ) cβ 1 + 1/μ u Γ Putting together these two estimates and choosing μ big gives (G.19) with 1 2α2 cα2 log γ(u) = cβ u and we readily deduce (4.13). G.3.3. The Case (a) in Theorem 7: Finally, consider the case β = 2. We have
∞ 1 (G.24) JuΓ = 2α2 cα2 dx √ 2 x1+α (x1+α e−x /2 / 2π + cα θα uα /σ α ) Γ 2
Suppose that Γ is large enough for x → x2+2α e−x /2 to be decreasing on [Γ ∞). For v > 0 small enough, √ there is a uniquenumber Hv = x > Γ such 2 that cα θα vα /σ α = x1+α e−x /2 / 2π, so in fact Hv ∼ 2α log(1/v) when v → 0. Then
Hu x2 /2 e 2α2 cα σ α ∞ 1 (G.25) JuΓ ≤ K dx + dx 1+α x2+2α θα uα Γ Hu x ≤
K 2αcα σ α 2αcα σ α + ≤ (1 + o(1)) Huα θα uα Huα θα uα (2α)α/2 (log(1/u))α/2
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√ 2 On the other hand, if μ > 1 and x > Huμ , we have x1+α e−x /2 / 2π + cα θα uα /σ α < +cα θα uα (1 + μα )/σ α , hence,
1 2α2 cα σ α Huϕ (G.26) JuΓ > dx α α 1+α θ u x (1 + μα ) Γ ≥
2αcα σ α (1 + o(1)) θα uα (2α)α/2 (log(1/u))α/2 (1 + μα )
So again we see, by choosing μ close to 1, that the desired result holds with (G.27)
γ(u) =
2αcα σ α θα uα (2α)α/2 (log(1/u))α/2
and we deduce (4.11). REFERENCES AÏT-SAHALIA, Y. (2004): “Disentangling Diffusion From Jumps,” Journal of Financial Economics, 74, 487–528. [729] AÏT-SAHALIA, Y., AND J. JACOD (2007): “Volatility Estimators for Discretely Sampled Lévy Processes,” The Annals of Statistics, 35, 355–392. [729,736,742,745] ANDERSEN, T. G., T. BOLLERSLEV, F. X. DIEBOLD, AND P. LABYS (2003): “Modeling and Forecasting Realized Volatility,” Econometrica, 71, 579–625. [729] BARNDORFF-NIELSEN, O. E., AND N. SHEPHARD (2004): “Power and Bipower Variation With Stochastic Volatility and Jumps” (With Discussion), Journal of Financial Econometrics, 2, 1–48. [729] BERGSTRØM, H. (1952): “On Some Expansions of Stable Distribution Functions,” Arkiv für Mathematik, 2, 375–378. [732] BROCKWELL, P., AND B. BROWN (1980): “High-Efficiency Estimation for the Positive Stable Laws,” Journal of the American Statistical Association, 76, 626–631. [731] DUMOUCHEL, W. H. (1971): “Stable Distributions in Statistical Inference,” Ph.D. Thesis, Department of Statistics, Yale University. [731] (1973a): “On the Asymptotic Normality of the Maximum-Likelihood Estimator When Sampling From a Stable Distribution,” The Annals of Statistics, 1, 948–957. [733,737] (1973b): “Stable Distributions in Statistical Inference: 1. Symmetric Stable Distributions Compared to Other Symmetric Long-Tailed Distributions,” Journal of the American Statistical Association, 68, 469–477. [731] (1975): “Stable Distributions in Statistical Inference: 2. Information From Stably Distributed Samples,” Journal of the American Statistical Association, 70, 386–393. [731] HOFFMANN-JØRGENSEN, J. (1993): “Stable Densities,” Theory of Probability and Its Applications, 38, 350–355. [731] HÖPFNER, R., AND J. JACOD (1994): “Some Remarks on the Joint Estimation of the Index and the Scale Parameters for Stable Processes,” in Proceedings of the Fifth Prague Symposium on Asymptotic Statistics, ed. by P. Mandl and M. Huskova. Heidelberg: Physica-Verlag, 273–284. [737] JACOD, J., AND A. N. SHIRYAEV (2003): Limit Theorems for Stochastic Processes (Second Ed.). New York: Springer-Verlag. [732] MANCINI, C. (2001): “Disentangling the Jumps of the Diffusion in a Geometric Jumping Brownian Motion,” Giornale dell’Istituto Italiano degli Attuari, LXIV, 19–47. [729]
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MYKLAND, P. A., AND L. ZHANG (2006): “ANOVA for Diffusions and Itô, Processes,” The Annals of Statistics, 34, 1931–1963. [729] NOLAN, J. P. (1997): “Numerical Computation of Stable Densities and Distribution Functions,” Communications in Statistics—Stochastic Models, 13, 759–774. [731] (2001): “Maximum Likelihood Estimation and Diagnostics for Stable Distributions,” in Lévy Processes: Theory and Applications, ed. by O. E. Barndorff-Nielsen, T. Mikosch, and S. I. Resnick. Boston: Birkhäuser, 379–400. [731] RACHEV, S., AND S. MITTNIK (2000): Stable Paretian Models in Finance. Chichester, U.K.: Wiley. [728] WOERNER, J. H. (2004): “Estimating the Skewness in Discretely Observed Lévy Processes,” Econometric Theory, 20, 927–942. [728] ZOLOTAREV, V. M. (1986): One-Dimensional Stable Distributions, Vol. 65. Providence, RI: Translations of Mathematical Monographs, American Mathematical Society. [732] (1995): “On Representation of Densities of Stable Laws by Special Functions,” Theory of Probability and Its Applications, 39, 354–362. [731]
Econometrica, Vol. 76, No. 4 (July, 2008), 763–814
ASYMPTOTIC PROPERTIES FOR A CLASS OF PARTIALLY IDENTIFIED MODELS BY ARIE BERESTEANU AND FRANCESCA MOLINARI1 We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval-identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between √ the population identification region and the estimator, when properly normalized by n converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests. KEYWORDS: Partial identification, confidence collections, set valued random variables, support functions.
1. INTRODUCTION THIS PAPER CONTRIBUTES to the growing literature on inference for partially identified population features. These features include vectors of parameters or 1 We are grateful to Larry Blume for introducing us to the theory of set valued random variables and for many useful conversations, and to Ilya Molchanov for extremely valuable suggestions. We thank the co-editor, Whitney Newey, and three anonymous reviewers for comments that greatly improved the paper. We benefited from comments by L. Barseghyan, V. Chernozhukov, A. Guerdjikova, B. Hansen, S. Hoderlein, H. Hong, B. Honoré, J. Horowitz, N. Kiefer, E. Mammen, C. Manski, M. Marinacci, K. Mosler, U. Müller, M. Nielsen, S. Resnick, G. Samorodnitsky, E. Tamer, T. Tatur, T. Vogelsang, M. Wells, seminar participants at Columbia, Cornell (Economics and Statistics), Duke, Harvard–MIT, Mannheim, NYU, Princeton, Syracuse, Torino, UCL, Vanderbilt, Wisconsin, Yale, Northwestern University’s 2005 Conference on Incomplete Econometric Models, the 2005 Triangle Econometrics Conference, and the 2006 NY Econometrics Camp. We thank Jenny Law, Liyan Yang, and especially Bikramjit Das for able research assistance. Beresteanu gratefully acknowledges financial support from the Micro Incentives Research Center at Duke University and from NSF Grant SES-0617559. Molinari gratefully acknowledges financial support from the Center for Analytic Economics at Cornell University and from NSF Grant SES-0617482.
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of statistical functionals characterizing a population probability distribution of interest for which the sampling process and the maintained assumptions reveal that they lie in a set—their identification region—which is not necessarily a singleton (Manski (2003)). Much of this literature focuses on cases in which the population identification region is an interval on the real line (e.g., Manski (1989)) or can be defined as the set of minimizers of a criterion function (e.g., Chernozhukov, Hong, and Tamer (2007)). In these cases the analogy principle is applied to replace either the extreme points of the interval or the criterion function with their sample analogs to obtain an estimator for the identification region. Limit theorems for sequences of scalar or vector valued random variables are then employed to establish consistency of these estimators and to construct confidence sets that asymptotically cover each point in the identification region (Imbens and Manski (2004)) or the entire identification region (Chernozhukov, Hong, and Tamer (2007)) with at least a prespecified probability. In this paper, we introduce a novel approach for estimation and inference for a certain class of partially identified population features. The key insight that leads to our approach is the observation that, within this class, the compact and convex identification region2 of the vector of parameters (or statistical functionals) of interest is given by the “expectation” of a measurable mapping that associates a set (rather than a real number or a real vector) with each element of the sample space. In the mathematics literature, this measurable mapping is called a set valued random variable (SVRV). Just as one can think of the identification region of a parameter vector as a set of parameter vectors, one can think of an SVRV as a set of random variables (Aumann (1965)). We extend the analogy principle to SVRVs and estimate the identification region, which is the “expectation” of an SVRV, by its sample analog, which is a “sample average” of SVRVs. The expressions “expectation” and “sample average” used above are in quotation marks because when working with SVRVs, a particular expectation operator needs to be used—the Aumann expectation; similarly, a particular summation operator needs to be used—the Minkowski summation— so as to get the set analog of the sample average, which is called the Minkowski sample average.3 Approaching the problem from this perspective is beneficial because it allows the researcher to perform, in the space of sets, operations which are analogs to those widely used in the space of vectors. Moreover, convex compact sets can be represented as elements of a functional space through their support function (Rockafellar (1970, Chap. 13)). The support function of a compact convex SVRV is a continuous valued random variable indexed on the unit sphere, and the support function of a Minkowski sample average of SVRVs is the sample average of the support functions of the SVRVs. Hence 2
When the population identification region is not a convex set, our analysis applies to its convex hull. 3 These concepts are formalized in Section 2.
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we can derive the asymptotic properties of set estimators using laws of large numbers and central limit theorems for stochastic processes that are of familiar use in econometrics. This enables us to conduct inference for the entire identification region of a population feature of interest and for its subsets in a way that is completely analogous to how inference would be conducted if this feature were point identified. Overview Our methodology easily applies to the entire class of interval-identified scalar parameters. Moreover, it applies to the class of partially identified population features which have a compact and convex identification region that is equal to a transformation of the Aumann expectation of a properly defined SVRV. Examples of population features in this class include means and best linear predictors with interval outcome data, and parameter vectors of semiparametric binary models with interval regressor data (under the assumptions of Magnac and Maurin (2008)). The SVRVs whose Aumann expectation is equal to the identification region of the population feature of interest can be constructed from observable random variables. The mathematics literature on SVRVs provides laws of large numbers and central limit theorems for the Hausdorff distance between the Minkowski average of a random sample of SVRVs and their Aumann expectation. These limit theorems are obtained by exploiting the isometric embedding of the family of convex compact subsets of d with the Hausdorff metric into the Banach space of continuous functions on the unit sphere with the uniform metric, which is provided by the support function. This allows one to represent random sets as elements of a functional space, for which limit theorems are available. Using these results, we show that our estimator of the identification region, which is given by a√transformation of the Minkowski average of a random sample of SVRVs, is n-consistent, in the sense that the Hausdorff distance between the estimated set √ and the population identification region converges to zero at the rate Op (1/ n) This result does not depend on whether the random variables used to construct the SVRVs have a continuous or a discrete distribution.4 We then introduce two Wald-type statistics which, respectively, allow one to test (i) hypothesis about the entire identification region and (ii) its subsets. The first statistic is based on the Hausdorff distance. Because the Hausdorff distance between two nonempty compact sets A and B is equal to zero if and only if A = B, this distance is ideal to test a hypothesis which specifies the entire 4
The choice of the Hausdorff distance to establish consistency results is a natural one, as this distance is a generalization of the Euclidean distance, and it is widely used in the literature on estimation of partially identified models (e.g., Manski and Tamer (2002)).
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identification region of a vector of parameters.5 When the Hausdorff distance between our√estimator and the population identification region is properly normalized by n, it converges in distribution to a function of a Gaussian process whose covariance kernel depends on parameters of the population identification region which can be consistently estimated. Hence, our first test statistic is given by the Hausdorff distance between the estimated set and the hypothesized population identification region. The null hypothesis is rejected if this statistic exceeds a critical value. We show that this critical value can be consistently estimated using bootstrap methods. The second statistic is based on the directed Hausdorff distance. Because the directed Hausdorff distance from a nonempty compact set A to a nonempty compact set B is equal to zero if and only if A ⊆ B, this distance is ideal to test an inclusion hypothesis which specifies a subset of the identification region. √ Similarly to the test based on the Hausdorff distance, normalization by n of the directed Hausdorff distance from the population identification region to the estimator converges in distribution to a function of a Gaussian process. Hence, our second test statistic is given by the directed Hausdorff distance from the hypothesized subset of the population identification region to the estimator. Similarly to the previous case, the null hypothesis is rejected if this test statistic exceeds a critical value that can be consistently estimated using the bootstrap. We show that the tests that we propose are consistent against any fixed alternative. We then extend the notion of local alternatives to partially identified models, derive the asymptotic distribution of our tests against these local alternatives, and show that the tests are locally asymptotically unbiased. Our tests can be inverted to yield confidence statements about the true identification region or its subsets. In the case of inversion of the test based on the Hausdorff distance, we obtain what we call a confidence collection which is given by the collection of all sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test. Its main property is that (asymptotically) the population identification region is one of its elements with a prespecified confidence level (1 − α). In the case of inversion of the test based on the directed Hausdorff distance, we obtain what we call a directed confidence collection which is given by the collection of all sets that, when specified as a null hypothesis for a subset of values of the population identification region, cannot be rejected by our test. Also in this case the population identification region is (asymptotically) one of its elements with a prespecified confidence level (1 − α) Additionally, the union of the sets in the directed confidence collection has the property of covering (asymptotically) the true identification region with a prespecified confidence level (1 − α). This result establishes a clear connection between our Wald-type 5 This statistic can also be used to test hypotheses which specify the entire identification region of linear combinations of the partially identified population features.
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test statistic based on the directed Hausdorff distance and the quasi-likelihood ratio (QLR)-type inferential statistic proposed by Chernozhukov, Hong, and Tamer (2007), which allows one to construct confidence sets which asymptotically cover the true identification region with confidence level (1 − α). In the special case of interval identified scalar parameters, we establish the asymptotic equivalence between the square of our test statistic based on the directed Hausdorff distance, and the inferential statistic of Chernozhukov, Hong, and Tamer (2007). Hence, for the class of problems addressed in this paper, there is a complete analogy, at the level of estimation and inference, between the approach usually adopted for point-identified parameters and the approach that we propose for partially identified parameters. In particular, when point identified, the parameters of interest can be consistently estimated using a transformed sample average of the data. The resulting estimator has an asymptotically normal distribution. The confidence region for the parameter vector is given by a collection of vectors—that is, a collection of points in the relevant space—and can be obtained through the inversion of a properly specified test statistic. In the partially identified case, our results show that the identification region of each of these parameter vectors can be consistently estimated using a transformed Minkowski average of the data. The Hausdorff distance (and the directed Hausdorff distance) between the population identification region and its estimator has an asymptotic distribution which is a function of a Gaussian process. The confidence region for the population identification region is given by a collection of sets (rather than points) and can be obtained through the inversion of the test statistics that we propose. Our inferential approach targets the entire identification region of a partially identified population feature, and provides asymptotically exact size critical values with which to test hypotheses and construct confidence collections. However, there are applications in which the researcher wants to test hypotheses and construct confidence sets for the “true” value of the population feature of interest, following the insight of Imbens and Manski (2004). For this case, our methodology based on the directed Hausdorff distance provides conservative confidence sets that asymptotically cover each point in the identification region with a prespecified probability. Structure of the Paper In Section 2 we propose our test statistics, establish their properties, provide a consistent bootstrap procedure to estimate their limiting distributions, show how the test statistics can be inverted to obtain the confidence collection and the directed confidence collection, and provide a simple characterization of the collections. In Section 3 we apply our results to the problem of inference for interval-identified scalar parameters and give a step-by-step description of how our method can be implemented in practice. In Section 4 we apply our
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results to conduct inference for the best linear predictor parameters when only interval data are available for the outcome variable, but the covariates are perfectly observed. We show how to test linear restrictions on these parameters and we prove that the estimator of the identification region for a single component of this parameter vector can be computed using standard statistical packages. Section 5 presents Monte Carlo results evaluating the finite sample performance of our estimators and test statistics, and Section 6 concludes. An Appendix contains all the proofs. Related Literature Consistent estimators for specific partially identified population features have been proposed, among others, by Manski (1989), Horowitz and Manski (1997, 1998, 2000), Manski and Tamer (2002), Chernozhukov, Hong, and Tamer (2002, 2007), Honore and Tamer (2006), Andrews, Berry, and Jia (2004), and Chernozhukov, Hahn, and Newey (2005). The development of methodologies that allow for the construction of confidence regions for partially identified population features is a topic of current research. Horowitz and Manski (1998, 2000) considered the case in which the identification region of the parameter of interest is an interval whose lower and upper bounds can be estimated from sample data, and proposed confidence intervals that asymptotically cover the entire identification region with fixed probability. For the same class of problems, Imbens and Manski (2004) suggested shorter confidence intervals that (asymptotically) uniformly cover each point in the identification region, rather than the entire region, with at least a prespecified probability 1 − α. Chernozhukov, Hong, and Tamer (2002) were the first to address the problem of construction of confidence sets for identification regions of parameters obtained as the solution of the minimization of a criterion function. They provided methods to construct confidence sets that cover the entire identification region with probability asymptotically equal to 1 − α, as well as confidence sets that asymptotically cover each point in the identification region with at least probability 1 − α They also developed resampling methods to implement these procedures. Romano and Shaikh (2006) analyzed the same problem and proposed various subsampling procedures. Andrews, Berry, and Jia (2004) considered economic models of entry in which the equilibrium conditions place a set of inequality restrictions on the parameters of the model. These restrictions may only allow the researcher to identify a set of parameter values consistent with the observable data. They suggested a procedure to obtain confidence regions that asymptotically cover the identified set with at least probability 1 − α by looking directly at the distribution of the inequality constraints. Pakes, Porter, Ho, and Ishii (2005) considered single agent and multiple agent structural models in which again equilibrium conditions impose moment inequality restrictions on the parameters of interest. They suggested a conservative specification test for the value of the estimated parameters. Rosen
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(2006) considered related models in which the parameter of interest is partially identified by a finite number of moment inequalities. He established a connection between these models and the literature on multivariate one-sided hypothesis tests and showed that for this class of models, conservative confidence sets can be constructed by inverting a statistic that tests the hypothesis that a given element of the parameter space satisfies the restrictions of the model. Guggenberger, Hahn, and Kim (2006) further explored specification tests for parameter vectors obtained as the solution of moment inequalities. Molinari (2008b) considered misclassification models in which the identification region for the distribution of the misclassified variable can be calculated using a nonlinear programming estimator, the constraints of which depend on parameters that can be consistently estimated. She proposed conservative confidence sets given by the union of the identification regions obtained by replacing the estimated parameters with the elements of their Wald confidence ellipsoid. Galichon and Henry (2006) proposed a specification test for partially identified structural models based on an extension of the Kolmogorov–Smirnov test for Choquet capacities. 2. HYPOTHESIS TESTING AND CONFIDENCE COLLECTIONS 2.1. Preliminaries We begin this section with some preliminary definitions that prove useful in what follows.6 Throughout the paper (with a few exceptions), we reserve the use of capital Latin letters to sets and SVRVs; we use lowercase Latin letters for random variables and boldface lowercase Latin letters for random vectors. We denote sets of parameters by capital Greek letters, scalar valued parameters by lowercase Greek letters, and vector valued parameters by boldface lowercase Greek letters. We denote by · the Euclidean norm, by · · the inner product in d by Sd−1 = {p ∈ d : p = 1} the unit sphere in d , and by C(Sd−1 ) the set of continuous functions from Sd−1 to . For h ∈ C(Sd−1 ), we let hC(Sd−1 ) = supp∈Sd−1 |h(p)| be the C(Sd−1 ) norm. We denote (g)+ ≡ max(0 g) and (g)− ≡ max(0 −g). Set Valued Random Variables, Distance Functions, and Support Functions Let (Ω A μ) be a probability space. Let d denote the Euclidean space, equipped with the Euclidean norm, and let K(d ) be the collection of all nonempty closed subsets of d .7 Denote by Kk (d ) the set of nonempty compact 6
Beresteanu and Molinari (2006) provided a succinct presentation of the theory of SVRVs, following closely the treatment in Molchanov (2005) (limit theorems for SVRVs are also discussed in Li, Ogura, and Kreinovich (2002)). The first self-contained treatment of the mathematical theory of SVRVs was given by Matheron (1975). 7 The theory of SVRVs generally applies to K(X), the space of closed subsets of a Banach space X (e.g., Molchanov (2005)). For the purposes of this paper it suffices to consider X = d , which simplifies the exposition.
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subsets of d , and by Kkc (d ) the set of nonempty compact and convex subsets of d . An SVRV is a measurable mapping F : Ω → K(d ) that associates a set to each point in the sample space. We can think of an SVRV as a bundle of random variables—its selections. A formal definition follows. DEFINITION 1: A map F : Ω → K(d ) is called a set-valued random variable (SVRV) if for each closed subset C of d , F −1 (C) = {ω ∈ Ω : F(ω) ∩ C = ∅} ∈ A. For any SVRV F , a (measurable) selection of F is a random vector f (taking values in d ) such that f(ω) ∈ F(ω) μ-a.s. We denote by S (F) the set of all selections from F . To measure the distance from one set to another, the distance between sets, and the norm of a set, we use the following: DEFINITION 2: Let A and B be two subsets of d . (a) The directed Hausdorff distance from A to B (or lower Hausdorff hemimetric) is denoted dH (A B) = sup inf a − b a∈A b∈B
(b) The Hausdorff distance8 between A and B (or Hausdorff metric) is denoted H(A B) = max{dH (A B) dH (B A)} (c) The Hausdorff norm of a set is denoted AH = H(A {0}) = sup{a : a ∈ A} To represent sets as elements of a functional space, we use the support function: DEFINITION 3: Let F ∈ K(d ). Then the support function of F at p ∈ d , denoted s(p F), is given by s(p F) = supf∈F p f. Expectation of SVRVs Denote by L1 = L1 (Ω d ) the space of A-measurable random variables with values in d such that the L1 -norm ξ1 = E[ξ] is finite. For an SVRV F defined on (Ω A) the family of all integrable selections of F is given by 8 If A and B are unbounded subsets of d , H(A B) may be infinite. However, (Kk (d ), H(· ·)) is a complete and separable metric space (Li, Ogura, and Kreinovich (2002, Theorems 1.1.2 and 1.1.3)). The same conclusion holds for (Kkc (d ) H(· ·)).
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S 1 (F) = S (F) ∩ L1 . Below we define integrably bounded SVRVs, and we define the Aumann expectation (Aumann (1965)) of an SVRV F , denoted E[F]. We reserve the notation E[·] for the expectation of random variables and random vectors. DEFINITION 4: An SVRV F : Ω → K(d ) is called integrably bounded if FH = sup{f : f ∈ F} has a finite expectation. DEFINITION 5: The Aumann expectation of an SVRV F is given by 1 E[F] = f dμ : f ∈ S (F) Ω
where
Ω
f dμ is taken coordinatewise. If F is integrably bounded, then E[F] = f dμ : f ∈ S (F) Ω
Clearly, since S (F) is nonempty (Aumann (1965); see also Li, Ogura, and Kreinovich (2002, Theorem 1.2.6)), the Aumann expectation of an integrably bounded SVRV is nonempty. Moreover, if F is an integrably bounded random compact set on a nonatomic probability space or if F is an integrably bounded random convex compact set, then E[F] is convex and coincides with E[co F], and E[s(p F)] = s(p E[F]) (Artstein (1974)). Linear Transformations Sections 3 and 4 discuss examples of partially identified parameters whose identification region is given by an Aumann expectation of SVRVs that can be constructed from observable random variables. Below we introduce novel general procedures based on the Hausdorff distance and on the directed Hausdorff distance to conduct inference for these identification regions and their subsets. Before getting to the details of our procedure, we observe that given a nonrandom finite matrix R of dimension l × d, if {F Fi : i ∈ N} are independent and identically distributed (i.i.d.) nonempty, compact valued SVRVs in Kk (d ) with E[F2H ] < ∞ and RF = {t ∈ l : t = Rf f ∈ S (F)}, then {RF RFi : i ∈ N} are i.i.d. nonempty, compact valued SVRVs in Kk (l ) with E[RF2H ] < ∞. It then follows from Theorem A.2 and Lemma A.1 in the Appendix that n √ 1 d nH RFi E[RF] → z R C(Sl−1 ) n i=1 n √ 1 d ndH RFi E[RF] → sup {z R (p)}+ n i=1 p∈Sl−1
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n √ 1 d ndH E[RF] RFi → sup {−z R (p)}+ n i=1 p∈Sl−1 where z R is a Gaussian random system with E[z R (p)] = 0 for all p ∈ Sl−1 and E[z R (p)z R (q)] = E[s(R p F)s(R q F)] − E[s(R p F)]E[s(R q F)] for all p q ∈ Sl−1 . In particular, R can be chosen to select specific components of the partially identified parameter vector (or linear functions of such components) so as to conduct inference on them. Below, for simplicity, we assume that R = I (the identity matrix). The results for R = I involve a straightforward modification. 2.2. Inference Based on the Hausdorff Distance Hypothesis Testing Given a nonempty, compact, and convex (nonrandom) set Ψ0 , to test the hypothesis H0 : E[F] = Ψ0
HA : E[F] = Ψ0
at a prespecified significance level α ∈ (0 1), we propose the following criterion: n √ 1 co Fi Ψ0 > cα reject H0 if nH n i=1 n √ 1 co Fi Ψ0 ≤ cα do not reject H0 if nH n i=1 where cα is chosen so that
Pr zC(Sd−1 ) > cα = α (2.1) and z is a Gaussian random system with E[z(p)] = 0 for all p ∈ Sd−1 and E[z(p)z(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . Observe that the test statistic uses the Hausdorff distance between Ψ0 and n n 1 1 co F , rather than the Hausdorff distance between Ψ and i 0 i=1 i=1 Fi . n n n Using co F to construct the test statistic greatly simplifies the compui i=1 tations, because the calculation of the Hausdorff distance between convex sets has been widely studied in computational geometry, and fast algorithms are easily implementable. At the n nsame time, by Shapley–Folkman’s theorem (Starr (1969)), H( n1 i=1 co Fi n1 i=1 Fi ) goes to zero9 at the rate n1 . Hence, by Theo n rem A.1 in the Appendix, n1 i=1 co Fi is a consistent estimator of E[F], and by 9 Minkowski averaging SVRVs which are compact valued but not necessarily convex is asymptotically “convexifying,” as noted by Artstein and Vitale (1975).
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n n √ √ Theorem A.2, nH( n1 i=1 co Fi E[F]) and nH( n1 i=1 Fi E[F]) have the n same asymptotic distribution. In what follows, we use F¯n to denote n1 i=1 co Fi . Since the limiting distribution of our test statistics depends on parameters to be estimated, we obtain the critical values using the following empirical bootstrap procedure: ALGORITHM 2.1: 1. Generate a bootstrap sample of size n, {Fi∗ : i = 1 n}, by drawing a random sample from the joint empirical distribution of {Fi : i = 1 n} with replacement. 2. Compute √ (2.2) rn∗ ≡ nH(F¯n∗ F¯n ) 3. Use the results of b repetitions of Steps 1 and 2 to compute the empirical distribution function of rn∗ at a point t, denoted by Jn (t). 4. Estimate the quantile cα defined in equation (2.1) by (2.3)
cˆαn = inf{t : Jn (t) ≥ 1 − α}
The results of Giné and Zinn (1990), along with an application of the continuous mapping theorem, guarantee the validity of this bootstrap procedure. In particular, the following result holds: PROPOSITION 2.1: Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued d SVRVs such that E[F2H ] < ∞. Then rn∗ → zC(Sd−1 ) , where rn∗ is defined in equation (2.2) and z is a Gaussian random system with E[z(p)] = 0 for all p ∈ Sd−1 and with covariance kernel E[z(p)z(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . If in addition Var(z(p)) > 0 for each p ∈ Sd−1 , then cˆαn = cα + op (1), where cˆαn is defined in (2.3). A consistent estimator of the critical value cα can be alternatively obtained using the following procedure. Simulate the distribution of the supremum of a Gaussian random system with mean function equal to zero for each p ∈ Sd−1 and covariance kernel equal to 1 1 1 s(p Fi )s(q Fi ) − s(p Fi ) s(q Fi ) γ(p ˆ q) = n i=1 n i=1 n i=1 n
n
n
for all p q ∈ Sd−1 . Observe that the supremum is to be taken over p ∈ Sd−1 , hence over a known set (the unit sphere) which does not need to be estimated. It then follows from the strong law of large numbers in Banach spaces of Mourier (1955) that as
γ(· ˆ ·) → E[s(· F)s(· F)] − E[s(· F)]E[s(· F)]
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By the same argument as in the proof of Proposition 2.1, if Var(z(p)) > 0 for each p ∈ Sd−1 , the critical values of the simulated distribution consistently estimate the critical values of zC(Sd−1 ) . We now show that our test is consistent against any fixed alternative hypothesis in HA . THEOREM 2.2: Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued SVRVs such that E[F2H ] < ∞, and let ΨA be a nonempty, compact, and convex valued set such that E[F] = ΨA = Ψ0 . Let Var(z(p)) > 0 for each p ∈ Sd−1 . Then √ lim Pr{ nH(F¯n Ψ0 ) > cˆαn } = 1 n→∞
We conclude the discussion of our test by determining its power against lo√ cal alternatives at distance (proportional to) 1/ n from the null hypothesis. Suppose we are interested in the power of our test of H0 against a sequence of nonempty, compact, and convex alternative sets {ΨAn }, (2.4)
ΨAn ∈ Kkc (d ) :
1 1 ΨAn ⊕ √ Δ1 = Ψ0 ⊕ √ Δ2 n n
where Δ1 and Δ2 are nonrandom nonempty, compact, and convex sets for which there exists a nonempty, compact, and convex set Δ3 such that Ψ0 = Δ1 ⊕ Δ3 . Let κ ≡ H(Δ1 Δ2 ) < ∞ and observe that by the properties of the Hausdorff distance between two sets10
√ √ 1 1 (2.5) nH(ΨAn Ψ0 ) = nH ΨAn ⊕ √ Δ1 Ψ0 ⊕ √ Δ1 = κ n n Clearly, for larger values of κ, the local alternatives get farther away from the null, with a resulting increase in the power of the test. This choice of local alternatives allows us to consider a large class of deviations from the null hypothesis. It encompasses the case in which Δ1 = {0}, Δ2 = {0}, and the sets ΨAn shrink (i.e., the null is a subset of the true identification region) and/or shift to become equal to Ψ0 , and the case in which Δ1 = {0}, Δ2 = {0}, and the sets ΨAn enlarge (i.e., the null is a superset of the true identification region) and/or shift to become equal to Ψ0 . The following theorem gives the asymptotic distribution of our test under these local alternatives and establishes its local asymptotic unbiasedness. THEOREM 2.3: Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued SVRVs such that E[F2H ] < ∞, let Ψ0 be a nonempty, compact, and convex valued set, and let {ΨAn } be the sequence of sets defined in (2.4). Then √ d nH(F¯n Ψ0 ) → wC(Sd−1 ) 10
See, for example, DeBlasi and Iervolino (1969).
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under ΨAn , where w is a Gaussian random system with E[w(p)] = s(p Δ2 ) − s(p Δ1 ) for all p ∈ Sd−1 and E[w(p)w(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . Moreover, under ΨAn the test is asymptotically locally unbiased, that is, √ lim Pr{ nH(F¯n Ψ0 ) > cˆαn } ≥ α
n→∞
Confidence Collections The “confidence collection” is the collection of all sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test. We denote this collection by CCn1−α . The confidence collection is based on exploiting the duality between confidence regions and hypothesis tests that is of familiar use for point-identified models, and it has the property that asymptotically the set E[F] is a member of such collection with a prespecified confidence level 1 − α. Inverting the test statistic described above, we construct CCn1−α as √ CCn1−α = {Ψ˜ ∈ Kkc (d ) : nH(F¯n Ψ˜ ) ≤ cˆαn } where cˆαn is defined in equation (2.3). Hence if the law of zC(Sd−1 ) is continuous, lim Pr{E[F] ∈ CCn1−α } = 1 − α
n→∞
In practice, it can be difficult to characterize all the sets that belong to the confidence collection. However, the union of all sets in CCn1−α can be calculated in a particularly simple way. The following theorem shows how. THEOREM 2.4: Let Un = Then Un = F¯n ⊕ Bcˆαn .
˜ ˜ {Ψ : Ψ ∈ CCn1−α } and Bcˆαn = {b ∈ d : b ≤
cˆαn √ n
}.
An interesting consequence of Theorem 2.4 is that the union of all sets in CCn1−α is also included in CCn1−α and thus represents the largest set that cannot be rejected as a null hypothesis. 2.3. Inference Based on the Directed Hausdorff Distance Hypothesis Testing Suppose that a researcher wants to test whether a certain set of values is contained in the identification region. To accommodate this case, we propose
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A. BERESTEANU AND F. MOLINARI
the following hypothesis, which specifies a subset of the identification region11 : H0 : Ψ0 ⊆ E[F]
HA : Ψ0 E[F]
where Ψ0 is a nonempty, compact, and convex (nonrandom) set. We call this an inclusion test. Using the triangle inequality and the fact that under the null hypothesis dH (Ψ0 E[F]) = 0, we obtain dH (Ψ0 F¯n ) ≤ dH (Ψ0 E[F]) + dH (E[F] F¯n ) n 1 = sup − s(p co Fi ) − E[s(p F)] n i=1 p∈Sd−1 +
where the equality follows from Lemma A.1 in the Appendix. Under the same assumptions of Theorem A.2, it follows that (2.6)
√ d ndH (E[F] F¯n ) → sup {−z(p)}+ p∈Sd−1
where z is a Gaussian random system with E[z(p)] = 0 for all p ∈ Sd−1 and E[z(p)z(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . Hence we can use the criterion √ reject H0 if ndH (Ψ0 F¯n ) > c˜α √ do not reject H0 if ndH (Ψ0 F¯n ) ≤ c˜α where c˜α is chosen so that Pr sup {−z(p)}+ > c˜α = α (2.7) p∈Sd−1
By construction, denoting by c˜αn an estimator of c˜α obtained through a bootstrap procedure similar to the one in Algorithm 2.1, √
√ lim Pr{ ndH (Ψ0 F¯n ) > c˜αn } ≤ lim Pr ndH (E[F] F¯n ) > c˜αn = α n→∞
n→∞
Because Ψ0 = E[F] is contained in H0 , our test’s significance level is asymptotically equal to α. Our testing procedure preserves the property of rejecting a false null hypothesis with probability approaching 1 as the sample size increases. In particular, we have the following result: For the case that one wants to test H0 : E[F] ⊆ Ψ0 against HA : E[F] Ψ0 , similar algebra to √ what follows in √ this section gives that we can use the criterion reject H0 if ndH (F¯n Ψ0 ) > c˘α , do not reject H0 if ndH (F¯n Ψ0 ) ≤ c˘α , where c˘α is chosen so that Pr{supp∈Sd−1 {z(p)}+ > c˘α } = α and √ supp∈Sd−1 {z(p)}+ is the limiting distribution of ndH (F¯n E[F]). 11
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COROLLARY 2.5: Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued SVRVs such that E[F2H ] < ∞ and suppose that Ψ0 E[F]. Let α < 12 and Var(z(p)) > 0 for each p ∈ Sd−1 , where z(p) is defined in equation (2.6). Then √ lim Pr{ ndH (Ψ0 F¯n ) > c˜αn } = 1 n→∞
The asymptotic distribution of this test under local alternatives at distance √ (proportional to) 1/ n from the null hypothesis follows from Theorem 2.3. In particular, suppose we are interested in the power of our test of H0 against a sequence of nonempty, compact, and convex alternative sets {ΨAn }, ΨAn ∈ Kkc (d ) :
1 ΨAn ⊕ √ Δ1 = Ψ0 n
where Δ1 is a nonrandom nonempty, compact, and convex set for which there exists a nonempty, compact and convex set Δ3 such that Ψ0 = Δ1 ⊕ Δ3 . Then (2.8)
√ d ndH (Ψ0 F¯n ) → sup {−w(p)}+ p∈Sd−1
under ΨAn , where w is a Gaussian random system with E[w(p)] = −s(p Δ1 ) for all p ∈ Sd−1 and E[w(p)w(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . When additionally Δ1 is such that s(p Δ1 ) ≥ 0 for all p ∈ Sd−1 , so that ΨAn ⊆ Ψ0 , it is easy to see that the test is asymptotically locally unbiased. We return to the implications of this result when we discuss the inversion of this test to obtain confidence sets. Directed Confidence Collections The “directed confidence collection” is the collection of all sets that, when specified as a null hypothesis for a possible subset of the population identification region, cannot be rejected by our test. The term “directed” is to emphasize the connection with the test statistic that uses the directed Hausdorff distance. We denote this collection by DCCn1−α : √ DCCn1−α = {Ψ˜ ∈ Kkc (d ) : ndH (Ψ˜ F¯n ) ≤ c˜αn } If the law of supp∈Sd−1 {−z(p)}+ is continuous, then for each Ψ˜ ⊆ E[F], lim Pr{Ψ˜ ∈ DCCn1−α } ≥ 1 − α
n→∞
with equality for Ψ˜ = E[F]. Similarly to the result presented in Theorem 2.4, the union of the sets in DCCn1−α can be calculated in a particularly simple way and it represents the largest set that cannot be rejected as a null hypothesis in the inclusion test.
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COROLLARY 2.6: Let DU n = {Ψ˜ : Ψ˜ ∈ DCCn1−α } and Bc˜αn = {b ∈ d : b ≤ c√˜αnn }. Then DU n = F¯n ⊕ Bc˜αn ⊂ Un . Confidence Sets Chernozhukov, Hong, and Tamer (2007) proposed criterion-function based (QLR-type) confidence sets that cover the true identification region with probability asymptotically equal to 1 − α. We show here that for the class of problems addressed in this paper, our Wald-type confidence set DU n possesses (asymptotically) the same coverage property. PROPOSITION 2.7: Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued SVRVs such that E[F2H ] < ∞. Then lim Pr{E[F] ⊆ DU n } = 1 − α
n→∞
This proposition also implies that our results for the local power of the inclusion test based on the directed Hausdorff distance can be translated into results for coverage of a false local region. In particular, let 1 Ψ0n = E[F] ⊕ √ Δ1 n √ d Then ndH (Ψ0n F¯n ) → supp∈Sd−1 {−w(p)}+ , where the Gaussian process w(·) is defined after equation (2.8). Hence it follows from the discussion after equation (2.8) that whenever Δ1 is such that s(p Δ1 ) ≥ 0 for all p ∈ Sd−1 , √ limn→∞ Pr{ ndH (Ψ0n F¯n ) > c˜αn } ≥ α, and in this case lim Pr{Ψ0n ⊆ DU n } ≤ 1 − α
n→∞
Testing Procedure and Confidence Sets for Points in the Identification Region The inferential approach in this section targets the entire identification region of a partially identified population feature, and provides asymptotically exact significance level with which to test hypotheses and construct confidence collections. However, there are applications in which the researcher is interested in testing hypotheses and constructing confidence sets for the “true” value of the population feature, following the insight of Imbens and Manski (2004). For this case, our test statistic based on the directed Hausdorff distance allows one to conduct conservative tests of hypotheses and construct conservative confidence sets that asymptotically cover each point in the identification region with a prespecified probability. In particular, given a real valued vector ψ0 and a prespecified significance level α ∈ (0 1), suppose that one wants to / E[F] (i.e., {ψ0 } E[F]). test H0 : ψ0 ∈ E[F] (i.e., {ψ0 } ⊆ E[F]) against HA : ψ0 ∈
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
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√ Then one can reject H0 against HA if ndH ({ψ0 } F¯n ) > c˜α , and fail to reject otherwise, where c˜α is given in equation (2.7). This test preserves the property of rejecting a false null hypothesis with probability approaching 1 as the sample size increases. The confidence set DU n introduced in Corollary 2.6 has the property that for each ψ˜ ∈ E[F], limn→∞ Pr{ψ˜ ∈ DU n } ≥ 1 − α. Clearly this confidence set is conservative, as illustrated by Imbens and Manski (2004, Lemma 1), because DU n covers the entire identification region with probability asymptotically equal to 1 − α. 3. INFERENCE FOR INTERVAL-IDENTIFIED PARAMETERS The results of Section 2 can be extended to the entire class of models that give interval bounds (in ) for the parameter of interest (Manski (2003) gave a comprehensive presentation of a large group of such problems). The only requirement is that there is joint asymptotic normality of the endpoints of the interval and that the covariance matrix of the limiting distribution can be consistently estimated. This result is useful in practice, as many applications of partially identified models fall into this category.12 It also helps us relate the theory of SVRVs to the well-known laws of large numbers (LLNs) and central limit theorems (CLTs) for scalar valued random variables. Here we specialize the limiting distribution of our test statistics for the case of interval identification, and detail a step-by-step procedure to test hypothesis and construct confidence collections. Suppose that the population identification region for a scalar valued parameter of interest is given by the interval Ψ ≡ [ψL ψU ]. Let Y¯ n ≡ [y¯nL y¯nU ] denote the estimated identification region and denote by an a growing sequence (at a possibly nonparametric rate). Then if
y¯nL − ψL d z−1 an (3.1) ∼ N(0 Π) → z1 y¯nU − ψU simple algebra and the continuous mapping theorem imply that (3.2)
d an H(Y¯ n Ψ ) = an max{|y¯nL − ψL | |y¯nU − ψU |} → max{|z−1 | |z1 |}
(3.3)
an dH (Ψ Y¯ n ) d
= an max{(y¯nL − ψL )+ (y¯nU − ψU )− } → max{(z−1 )+ (z1 )− } The result in equation (3.3) allows us to establish the asymptotic equivalence of the square of our test statistic based on the directed Hausdorff distance and 12 A few examples include Manski, Sandefur, McLanahan, and Powers (1992), Hotz, Mullin, and Sanders (1997), Manski and Nagin (1998), Ginther (2000), Manski and Pepper (2000), Pepper (2000, 2003), Haile and Tamer (2003), Scharfstein, Manski, and Anthony (2004), GonzalezLuna (2005), Molinari (2008a), Dominitz and Sherman (2006).
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the criterion-function based inferential statistic proposed by Chernozhukov, Hong, and Tamer (2007). While the result that we give is mathematically simple and does not need a formal proof, we view it as conceptually important. It establishes, for the interval-identified case, an asymptotic equivalence result which corresponds, in the point-identified case, to the asymptotic equivalence between the distributions of the QLR and the Wald test statistics. We therefore place this result in a formal theorem. THEOREM 3.1: Let Ψ ≡ [ψL ψU ] and Y¯ n ≡ [y¯nL y¯nU ]. Denote by an a growing sequence. Then if equation (3.1) holds, (3.4)
d a2n [dH (Ψ Y¯ n )]2 → max{(z−1 )2+ (z1 )2− }
(3.5)
a2n sup[(y¯nL − ψ)2+ + (y¯nU − ψ)2− ] → max{(z−1 )2+ (z1 )2− }
d
ψ∈Ψ
The result in equation (3.5) is due to Chernozhukov, Hong, and Tamer (2002). ˆ the quantiles of the cumulative Replacing Π with a consistent estimator Π, distribution function of each of the random variables appearing on the righthand side of the limits in equations (3.2) and (3.3) can be easily estimated using the following procedure, which also details how to test the hypothesis about Ψ . Algorithm for Estimation of the Critical Values and Hypothesis Testing 1. Suppose H0 : Ψ = Ψ0 , HA : Ψ = Ψ0 . Then: (a) Draw a large random sample of pairs (zˆ −1 zˆ 1 ) from the distribution ˆ For each pair compute r ∗ = max{|ˆz−1 | |ˆz1 |}. N(0 Π). (b) Use the results of step (a) to compute the empirical distribution function ˆ of r ∗ , J(·). (c) Estimate the quantile cα : Pr{max{|z−1 | |z1 |} > cα } = α by ˆ ≥ 1 − α} cˆαn = inf{t : J(t) (d) Calculate an H(Y¯ n Ψ0 ) = an max{|y¯nL − ψ0L | |y¯nU − ψ0U |}. (e) If an H(Y¯ n Ψ0 ) > cˆαn , reject H0 at the 100α% level; otherwise fail to reject. 2. Suppose H0 : Ψ0 ⊆ Ψ , HA : Ψ Ψ0 . Then: (a) Draw a large random sample of pairs (zˆ −1 zˆ 1 ) from the distribution ˆ For each pair compute r˜∗ = max{(zˆ −1 )+ (ˆz1 )− }. N(0 Π). (b) Use the results of step (a) to compute the empirical distribution function ˜ of r˜∗ , J(·). (c) Estimate the quantile c˜α : Pr{max{(z−1 )+ (z1 )− } > c˜α } = α by ˜ ≥ 1 − α} c˜αn = inf{t : J(t)
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
781
(d) Calculate an dH (Ψ0 Y¯ n ) = an max{(y¯nL − ψ0L )+ (y¯nU − ψ0U )− }. (e) If an dH (Ψ0 Y¯ n ) > c˜αn , reject H0 at the 100α% level, otherwise fail to reject. Construction of CCn1−α , DCCn1−α , Un , and DU n , and Choice Among Them In the case of interval-identified parameters, CCn1−α , DCCn1−α , Un , and DU n are extremely easy to construct. In particular cˆαn cˆαn y¯nL + CCn1−α = [ψ0L ψ0U ] : ψ0L ≤ ψ0U ψ0L ∈ y¯nL − an an cˆαn cˆαn ψ0U ∈ y¯nU − y¯nU + an an c˜αn c˜αn DCCn1−α = [ψ0L ψ0U ] : y¯nL − ≤ ψ0L ≤ ψ0U ≤ y¯nU + an an cˆαn cˆαn Un = y¯nL − y¯nU + an an c˜αn c˜αn DU n = y¯nL − y¯nU + an an When the researcher is not comfortable conjecturing the entire identification region of the parameter of interest, but only a subset of it, DU n is the proper confidence set to report. This set, obtained through the inversion of the Wald statistic based on the directed Hausdorff distance, answers the question “What are the values that cannot be rejected as subsets of the identification region, given the available data and the maintained assumptions?” When the researcher is interested in answering the question “What are the sets that can be equal to the entire identification region of the parameter of interest, given the available data and the maintained assumptions?,” the proper confidence statement to make is based on CCn1−α . This is the collection of sets obtained through the inversion of the Wald statistic based on the Hausdorff distance. While DU n is a smaller confidence set than Un (the union of the intervals in CCn1−α ), it is important to observe that not all proper subsets of DU n are elements of CCn1−α . A proper subset of DU n might be rejected as a null hypothesis for the entire identification region, for example, because it might be too small and therefore not be an element of CCn1−α . EXAMPLE —Population Mean With Interval Data: Suppose that one is interested in the population mean of a random variable y, E(y). Suppose further that one does not observe the realizations of y, but rather the realizations of two real valued random variables yL , yU such that Pr{yL ≤ y ≤ yU } = 1. Manski (1989) showed that [E(yL ) E(yU )] is the sharp bound for
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E[y]. Hence the results in equations (3.2) and (3.3) directly apply here, with n n Y¯ n = [ n1 i=1 yiL n1 i=1 yiU ]. To exemplify our approach in the simplest possible setting, we rederive the result in equations (3.2) and (3.3) using the language of SVRVs. Let Y = [yL yU ]. Assume that {(yiL yiU )}ni=1 are i.i.d. random vectors, let n Yi = [yiL yiU ], and denote Y¯ n ≡ n1 i=1 Yi . Then we have the following result: THEOREM 3.2: Let {(yiL yiU )}ni=1 be i.i.d. real valued random vectors such that Pr{yiL ≤ yi ≤ yiU } = 1 and E(|yL |) < ∞, E(|yU |) < ∞. as (i) Then E[Y ] = [E(yL ) E(yU )] and H(Y¯ n E[Y ]) → 0 as n → ∞. 2 2 (ii) If E(yL ) < ∞, E(yU ) < ∞, then √ d nH(Y¯ n E(Y )) → max{|z−1 | |z1 |} √ d ndH (E(Y ) Y¯ n ) → max{(z−1 )+ (z1 )− } where
z−1 z1
∼N
Var(yL ) 0 Cov(yL yU ) 0 Cov(yL yU ) Var(yU )
4. BEST LINEAR PREDICTION WITH INTERVAL OUTCOME DATA Suppose that one is interested in the parameters of the best linear predictor (BLP) of a random variable y conditional on a random vector x. Suppose that one does not observe the realizations of y, but rather the realizations of two real valued random variables yL , yU such that Pr{yL ≤ y ≤ yU } = 1. We remark that best linear prediction finds the linear function that minimizes square loss, but does not impose that such linear function is best nonparametric (Manski (2003, pp. 56–58)). In other words, we are not restricting the conditional expectation of y given x to be linear. Let Y = [yL yU ]. Throughout this section, we maintain the following assumption: ASSUMPTION 4.1: Let (y yL yU x) be a random vector in × × × d such that Pr{yL ≤ y ≤ yU } = 1. The researcher observes a random sample {(yiL yiU xi ) : i = 1 n} from the joint distribution of (yL yU x). The proofs of the propositions and theorems for this section, given in the Appendix, consider the general case that x ∈ d . To simplify the notation, in this section we introduce ideas restricting attention to the case that x ∈ (though our assumptions are written for the general case d ≥ 1). Let 1 x¯ 1 E(x) ˆ Σn ≡ Σ≡ x¯ x2 E(x) E(x2 )
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
where x¯ =
1 n
n i=1
xi , x2 =
1 n
n i=1
783
x2i . Assume the following:
ASSUMPTION 4.2: E(|yL |) < ∞, E(|yU |) < ∞, E(|yL xk |) < ∞, and E(|yU × xk |) < ∞, k = 1 d. ASSUMPTION 4.3: Σ is of full rank. Then in the point-identified case the population best linear predictor [θ1 θ2 ] solves the equations E(y) = θ1 + θ2 E(x) E(xy) = θ1 E(x) + θ2 E(x2 ) and its sample analog [θˆ 1 θˆ 2 ] solves the equations 1 1 yi = θˆ 1 + θˆ 2 xi n i=1 n i=1 n
n
1 1 1 2 xi yi = θˆ 1 xi + θˆ 2 x n i=1 n i=1 n i=1 i n
n
n
In the interval outcomes case Y is an SVRV. We first introduce some additional notation to accommodate set valued variables. Let y(ω) (4.1) G(ω) = : y(ω) ∈ Y (ω) x(ω)y(ω) Let Gi be the mapping defined as in (4.1) using (yiL yiU xi ). Lemma A.4 in the Appendix shows that G and Gi , i ∈ N, are SVRVs. We define the population set valued best linear predictor as (4.2)
Θ = Σ−1 E[G] −1 θ1 1 E(x) E(y) y θ1 1 : = ∈ S (G) = θ2 θ2 E(x) E(x2 ) E(xy) xy
Before proceeding to deriving the estimator of Θ and its asymptotic properties, we show that the identification region Θ defined in (4.2) is identical to the identification region for the BLP obtained following the approach in Manski (2003) and denoted by ΘM : M M θ1 θ1 M 2 (4.3) : M = arg min (y − θ1 − θ2 x) dη η ∈ Pyx Θ = θ2M θ2
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A. BERESTEANU AND F. MOLINARI
where (4.4)
Pyx = η : Pr(yL ≤ t x ≤ x0 ) ≥ η((−∞ t] (−∞ x0 ]) ≥ Pr(yU ≤ t x ≤ x0 )∀t ∈ ∀x0 ∈
η((−∞ +∞) (−∞ x0 ]) = Pr(x ≤ x0 ) ∀x0 ∈ PROPOSITION 4.1: Let Assumptions 4.1–4.3 hold. Let Θ and ΘM be defined, respectively, as in (4.2) and (4.3)–(4.4). Then Θ = ΘM . Given these preliminaries, we apply the analogy principle and define the sample analog of Θ as ¯ Θˆ n = Σˆ −1 n Gn ¯ n = 1 n Gi . where G i=1 n Observe that by construction, Θ is a convex set. Lemma A.8 in the Appendix describes further its geometry, showing that when x has an absolutely continuous distribution and G is integrably bounded, Θ is a strictly convex set, that is, it does not have a flat face. The estimated set Θˆ n is a convex polytope, because it is given by a finite Minkowski sum of segments in d . Theorem 4.2 below shows that under mild regularity conditions on the moments of (yL yU x) as in Assumption 4.2, Θˆ n is a consistent estimator of Θ. Under the additional Assumption 4.4, this convergence occurs at the rate Op ( √1n ). (4.5)
ASSUMPTION 4.4: E(|yL |2 ) < ∞, E(|yU |2 ) < ∞, E(|yL xk |2 ) < ∞, and E(|yU × xk |2 ) < ∞, E(|xk |4 ) < ∞, k = 1 d. THEOREM 4.2: Let Assumptions 4.1, 4.2, and 4.3 hold. Define Θ and Θˆ n as in as (4.2) and (4.5), respectively. Then H(Θˆ n Θ) → 0. If in addition Assumption 4.4 holds, then H(Θˆ n Θ) = Op ( √1n ). The proof of Theorem 4.2 is based on an extension of Slutsky’s theorem to SVRVs, which we provide in √ Lemma A.6 in the Appendix. The rate of convergence that we obtain is 1/ n, irrespective of whether x has a continuous or a discrete distribution. To derive the asymptotic distribution of H(Θˆ n Θ), we impose an additional condition13 on the distribution of x: 13 When x has a discrete distribution, the population identification region Θ is a polytope, given by a Minkowski sum of segments. As a result, when x is discrete, the support function of Θ is not differentiable and the functional delta method (which we use when x is absolutely continuous) cannot be applied. In this case, tedious calculations allow one to express the extreme points of Θ as functions of moments of (yL yU x). Hence the exact asymptotic distribution of H(Θˆ n Θ) can be derived.
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ASSUMPTION 4.5: The distribution of x is absolutely continuous with respect to Lebesgue measure on d . THEOREM 4.3: Let Assumptions 4.1, 4.3, 4.4, and 4.5 hold. √ √ d d nH(Θˆ n Θ) → vC(Sd ) and ndH (Θ Θˆ n ) → supp∈Sd {−v(p)}+ , (i) Then where v is a linear function of a vector Gaussian random system with E[v(p)] = 0 for all p ∈ Sd and (4.6)
E[v(p)v(q)] = E[s(p Σ−1 G)s(q Σ−1 G)] − E[s(p Σ−1 G)]E[s(q Σ−1 G)] − ξp κqp − κpq ξq + ξp Vpq ξq
for all p q ∈ Sd . The singleton ξp = Θ ∩ {ϑ ∈ d+1 : ϑ p = s(p Θ)}, and the matrix Vpq and the vector κpq are given, respectively, in equations (A.7) and (A.8). (ii) Assume in addition that Var(yL |x), Var(yU |x) ≥ σ 2 > 0 P(x)-a.s. Then Var(v(p)) > 0 for each p ∈ Sd , and therefore the laws of vC(Sd ) and supp∈Sd {−v(p)}+ are absolutely continuous with respect to Lebesgue measure, respectively, on + and ++ . The difficulty in obtaining the asymptotic distribution of the statistics in Theorem 4.3 can be explained by recalling how one would proceed in the pointidentified case. When Θˆ n and Θ are singletons,
√ √ θ1 y¯ −1 ˆ ˆ ˆ − Σn n(θn − θ) = Σn n xy θ2 with the second expression giving the product of a random matrix converging in probability to a nonsingular matrix, and the sample average of a mean zero i.i.d. random vector converging to a multivariate normal distribution. In this case, an application of Slutsky theorem delivers the desired result. In our case, the Aumann expectation of the random set given by Σˆ n Θ is not equal to ΣΘ, and therefore a simple Slutsky-type result which extends the one that would be applied in the point-identified case (and which we provide in the Appendix, Lemma A.9) does not suffice for obtaining the limiting distribution of our statistics.14 Hence, to prove Theorem 4.3 we proceed in steps. We start by looking at the difference between the support functions of Θˆ n and Θ. We rewrite these support functions as the sum of two (not necessarily independent) elements, and use Skorokhod representation theorem and an application of the delta method for C(Sd ) valued random variables to derive their joint asymptotic distribution. However, the use of this functional delta method requires 14 For a scalar random variable σˆ n such that Pr(σˆ n ≥ 0) = 1, E[σˆ n Θ] = E[σˆ n ]Θ (Molchanov (2005, Theorem 2.1.48)) and therefore standard arguments can be applied easily.
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differentiability of the support function of Θ; Lemma A.8 in the Appendix establishes this condition when x has an absolutely continuous distribution.15 Consider now the case that one is interested in a subset of the parameters of the BLP and in testing linear restrictions on Θ. Let R denote a matrix of linear restrictions, including the special case that R projects the set Θ on one of its components. COROLLARY 4.4: Let Assumptions 4.1, 4.3, 4.4, and 4.5 hold. Let R be a nonrandom full-rank finite matrix of dimension l × (d + 1) with l < (d + 1). √ √ d d nH(RΘˆ n RΘ) → vR C(Sl−1 ) and ndH (RΘ RΘˆ n ) → (i) Then R R supp∈Sl−1 {−v (p)}+ , where v is a linear function of a vector Gaussian random system with E[vR (p)] = 0 for all p ∈ Sl−1 and E[vR (p)vR (q)] = E[s(R p Σ−1 G)s(R q Σ−1 G)] − E[s(R p Σ−1 G)]E[s(R q Σ−1 G)] − ξR p κR qR p − κR pR q ξR q + ξR p VR pR q ξR q where for p, q ∈ Sl−1 the singleton ξR p = Θ ∩ {ϑ ∈ d+1 : ϑ R p = s(R p Θ)}, and the matrix VR pR q and the vector κR pR q are given, respectively, in equations (A.7) and (A.8). (ii) Let R project the set Θ on one of its components. Without loss of generality, assume R = [ 0 0 · · · 0 1 ]. Then
√ d nH(RΘˆ n RΘ) → max |v(−R)| |v(R)|
√ d ndH (RΘ RΘˆ n ) → max (v(−R))+ (v(R)− ) where v is defined in Theorem 4.3. Often empirical researchers are particularly interested in estimation and inference for a single component of the BLP parameter vector. Corollary 4.4 establishes the asymptotic distribution of our test statistics for this case. In terms of computation of the bounds, our methodology provides substantial advantages. In particular, without loss of generality, for i = 1 n denote by xi = [1 xi1 xid−1 xid ] = [xi1 xid ] and by x˜ id the residuals obtained after projecting xd on the other covariates x1 . Then the following result holds: 15 Beresteanu and Molinari (2006, Corollary 5.4) gave an extremely simple approximation to the distribution in Theorem 4.3 based on sample size adjustments which allow one to ignore the randomness in Σˆ n at the cost of a slower rate of convergence.
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COROLLARY 4.5: Let Assumptions 4.1–4.3 hold. Let Θˆ d+1n = {θd+1 ∈ : [ θ1 θd+1 ] ∈ Θˆ n }. Then n n 1 ˆ Θd+1n = n 2 min{x˜ id yiL x˜ id yiU } max{x˜ id yiL x˜ id yiU } ˜ id i=1 i=1 x i=1 The practical relevance of this result is that the calculation of the estimator of the identification region of the BLP coefficient for a single variable can be carried out using standard statistical packages.16 This further facilitates implementation of the bootstrap procedure that we propose in Algorithm 4.2 below. Similarly to the discussion in Section 2.2, the asymptotic distributions of our test statistics depend on parameters to be estimated. Hence we again obtain the critical values using bootstrap procedures. Here we detail a bootstrap procedure for the approximation of the critical values of the limiting distribution √ √ of nH(Θˆ n Θ). A similar procedure can be applied for ndH (Θ Θˆ n ). ALGORITHM 4.2: ∗ ∗ yiU x∗i ) : i = 1 n}, by 1. Generate a bootstrap sample of size n, {(yiL drawing a random sample from the joint empirical distribution of the vector {(yiL yiU xi ) : i = 1 n} with replacement. Use this sample to construct ∗ ˆ −1∗ bootstrap versions of Gi and Σˆ −1 n denoted Gi and Σn . 2. Compute √ ¯∗ ˆ (4.7) rn∗ ≡ nH(Σˆ −1∗ n Gn Θn ) 3. Use the results of b repetitions of Steps 1 and 2 to compute the empirical distribution of rn∗ at a point t, denoted by Jn (t). 4. Estimate the critical value cαBLP such that Pr{vC(Sd ) > cαBLP } = α by (4.8)
BLP = inf{t : Jn (t) ≥ 1 − α} cˆαn
The asymptotic validity of this procedure follows by an application of the delta method for the bootstrap. In particular, the following result holds: d
PROPOSITION 4.6: Let the assumptions of Theorem 4.3(i) hold. Then rn∗ → vC(Sd ) , where rn∗ is defined in equation (4.7) and the random variable v is given in BLP = Theorem 4.3. If in addition the assumptions of Theorem 4.3(ii) hold, then cˆαn BLP BLP cα + op (1) where cˆαn is defined in (4.8). 16 We are grateful to an anonymous referee for suggesting this result. The proof of this result is based on the Frisch–Waugh–Lovell theorem (see Magnac and Maurin (2008) for a related use of this theorem). If one is interested in a subset of the parameters of the BLP of dimension k ≥ 2, this theorem can again be applied, and it again yields computational advantages. Stoye (2007) proposed computationally equivalent estimators to those in Corollary 4.5, without using the Frisch–Waugh–Lovell theorem.
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Alternatively we could obtain consistent estimates of cαBLP by simulating the distribution of the supremum of a linear function of a vector Gaussian random system with mean function equal to zero for each p ∈ Sd and with a covariance kernel which consistently estimates the covariance kernel in (4.6). Beresteanu and Molinari (2006, Proposition 5.6) established the asymptotic validity of this procedure. 5. MONTE CARLO RESULTS In this section we conduct a series of Monte Carlo experiments to evaluate the properties of the test proposed in Section 2.2, as applied to the problem of inference for the mean and for the parameters of the BLP with interval outcome data. We use the same data set as in Chernozhukov, Hong, and Tamer (2002) to conduct inference for (i) the mean (logarithm of) wage and (ii) the returns to education for men between the ages of 20 and 50. The data are taken from the March 2000 wave of the Current Population Survey (CPS), and contain 13,290 observations on income and education. The wage variable is artificially bracketed to create interval valued outcomes. A detailed description of the data construction procedure appears in Chernozhukov, Hong, and Tamer (2002, Sec. 4). Denote the logarithm of the lower bound of the wage, the logarithm of the upper bound of the wage, and the level of education in years, by yL , yU , and x, respectively. We treat the empirical joint distribution of (yL yU x) in the CPS sample as the population distribution and draw small samples from it. The first Monte Carlo experiment looks at the asymptotic properties of the estimator for the population mean with interval data (see Section 3). Denote by Y = [yL yU ] the interval valued log of wages. The population identification region is Ψ0 = E(Y ) = [44347 49674] We draw 25,000 small samples of sizes 100, 200, 500, 1000 and 2000 from the CPS “population,” and use the statistic based on the Hausdorff distance to test H0 : E[Y ] = Ψ0 , HA : E[Y ] = Ψ0 . For each sample, we estimate the critical value cα , α = 005, implementing Algorithm 2.1 with 2000 bootstrap repetitions.17 We √ . build the local alternatives using equation (2.4) with Δ1 = {0} and Δ2 = δ {05} n {05} More specifically, the local alternatives are defined as ΨAn (δ) = Ψ0 δ √n for δ ∈ {0 12 1 2 4 8 16}, where 05 is the width (approximated to the first decimal point) of E(Y ). Fortran 90 code for computing the Minkowski sample average of intervals in and 2 , the Hausdorff distance, and the directed Hausdorff distance between polytopes in 2 , and for implementing Algorithm 2.1 and Algorithm 4.2 is available upon request from the authors. 17
789
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS TABLE I
REJECTION RATES OF THE TEST BASED ON THE DIRECTED HAUSDORFF DISTANCE, AND OF THE TEST BASED ON THE HAUSDORFF DISTANCE AGAINST LOCAL ALTERNATIVES (NOMINAL LEVEL = 005) − E(y)
Sample Size
n = 100 n = 200 n = 500 n = 1000 n = 2000
√ nH(Y¯ n Ψ0 )
√ ndH (Ψ0 Y¯ n )
δ=0
0.5
1
2
4
8
16
0.091 0.076 0.065 0.061 0.060
0.046 0.047 0.049 0.049 0.051
0.061 0.058 0.059 0.059 0.061
0.092 0.087 0.085 0.087 0.086
0.212 0.200 0.190 0.190 0.190
0.580 0.577 0.570 0.565 0.568
0.979 0.985 0.989 0.989 0.990
1.000 1.000 1.000 1.000 1.000
In terms of equation (2.5), the Hausdorff distance between the local alternative and the null is κ = 12 δ. Results for this Monte Carlo experiment appear in Table I, columns 2–8.18 The empirical size of our test (column 2) is quite close to the nominal level of 005. As δ (and therefore κ) increases, the rejection rates increase for each given sample size, and for a given δ, the rejection rates are stable across sample sizes. These results are invariant with the width of E(Y ). We then run a similar bootstrap procedure to estimate c˜α , α = 005, and use the statistic based on the directed Hausdorff distance to test H0 : Ψ0 ⊆ E[Y ], HA : Ψ0 E[Y ]. Table I, column 1, reports the rejection rate of this test for different sample sizes, when the nominal level is 005. For small samples (e.g., n = 100) the test is a little oversized. However, as n increases, this size distortion decreases and the rejection rate gets close to its nominal level.19 The second Monte Carlo experiment uses both the interval valued information on the logarithm of wage and the information about education. Here we implement the test based on the Hausdorff distance at the 005 level for the best linear predictor of the logarithm of wage given education. Figure 1 depicts Θ, the population identification region of the parameters of the BLP given in equation (4.2). To trace the increase in power as we get farther away from the null, we again use a sequence of local alternatives. The local alternatives are defined √ as ΘAn (δ) = Θ0 ⊕ δ {(303)} for δ ∈ {0 12 1 2 4 8 16}, where Θ0 = Θ is the polyn tope in Figure 1, and the vector (3 03) gives the width (approximated to the 18 We also conducted Monte Carlo experiments in which the critical value cα is obtained by simulations. We considered both the case in which the covariance kernel was known (i.e., it was the covariance kernel obtained from the entire CPS population) and the case in which it was replaced by a consistent estimator. The results are comparable to those reported in Table I. 19 We observe that inversion of this test leads to the construction of the confidence set DU n , and therefore the results in Table I, column 1, can be used to infer the coverage properties of DU n .
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FIGURE 1.—The population identification region of the parameters of the BLP.
first decimal point) of the projection of Θ on each axis. In terms of equation (2.5), the distance between the local alternative and the null is κ = 3015δ. We draw 25,000 small samples of sizes 100, 200, 500, 1000, and 2000 from the CPS population. For each sample, we estimate the critical value cα , α = 005, implementing Algorithm 4.2 with 2000 bootstrap repetitions. The rejection rates of the null and local alternatives are shown in Table II, columns 2–8. As the second column of this table shows, for small samples (e.g., n = 100) the test is a little oversized; however, as n increases, this size distortion disappears. As δ (and therefore κ) increases, the rejection rates increase for each given sample size. TABLE II REJECTION RATES OF THE TEST BASED ON THE DIRECTED HAUSDORFF DISTANCE AND OF THE TEST BASED ON THE HAUSDORFF DISTANCE AGAINST LOCAL ALTERNATIVES (NOMINAL LEVEL = 005) − BLP √
Sample Size
n = 100 n = 200 n = 500 n = 1000 n = 2000
nH(Θˆ n Θ0 )
√ ndH (Θ0 Θˆ n )
δ=0
0.5
1
2
4
8
16
0.077 0.076 0.066 0.062 0.060
0.062 0.068 0.062 0.059 0.057
0.074 0.078 0.070 0.067 0.061
0.094 0.095 0.085 0.081 0.072
0.151 0.150 0.135 0.129 0.121
0.352 0.349 0.336 0.330 0.321
0.867 0.880 0.889 0.896 0.900
1.000 1.000 1.000 1.000 1.000
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We then run a similar bootstrap procedure to estimate c˜αBLP , α = 005, and use the statistic based on the directed Hausdorff distance to test H0 : Θ0 ⊆ Θ, HA : Θ0 Θ. Table II, column 1, reports the rejection rate of this test for different sample sizes, when the nominal level is 005. Again, for small samples (e.g., n = 100) the test is a little oversized. However, as n increases, this size distortion decreases and the rejection rate gets close to its nominal level.20 In summary, our Monte Carlo experiments show that the test described in Section 2.2 performs well even with samples of size as small as 100. The rejection rates of the null are very close to 005. The power against the local alternatives grows rapidly as the alternatives get far away from the null. Similarly, the rejection rate of the test described in Section 2.3 is close to its nominal level even with small samples. 6. CONCLUSIONS This paper has introduced a methodology to conduct estimation and inference for partially identified population features in a completely analogous way to how estimation and inference would be conducted if the population features were point identified. We have shown that for a certain class of partially identified population features, which include means and best linear predictors with interval outcome data, and can be easily extended to parameter vectors characterizing semiparametric binary models with interval regressor data, the identification region is given by a transformation of the Aumann expectation of an SVRV. Extending √ the analogy principle to SVRVs, we proved that this expectation can be n-consistently estimated (with respect to the Hausdorff distance) by a transformation of the Minkowski average of a sample of SVRVs which can be constructed from observable random variables. When the Hausdorff distance and the directed Hausdorff distance between the population √ identification region and the proposed estimator are normalized by n, these statistics converge in distribution to different functions of the same Gaussian process, whose covariance kernel depends on parameters of the population identification region. We introduced consistent bootstrap procedures to estimate the quantiles of these distributions. The asymptotic distribution results have allowed us to introduce procedures to test, respectively, whether the population identification region is equal to a particular set or whether a certain set of values is included in the population identification region. These test procedures are consistent against any fixed alternative and are locally asymptotically unbiased. A Monte Carlo exercise showed that our tests perform well even in small samples. The test statistic based on the Hausdorff distance can be inverted to construct a confidence collection for the population identification region. The confidence collection is given by the collection of all sets that, when specified as 20
As before, the results in Table II, column 1, can be used to infer the coverage properties of
DU n .
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a null hypothesis for the true value of the population identification region, cannot be rejected by our test. Its main property is that (asymptotically) the population identification region is one of its elements with a prespecified confidence level 1 − α. The test statistic based on the directed Hausdorff distance can be inverted to construct a directed confidence collection, given by the collection of all sets that, when specified as a null hypothesis for a subset of values of the population identification region, cannot be rejected by our test. Its main property is that (asymptotically) the union of the sets comprising it covers the population identification region with a prespecified confidence level 1 − α. This result establishes a clear connection, within the class of models studied in this paper, between our Wald-type confidence sets, and the QLR-type confidence sets proposed by Chernozhukov, Hong, and Tamer (2007). Dept. of Economics, Duke University, Durham, NC 27708-0097, U.S.A.;
[email protected] and Dept. of Economics, Cornell University, 492 Uris Hall, Ithaca, NY 14853-7601, U.S.A.;
[email protected]. Manuscript received June, 2006; final revision received November, 2007.
APPENDIX: LEMMAS AND PROOFS Preliminaries We first define notation and state limit theorems that we use throughout the appendix. In this appendix, we use the capital Greek letters Π and Σ to denote matrices, and we use other capital Greek letters to denote sets of parameters. It will be obvious from the context whether a capital Greek letter refers to a matrix or to a set of parameters. For a finite d × d matrix Π and a set A ⊂ d , let ΠA = {r ∈ d : r = Πa a ∈ A} and observe that s(p ΠA) = s(Π p A), where Π denotes the transposed matrix. We denote the matrix norm induced√by the Euclidean vector norm (i.e., the 2-norm) as Π = maxp=1 Πp = λmax , where λmax is the largest eigenvalue of Π Π. This matrix norm is compatible with its underlying vector norm (i.e., the Euclidean norm), so that Πa ≤ Πa, and is a continuous function of the elements of the matrix. We let ⇒ denote weak convergence. An SVRV F : Ω → Kkc (d ) can be represented through the support function of its realizations.21 Sublinearity of the support function implies that when considering the support function of a set, it suffices to restrict attention to vectors p ∈ Sd−1 . This results in a C(Sd−1 ) valued random variable. Hörmander’s 21 Similarly, a Minkowski average of SVRVs Fi : Ω → Kkc (d ), i = 1 n, can be represented through the sample average of the corresponding support functions s(p Fi ), i = 1 n.
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embedding theorem (Li, Ogura, and Kreinovich (2002, Theorem 1.1.12)) ensures that (Kkc (d ) H(· ·)) can be isometrically embedded into a closed convex cone in C(Sd−1 ). In particular, for any F1 , F2 : Ω → Kkc (d ), (A.1)
H(F1 F2 ) = s(· F1 ) − s(· F2 )C(Sd−1 )
Let B (K(d )) be the Borel field of K(d ) with respect to the Hausdorff metric H. Then it follows from Definition 1 that an SVRV F : Ω → K(d ) is B (K(d ))-measurable (this result is stated in Molchanov (2005, Theorem 1.2.3); for a proof see Li, Ogura, and Kreinovich (2002, Theorem 1.2.3)). Define AF = σ{F −1 (A) : A ∈ B (K(d ))} to be the σ-algebra generated by inverse images of sets in B (K(d )). Also let μF (A) = μ(F −1 (A)) for any A ∈ B (K(d )) denote the distribution of F . DEFINITION 6: Let F1 and F2 be two B (K(d ))-measurable SVRVs defined on the same measurable space Ω. F1 and F2 are independent if AF1 and AF2 are independent; F1 and F2 are identically distributed if μF1 and μF2 are identical. Many of the results that we obtain in this paper are based on a strong law of large numbers and on a central limit theorem for SVRVs which parallel the classic results for random variables. We state these results here for the reader’s convenience. THEOREM A.1—(SLLN): Let {F Fi : i ∈ N} be i.i.d. nonempty, integrably bounded, compact valued SVRVs. Then n 1 as Fi E[F] → 0 as n → ∞ H n i=1 For the proof, see Artstein and Vitale (1975). THEOREM A.2—(CLT): Let {F Fi : i ∈ N} be i.i.d. nonempty, compact valued SVRVs such that E[F2H ] < ∞. Then n √ 1 d nH Fi E[F] → zC(Sd−1 ) n i=1 where z is a Gaussian random system with E[z(p)] = 0 for all p ∈ Sd−1 and E[z(p)z(q)] = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)] for all p q ∈ Sd−1 . For the proof, see Giné, Hahn, and Zinn (2006).
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LEMMA A.1: Given any two compact convex sets K L ⊂ d , dH (K L) = sup{s(p K) − s(p L)} p∈Bd
= max 0 sup {s(p K) − s(p L)} p∈Sd−1
where Bd = {p ∈ d : p ≤ 1}. PROOF: By definition, dH (K L) = inf{c > 0 : K ⊂ L ⊕ cBd }. Let dH (K L) ≤ α. Then K ⊂ L ⊕ αBd , which implies that s(p K) ≤ s(p L ⊕ αBd ) = s(p L) + α for all p ∈ Bd This argument can be reversed to show that if supp∈Bd {s(p K) − s(p L)} ≤ α, then dH (K L) ≤ α. Hence, dH (K L) = supp∈Bd {s(p K) − s(p L)}. To see that supp∈Bd {s(p K) − s(p L)} = max{0 supp∈Sd−1 {s(p K) − s(p L)}}, observe that if s(p K) − s(p L) ≤ 0 for all p ∈ Bd , then supp∈Bd {s(p K) − s(p L)} = 0 and max{0 supp∈Sd−1 {s(p K) − s(p L)}} = 0. (Observe incidentally that dH (K L) = 0 and s(p K) ≤ s(p L) for every p ∈ Bd if and only if K ⊆ L.) Suppose now that there exists at least a p ∈ Bd such that s(p K) − s(p L) > 0. Let p˜ ∈ arg supp∈B {s(p K) − s(p L)}. Then p˜ ∈ Sd−1 To see why this is the case, suppose by contradiction that ˜ ˜ < 1. Then p˜ ∈ / Sd−1 . Let p = pp ∈ Sd−1 and observe that we are assuming p ˜ 1 ˜ K) − s(p ˜ L) < p ˜ K) − s(p ˜ L)] = s(p K) − s(p L), which 0 < s(p [s(p ˜ leads to a contradiction. Q.E.D. A.1. Proofs for Section 2.2 PROOF OF PROPOSITION 2.1: To establish the asymptotic validity of this procedure, observe that by Theorem 2.4 in Giné and Zinn (1990), n n √ 1 1 ∗ n s(· co Fi ) − s(· co Fi ) ⇒ z(·) n i=1 n i=1 as a sequence of processes indexed by p ∈ Sd−1 . Observing that for each g ∈ C(), the functional on C(Sd−1 ) defined by h(x) = g(xC(Sd−1 ) ) belongs to C(C(Sd−1 ) · C(Sd−1 ) ), the result follows by the continuous mapping theorem using standard arguments (e.g., Politis, Romano, and Wolf (1999, Chap. 1)). The processes considered in this paper are separable with bounded realizations (a consequence of the fact that the support function of a bounded set F ∈ Kk (d ) is Lipschitz with Lipschitz constant FH , (Molchanov (2005, Theorem F.1))). Hence, it follows from Theorem 1 in Tsirel’son (1975) and from the corollary in Lifshits (1982) that Var(z(p)) > 0 for each p ∈ Sd−1 is a sufficient condition for the law of zC(Sd−1 ) to be absolutely continuous with
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
795
respect to Lebesgue measure on + . The result then follows by standard arguments. Q.E.D. PROOF OF THEOREM 2.2: By triangle inequality, H(ΨA Ψ0 ) ≤ H(F¯n Ψ0 ) + H(F¯n ΨA ). Hence, √ lim Pr{ nH(F¯n Ψ0 ) > cˆαn } n→∞
√ √ ≥ lim Pr{ nH(ΨA Ψ0 ) − nH(F¯n ΨA ) > cˆαn } n→∞ cα + op (1) = lim Pr H(ΨA Ψ0 ) > H(F¯n ΨA ) + √ n→∞ n = Pr{H(ΨA Ψ0 ) > 0} = 1
where the second to last equality follows from Theorem A.1 and because p cˆαn → cα < ∞ (by Proposition 2.1), and the last equality follows because H(ΨA Ψ0 ) is a positive constant. Q.E.D. √ By Hörmander’s embedding theorem, nH(F¯n PROOF √ OF THEOREM 2.3: Ψ0 ) = n supp∈Sd−1 |s(p F¯n ) − s(p Ψ0 )|. Let h(·) = s(· F) − s(· Ψ0 )
1 1 = s(· F) − s · ΨAn ⊕ √ Δ1 + s · ΨAn ⊕ √ Δ1 − s(· Ψ0 ) n n
1 = s(· F) − s(· ΨAn ) − s · √ Δ1 n
1 + s · Ψ0 ⊕ √ Δ2 − s(· Ψ0 ) n
1 1 = s(· F) − s(· ΨAn ) + s · √ Δ2 − s · √ Δ1 n n Similarly, hk (·) = s(· Fk ) − s(· Ψ0 ) = s(· Fk ) − s(· ΨAn ) + s(· √1n Δ2 ) − s(· √1n Δ1 ). Hence under the local alternative, h, h1 , h2 are C(Sd−1 ) valued i.i.d. random variables with E[h(p)] = √1n s(p Δ2 ) − √1n s(p Δ1 ). The limiting √ distribution of nH(F¯n Ψ0 ) then follows from Proposition 3.1.9 in Li, Ogura, and Kreinovich (2002). The above result and Proposition 2.1 imply that √ √ lim Pr( nH(F¯n Ψ0 ) > cˆαn ) = lim Pr( nH(F¯n Ψ0 ) + op (1) > cα ) n→∞
n→∞
= Pr{wC(Sd−1 ) > cα }
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Recall that z is a Gaussian random system with E[z(p)] = 0 ∀p ∈ Sd−1 and Cov(z(p) z(q)) = E[s(p F)s(q F)] − E[s(p F)]E[s(q F)]. Also, w(·) is a d Gaussian random system with w = z + τ, where τ(p) = s(p Δ2 ) − s(p Δ1 ) ∀p ∈ d Sd−1 and = means equivalent in distribution. So by Anderson’s lemma (see, e.g., Molchanov (2005, Theorem 3.1.12)), for any finite set Sj , Pr sup |w(p)| > cα ≥ Pr sup |z(p)| > cα p∈Sj
p∈Sj
The metric space (Sd−1 · ·), where · · denotes the usual Euclidean distance in d , is separable. Also z(·) and w(·) are uniformly continuous in probability with respect to · · (a consequence of the fact that the support function of a closed bounded set F is Lipschitz with Lipschitz constant FH , (Molchanov (2005, Theorem F.1))). Thus both have separable versions. Without loss of generality we may assume that z(·) and w(·) are separable. The result then follows from the same argument as in Section 6 of Andrews (1997, p. 1114). Q.E.D. PROOF OF THEOREM 2.4: We first show that F¯n ⊕ Bcˆαn ⊂ Un . This follows because F¯n ⊕ Bcˆαn is a convex compact set (the Minkowski sum of convex sets √ √ is convex) and because nH(F¯n F¯n ⊕ Bcˆαn ) = nH({0} Bcˆαn ) = cˆαn , where the last equality follows from the definition of Bcˆαn . Hence F¯n ⊕ Bcˆαn ∈ CCn1−α . We now show that Un ⊂ F¯n ⊕ Bcˆαn . Take any ψ ∈ Un . Then by definition ∃Ψ˜ ∈ √ CCn1−α such that ψ ∈ Ψ˜ and nH(F¯n Ψ˜ ) ≤ cˆαn . Choose ˜f ∈ F¯n such that ˜f = ˜ and b˜ ∈ Bcˆαn arg inff∈F¯n ψ − f. Let b˜ = ψ − ˜f; then by construction ψ = ˜f + b, because cˆαn ˜ = ψ − ˜f = inf ψ − f ≤ sup inf g − f ≤ √ b ¯ f∈F¯n f∈ F n n g∈Ψ˜ where the last inequality follows because Bcˆαn .
√ nH(F¯n Ψ˜ ) ≤ cˆαn . Hence ψ ∈ F¯n ⊕ Q.E.D.
A.2. Proofs for Section 2.3 PROOF OF COROLLARY 2.5: By the triangle inequality, dH (Ψ0 E[F]) ≤ dH (Ψ0 F¯n ) + dH (F¯n E[F]), and the result follows from a similar argument as in the proof of Theorem 2.2 because dH (Ψ0 E[F]) is a positive constant p when Ψ0 E[F], dH (F¯n E[F]) → 0 by Theorem A.1, and by Theorem 2 in Lifshits (1982), if Var(z(p)) > 0 for each p ∈ Sd−1 , the law of supp∈Sd−1 {−z(p)}+ is absolutely continuous with respect to Lebesgue measure on ++ , so that Q.E.D. c˜αn = c˜α + op (1).
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PROOF OF COROLLARY 2.6: The fact that DU n = F¯n ⊕ Bc˜αn follows from a similar argument as in the proof of Theorem 2.4. The fact that DU n ⊂ Un is an obvious consequence of the fact that for any two compact sets A and B, Q.E.D. H(A B) ≥ dH (A B). PROOF OF PROPOSITION 2.7: We show that E[F] ∈ DCCn1−α if and only if E[F] ⊆ DU n . Trivially, E[F] ∈ DCCn1−α ⇒ E[F] ⊆ DU n . Next, given any two subsets A and B of d , let Bc denote the c-envelope (or parallel set) of B, that ˜ ≤ c}. Because dH (A B) = inf{c > 0 : A ⊂ Bc }, it is, Bc = {b˜ ∈ d : infb∈B b − b follows that E[F] ⊆ DU n
⇐⇒
E[F] ⊆ F¯n ⊕ Bc˜αn
⇒
E[F] ∈ DCCn1−α
⇒
c˜αn dH (E[F] F¯n ) ≤ √ n
Therefore, lim Pr{E[F] ⊆ DU n } = lim Pr{E[F] ∈ DCCn1−α } = 1 − α
n→∞
n→∞
Q.E.D.
A.3. Proofs for Section 3 PROOF OF THEOREM 3.2: This result can be easily proved by using the following lemmas. LEMMA A.2: Let Y = [yL yU ], where yL yU are real valued random variables such that Pr(yL ≤ yU ) = 1. Then Y is a compact valued nonempty SVRV. PROOF: Theorem 1.2.5 in Molchanov (2005) implies that Y is an SVRV iff s(p Y ) is a random variable for each p ∈ {−1 1}. Since yL (ω) ≤ yU (ω) μ-a.s., yU (ω) if p = 1, s(p Y (ω)) = max{pyL (ω) pyU (ω)} = −yL (ω) if p = −1. From the fact that yL and yU are random variables, also s(p Y ) is a random variable and the claim follows. The fact that Y takes almost surely compact values follows because yL and yU are real valued random variables. The fact Q.E.D. that Y is nonempty follows because Pr(yL ≤ yU ) = 1. LEMMA A.3: Given an i.i.d. sequence {yiL yiU }ni=1 , where yiL yiU are real valued random variables such that Pr(yiL ≤ yiU ) = 1 for each i, let Yi = [yiL yiU ]. Then {Yi : i ∈ N} are i.i.d. SVRVs.
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PROOF: (i) {Yi }ni=1 are identically distributed. For A ∈ B (K()), μYi (A) = μ(Yi−1 (A)) = Pr{[yiL yiU ] ∩ A = ∅}. Since {(yiL yiU )}ni=1 are i.i.d., μYi = μYj for all i and j. (ii) {Yi }ni=1 are independent. For all C1 Cn in Kk (), μ(Y1−1 (C1 ) Yn−1 (Cn )) = μ(Y1 ∩ C1 = ∅ Yn ∩ Cn = ∅) = Pr{[y1L y1U ] ∩ C1 = ∅ [ynL ynU ] ∩ Cn = ∅} =
n
Pr{[yiL yiU ] ∩ Ci = ∅}
i=1
where the last equality comes from the fact that {(yiL yiU )}ni=1 are independent. The result then follows by Proposition 1.1.19 in Molchanov (2005). Q.E.D. To complete the proof of Theorem 3.2: (i) Let y(ω) ∈ Y (ω) = [yL (ω) yU (ω)] μ-a.s. Then Ω y dμ ∈ [E(yL ) E(yU )]. This implies E[Y ] ⊂ [E(yL ) E(yU )]. Conversely, since [E(yL ) E(yU )] is a convex set, any b ∈ [E(yL ) E(yU )] can be written as b = αE(yL ) + (1 − α)E(yU ) for some α ∈ [0 1]. Define yb (ω) = αyL (ω) + (1 − α)yU (ω). Then Ω yb dμ = b and yb ∈ S (Y ), and therefore [E(yL ) E(yU )] ⊂ E[Y ]. Lemmas A.2 and A.3 show that {Y Yi : i ∈ N} are i.i.d. compact valued nonempty SVRVs. To verify that the SVRVs are integrably bounded, observe that Y H dμ ≤ E(|yL |) + E(|yU |) < ∞, where the last inequality follows from the assumptions in Theorem 3.2. All the conditions of Theorem A.1 are therefore satisfied. (ii) Our random sets are i.i.d., compact valued (i.e., in2Kk ()), and nonempty by Lemmas A.2 and A.3. We know that Y (ω)H dμ ≤ Ω yL (ω)2 dμ + Ω 2 y (ω) dμ, which is finite by assumption. Therefore, all the conditions Ω U in Theorem A.2 are satisfied. By the definition of the support function, s(p Y )(ω) = yU (ω) if p = 1, s(p Y )(ω) = −yL (ω) if p = −1. The covariance kernel in the theorem follows by simple algebra. Q.E.D. A.4. Proofs for Section 4 In the text we defined Θ, ΘM , and the relevant set valued random variables for the simple case x ∈ . Here we extend these definitions for the case of a column vector x ∈ d , and then prove all the results for this more general case. Let 1 E(x ) 1 x ˆ Σn ≡ Σ≡ x xx E(x) E(xx ) (A.2)
G(ω) =
y(ω) : y(ω) ∈ Y (ω) x(ω)y(ω)
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ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
Gi (ω) =
(A.3)
yi (ω) : yi (ω) ∈ Yi (ω) xi (ω)yi (ω)
Θ = Σ−1 E[G] −1 E(y) y 1 E(x ) d+1 1 = θ ∈ :θ = ∈ S (G) E(xy) xy E(x) E(xx )
and (A.4)
Θ = θ ∈ M
M
d+1
: θ = arg min M
2
(y − θ1 − θ x) dη η ∈ Pyx 2
where Pyx = η : Pr(yL ≤ t x ≤ x0 ) ≥ η((−∞ t] (−∞ x0 ]) ≥ Pr(yU ≤ t x ≤ x0 ) ∀t ∈ ∀x0 ∈ d
η((−∞ +∞) (−∞ x0 ]) = Pr(x ≤ x0 ) ∀x0 ∈ d θ = [ θ1 θ2 ] and the notation x ≤ x0 indicates that each element of x is less than or equal to the corresponding element of x0 . LEMMA A.4: G and {Gi }ni=1 as defined in equation (A.2) are nonempty compact valued SVRVs. PROOF: By Lemma A.2, Y is a nonempty, compact valued SVRV. By Theorem 1.2.7 in Li, Ogura, and Kreinovich (2002) there is a countable selection {fi }∞ i=1 such that fi (ω) ∈ Y (ω), fi ∈ S (Y ) for all i, and cl{fi (ω)} = Y (ω) μ-a.s. Therefore, it is easy to see that for each λ ∈ , λY is an SVRV and for each set A, Y ⊕ A is an SVRV. Observe that x is a random vector defined on the same probability space as Y . Then (xY )(ω) = {x(ω)f (ω) : f (ω) ∈ Y (ω)} =
cl{x(ω)fi (ω)} μ-a.s. Hence, xffii spans the set G as defined in (A.2) and therefore G is an SVRV by Theorem 1.2.7 in Li, Ogura, and Kreinovich (2002). The fact that G is nonempty and compact valued follows from the same arguments as in Lemma A.2. Q.E.D. LEMMA A.5: Under Assumption 4.1, {Gi }ni=1 as defined in equation (A.2) is a sequence of i.i.d. SVRVs. PROOF: (i) {Gi }ni=1 are identically distributed. For A ∈ B (K(d+1 )), μGi (A) = μ(G−1 i (A))
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⎛ ⎜ = μ⎜ ⎝ ω : ∃yi (ω) ∈ Yi (ω) (a1 ad+1 ) ∈ A s.t. ⎡
⎤⎞ yi−1 (a1 ) = ω ⎢ (x1i yi )−1 (a2 ) = ω ⎥ ⎟ ⎢ ⎥⎟ ⎢ ⎥⎠ ⎣ ⎦ (xdi yi )−1 (ad+1 ) = ω Since {(Yi xi )}ni=1 are identically distributed, μGi = μGj for all i and j. (ii) {Gi }ni=1 are independent. For all sets C1 Cn in Kk (d+1 ), −1 μ(G−1 1 (C1 ) Gn (Cn )) = μ(G1 ∩ C1 = ∅ Gn ∩ Cn = ∅) y1 = Pr : y1 ∈ [y1L y1U ] ∩ C1 = ∅ x1 y1
yn : yn ∈ [ynL ynU ] ∩ Cn = ∅ xn yn n yi = : yi ∈ [yiL yiU ] ∩ Ci Pr xi yi i=1
where the last equality comes from the fact that {(yiL yiU xi )}ni=1 are independent. The result then follows by Proposition 1.1.19 in Molchanov (2005). Q.E.D. PROOF OF PROPOSITION 4.1: With x a column vector in d , the sets Θ and Θ are convex and compact subsets of d+1 . By assumption, Θ and ΘM are also nonempty, because the set G is integrably bounded. For any vector θ ∈ Θ or θM ∈ ΘM , let the first entry of such vector correspond to the constant term and be denoted, respectively, by θ1 and θ1M , and let the remaining d entries M ⊂ Θ. be denoted, respectively, by θ2 and θM 2 . We start by showing that Θ M M M Pick a vector [ θ1 (θ2 ) ] ∈ Θ . Then there exists a distribution for (y x) M with x-marginal P(x) denoted ηM (θM 0 such that [ θ1 2 ) ] is a minimizer of M the problem in (A.4). Let (y x) be a random vector with distribution ηM 0 . It follows that y M (ω) ∈ Y (ω) μ-a.s., and therefore [ y M (ω) x y M (ω) ] ∈ G(ω) μ-a.s., from which [ θ1M (θM 2 ) ] ∈ Θ. Conversely, pick a vector [ θ1 (θ2 ) ] ∈ Θ. Then there exists a random vector [ y yx ] ∈ S 1 (G) such that M
−1 1 E(x ) E(y) θ1 = θ2 E(xy) E(x) E(xx )
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801
We show that the corresponding vector (y x) has an admissible probability distribution η ∈ Pyx . Because (y x) is a selection from (Y x), it follows from Theorem 2.1 in Artstein (1983) that for any t ∈ , x0 ∈ d , η(y ≤ t x ≤ x0 ) ≤ μ(Y ∩ (−∞ t] = ∅ x ≤ x0 ) = Pr(yL ≤ t x ≤ x0 ) η(y ≥ t x ≤ x0 ) ≤ μ(Y ∩ [t +∞) = ∅ x ≤ x0 ) = Pr(yU ≥ t x ≤ x0 ) The fact that the marginal of η(y x) is P(x) follows easily. Hence [ θ1 (θ2 ) ] ∈ ΘM . Q.E.D. PROOF OF THEOREM 4.2: Theorem 4.2 can be easily proved by using the following lemma. LEMMA A.6: Let {F Fi : i ∈ N} be i.i.d. nonempty compact valued, integrably bounded SVRVs. Let {Πi : i ∈ N} be i.i.d. random d × d matrices on (Ω A μ) as such that Πi → Π element-by-element, where Π is a d × d nonstochastic matrix with finite elements and Πi has finite elements with probability 1. Then H(Πn F¯n ΠE[F]) → 0 as
PROOF: This is a version of Slutsky’s theorem for d-dimensional SVRVs. We interpret vectors in d as being column vectors. Then H(Πn F¯n ΠE[F]) = max sup inf Πn fn − Πf sup inf Πn fn − Πf
f∈E[F] fn ∈F¯n
f∈E[F] fn ∈F¯n
≤ max sup inf (Πn fn − Πn f + Πn f − Πf) f∈E[F] fn ∈F¯n
sup inf (Πn fn − Πn f + Πn f − Πf)
f∈E[F] fn ∈F¯n
≤ Πn max sup inf fn − f sup inf fn − f fn ∈F¯n
f∈E[F]
f∈E[F] fn ∈F¯n
+ Πn − Π sup f = oas (1) f∈E[F] as
because Πn → Π < ∞ by Slutsky’s theorem, max{supfn ∈F¯n inff∈E[F] fn − f as as supf∈E[F] inffn ∈F¯n fn − f} → 0 by the LLN for SVRVs, Πn − Π → 0 by the continuous mapping theorem, and supf∈E[F] f < ∞ because all selections of an integrably bounded SVRV are integrable. Q.E.D. The proof of Theorem 4.2 follows directly from Lemma A.6, replacing −1 Σˆ −1 = Π, Fi = Gi , and F = G. This is because Lemmas A.4 and A.5 n = Πn , Σ
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show that {G Gi : i ∈ N} are i.i.d. nonempty compact valued SVRVs. To verify that these SVRVs are integrably bounded, observe that GH dμ ≤ E(|yL |) + d E(|yU |) + k=1 [E(|xk yL |) + E(|xk yU |)] < ∞, where the last inequality follows from the assumptions in Theorem 4.2. If in addition E(|yL |2 ) < ∞, E(|yU |2 ) < ∞, E(|xk yL |2 ) < ∞, E(|xk yU |2 ) < ∞, and E(|xk |4 ) < ∞, k = 1 d, then
1 max sup inf gn − g sup inf gn − g = Op √ ¯n n ¯ n g∈E[G] g∈E[G] gn ∈G gn ∈G by the CLT for SVRVs, because G2H dμ ≤ E(|yL |2 ) + E(|yU |2 ) + d 2 2 √1 k=1 [E(|xk yL | ) + E(|xk yU | )] < ∞ and Πn − Π = Op ( n ) by the continuous mapping theorem. Q.E.D. PROOF OF THEOREM 4.3: Before giving this proof, we introduce a definition and three lemmas. DEFINITION 7: Let w be a random vector in d . Its zonoid, Λw , is the Aumann expectation of the random segment in d with the endpoints being the origin (0) and the d-dimensional vector w. The lift zonoid, Λ˜ w , of w is the Aumann expectation of the random segment in d+1 with the endpoints being the origin and the (d + 1)-dimensional vector (1 w ). In the following lemma, for p ∈ [1 ∞), denote by LA = Lp (Ω A d ) the space of A-measurable random variables with values in d such that the Lp norm ξp = [E(ξp )]1/p is finite, and for an SVRV F defined on (Ω A) p p p denote by SA (F) the family of all selections of F from LA , so that SA (F) = p S (F) ∩ LA . p
LEMMA A.7: Let A be an integrably bounded SVRV defined on (Ω A μ). For each σ-algebra A0 ⊂ A there exists a unique integrable A0 -measurable SVRV B, denoted by B = E[A|A0 ] and called the conditional Aumann expectation of A, such that
SA1 0 (B) = cl{E[a|A0 ] : a ∈ SA1 (A)} where the closure is taken with respect to the norm in L1A0 . Since A is integrably bounded, so is B. For the proof, see Molchanov (2005, Theorem 2.1.46). Using the definition of a lift zonoid and the properties of conditional expectations of SVRV, we prove the following result: LEMMA A.8: Define Θ as in (4.2). Under Assumptions 4.2, 4.3, and 4.5, Θ is a strictly convex set.
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
803
PROOF: The random set G can be represented as the convex hull of points with coordinates [ y x y ] , where y(ω) ∈ Y (ω). Define ζ = y U − yL
η = [ yL
x yL ]
Then one can represent G as ˜ G = η ⊕ L
L˜ = co{0 [ ζ
x ζ ] }
Hence, by Theorem 2.1.17 in Molchanov (2005), E[G] is the sum of the expectation of η (a random vector) and the Aumann expectation of the random set L˜ being the convex hull of the origin (0) and [ ζ x ζ ] = ζ [ 1 x ] . If ζ were degenerate and identically equal to 1, then the Aumann expecta˜ would be the lift zonoid of x, Λ˜ x . In our case ζ is a nondegenerate tion E[L] random variable. By Theorem 2.1.47(ii) and (iii) in Molchanov (2005), we can use the properties of the conditional expectation of SVRVs to get ) * ˜ E[G|ζ] = E(η|ζ) ⊕ E[L|ζ] = E(η|ζ) ⊕ ζE co(0 [ 1 x ] )|ζ E[co(0 [ 1 x ] )|ζ] is a lift zonoid, and therefore E[G|ζ] is a (rescaled by ζ) lift zonoid shifted by the vector E(η|ζ). It then follows by Corollary 2.5 in Mosler (2002) that since x has an absolutely continuous distribution with respect to Lebesgue measure on d , E[G|ζ] is a strictly convex set P(ζ)a.s. Premultiplying G by the nonrandom matrix Σ−1 , we obtain that also E[Σ−1 G|ζ] is a strictly convex set P(ζ)-a.s. Therefore, by Corollary 1.7.3 in Schneider (1993), s(· E[Σ−1 G|ζ]) is Fréchet differentiable on d+1 \{0} P(ζ)-a.s. By Theorem 2.1.47(iv) in Molchanov (2005), s(p E[Σ−1 G|ζ]) = E[s(p Σ−1 G)|ζ]. Observing that by Corollary 1.7.3 in Schneider (1993) the gradient of s(p E[Σ−1 G|ζ]) is given by ∇s(p E[Σ−1 G|ζ])
= E[Σ−1 G|ζ] ∩ ϑ ∈ d+1 : ϑ p = s(p E[Σ−1 G|ζ]) we can apply Theorem 2.1.47(v) in Molchanov (2005) to the sets {0} and Σ−1 G (which is absolutely integrable because by assumption G is absolutely integrable; see the proof of Theorem 4.2) to obtain + , ) * * ) E +∇s(p E[Σ−1 G,ζ]) ≤ E E[Σ−1 G|ζ]H = E H(E[Σ−1 G|ζ] {0}) ) * * ) ≤ E H(Σ−1 G {0}) = E [Σ−1 G]H < ∞ Hence s(p Θ) = s(p E[Σ−1 G]) = E[s(p Σ−1 G)] = E[s(p Σ−1 G)|ζ] dP(ζ)
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A. BERESTEANU AND F. MOLINARI
is differentiable at p ∈ d+1 \{0}. The final result follows because the support set in each direction p ∈ Sd−1 of a set which has a differentiable support function contains only one point (Schneider (1993, Corollary 1.7.3). Q.E.D. LEMMA A.9: Let {F Fi : i ∈ N} be i.i.d. nonempty compact valued SVRVs such that E[F2H ] < ∞. Let {Πi : i ∈ N} be i.i.d. random d × d matrices on (Ω A μ) p such that Πi → Π element-by-element, where Π is a d × d nonstochastic matrix with finite elements and Πi has finite elements with probability 1. Then . √ n s(· Πn F¯n ) − s(· Πn E[F]) ⇒ z Π (·) where z Π is a Gaussian random system with E[z Π (p)] = 0 for all p ∈ Sd−1 , and E[z Π (p)z Π (q)] = E[s(p ΠF)s(q ΠF)] − E[s(p ΠF)]E[s(q ΠF)] for all p q ∈ Sd−1 . PROOF: It follows from the assumptions of the lemma that the sets {ΠF ΠFi : i ∈ N} are i.i.d. nonempty, compact valued SVRVs such that E[ΠF2H ] < ∞. By the central limit theorem for C(Sd−1 )-valued random variables in Li, Ogura, and Kreinovich (2002, Proposition 3.1.9), the assumptions of which are verified in Li, Ogura, and Kreinovich (2002, Proof of Theorem 3.1.8), it follows that . √ n s(· Π F¯n ) − s(· ΠE[F]) ≡ znΠ (·) ⇒ z Π (·) in C(Sd−1 ). By Skorokhod representation theorem there exist random elements d d znΠ (·) and z Π (·) with znΠ (·) = znΠ (·) and z Π (·) = z Π (·), such that znΠ (·) → z Π (·) a.s. By Theorem 1.8.12 of Schneider (1993), this convergence is uniform. p Q.E.D. Since Πn → Π, the result follows by standard arguments. The proof of Theorem 4.3 proceeds in steps. Step 1—Derivation of the fact that: √ n[s(· Θˆ n ) − s(· Θ)] .* A √ ) ¯ n ) − s(· Θ)) + s(· Σˆ −1 E[G]) − s(· Θ) = n (s(· Σ−1 G n A
where = means asymptotically equivalent in distribution: Using the definition of Θˆ n , √ n[s(· Θˆ n ) − s(· Θ)] . .* √ )ˆ −1 ˆ −1 ¯ = n s(· Σˆ −1 n Gn ) − s(· Σn E[G]) + s(· Σn E[G]) − s(· Θ)
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
805
By Lemma A.9, .A√ √ ˆ −1 ¯ ¯ n ) − s(· Θ)) n s(· Σˆ −1 n(s(· Σ−1 G n Gn ) − s(· Σn E[G]) = √ ¯ n )− Step 2 —Derivation of the asymptotic distribution of n [ (s(· Σ−1 G −1 d ˆ ˆ s(· Θ)) un (·) ] , where un (p) ≡ (Σn − Σ)Σn p, p ∈ S : Lemmas A.4 and A.5 show that {G Gi : i ∈ N} are i.i.d. nonempty compact valued SVRVs. Observe that by the assumptions of Theorem 4.3, G2H dμ ≤ d E(|yL |2 )+E(|yU |2 )+ k=1 [E(|xk yL |2 )+E(|xk yU |2 )] < ∞ and E(|x|4 ) < ∞. Using the same argument as in the proof of Lemma A.9, it follows by the central limit theorem for C(Sd ) valued random variables in Li, Ogura, and Kreinovich (2002, Proposition 3.1.9) and by the Cramér–Wold device that
Σ−1 ¯ n ) − s(· Θ) √ s(· Σ−1 G z (·) ⇒ n (A.5) un (·) u(·) as a sequence of processes indexed by p ∈ Sd , where for each p ∈ Sd , −1 ( z Σ (p) u(p) ) is a (d + 2)-dimensional normal random vector with −1 E[z Σ (p)] and E[u(p)] = 0. Regarding the covariance kernel, denote by {ρij }, i j = 1 d + 1, the elements of Σ−1 and let ⎡ ⎤ 1 x1i ··· xdi n ⎢ x1i x21i · · · x1i xdi ⎥ 1 ⎢ ⎥ ˆ Σn = Πi with Πi ≡ ⎢ (A.6) ⎥ ⎣ n i=1 ⎦ x2di xdi x1i xdi · · · Then
E
Σ−1 −1 z Σ (p) z (p) u(p) u(p) E[(s(p Σ−1 G))2 ] − E[s(p Σ−1 G)]2 = κpp
κpp Vpp
where Vpp = E[((Πi − Σ)Σ−1 p)((Πi − Σ)Σ−1 p) ] and κpp is a (d + 1) × 1 vector with first element κ1(pp) =
d+1 d+1 )-
.* E[xi−1 s(p Σ−1 G)] − s(p Θ)E(xi−1 ) ρik pk
k=1 i=2
and jth element (j = 2 d + 1) d+1 . ρ1l pl κj(pp) = E[xj−1 s(p Σ−1 G)] − s(p Θ)E(xj−1 ) l=1
806
A. BERESTEANU AND F. MOLINARI
+
d+1 d+1 -
E[xj−1 xi−1 s(p Σ−1 G)]
l=1 i=2
.
− s(p Θ)E(xj−1 xi−1 ) ρil pl with xj the jth element of x. For the next step in this proof it is also useful to observe that (A.7)
) −1 * −1 E z Σ (p)z Σ (q) = E[s(p Σ−1 G)s(q Σ−1 G)] − E[s(p Σ−1 G)]E[s(q Σ−1 G)] ) * E[u(p)u(q)] = Vpq = E ((Πi − Σ)Σ−1 p)((Πi − Σ)Σ−1 q) −1
and E[z Σ (p)u(q)] = κpq , where κpq is a (d + 1) × 1 vector with 1st and jth element (j = 2 d + 1) given, respectively, by (A.8)
κ1(pq) =
d+1 d+1 )-
.* E[xi−1 s(p Σ−1 G)] − s(p Θ)E(xi−1 ) ρik qk
k=1 i=2
κj(pq) =
d+1 .
-
E[xj−1 s(p Σ−1 G)] − s(p Θ)E(xj−1 )
ρ1l ql
l=1
d+1 d+1
+
E[xj−1 xi−1 s(p Σ−1 G)]
l=1 i=2
.
− s(p Θ)E(xj−1 xi−1 ) ρil ql
Step 3—Derivation of the asymptotic distribution of s(· Θ)]: Step 1 gives that
√ ¯ n[s(· Σˆ −1 n Gn ) −
√ ¯ n[s(· Σˆ −1 n Gn ) − s(· Θ)] .
A √ ¯ n ) − s(· Θ)) + s(· Σˆ −1 E[G]) − s(· Θ) = n (s(· Σ−1 G n Recall that by Lemma A.8, since x has an absolutely continuous distribution with respect to Lebesgue measure on d , Θ is a strictly convex set. Hence, its support set in the direction p ∈ Sd is the singleton ξp = Θ ∩ {ϑ ∈ d+1 : ϑ p = s(p Θ)}
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
807
By Corollary 1.7.3 of Schneider (1993), it follows that s(p Θ) is Fréchet differentiable at p ∈ d+1 \{0} with gradient equal to ξp and, therefore, the functional delta method (Van der Vaart (2000, Theorem 20.8)) implies that * √ ) n s(p Σˆ −1 n E[G]) − s(p Θ) √ = n[s(ΣΣˆ −1 n p Θ) − s(p Θ)] * √ ) = n s(p+(Σ − Σˆ n )Σˆ −1 n p Θ) − s(p Θ) . √ s p+ √1n [ n(Σ − Σˆ n )Σˆ −1 n p] Θ − s(p Θ) d −→ −ξp u(p) = 1 √ n
where u(p) is given in equation (A.5). It then follows by another application of the functional delta method that .* √ ) ¯ n ) − s(· Θ)) + s(· Σˆ −1 E[G]) − s(· Θ) n (s(· Σ−1 G n −1
⇒ z Σ (·) − ξ· u(·) −1
as a sequence of processes indexed by p ∈ Sd , with z Σ (p) and u(p) given in −1 equation (A.5) and such that E[z Σ (p) − ξp u(p)] = 0 and )- −1 .- −1 .* E z Σ (p) − ξp u(p) z Σ (q) − ξq u(q) = E[s(p Σ−1 G)s(q Σ−1 G)] − E[s(p Σ−1 G)]E[s(q Σ−1 G)] − ξp κqp − κpq ξq + ξp Vpq ξq where Vpq and κpq are given in (A.7) and (A.8). √ By Hörmander’s embedding theorem and by Lemma A.1, nH(Θˆ n Θ) = √ √ √ ¯ n supp∈Sd |s(p Σˆ −1 ndH (Θ Θˆ n ) = n supp∈Sd [−(s(p n Gn ) − s(p Θ)| and Σ−1 ¯ (p) − ξp u(p) for each p ∈ Sd , the Σˆ −1 n Gn ) − s(p Θ))]+ . Letting v(p) ≡ z result follows by the continuous mapping theorem. The fact that Σ is of full rank implies that Σ−1 p = 0 for each p ∈ Sd . Let 0 / 1 −1 (A.9) f (xi ) = p Σ xi Then
/ −1
s(p Σ Gi ) = sup
p Σ
−1
yi ∈ xi yi
yi xi yi
0 :
yi yi (ω) ∈ Yi (ω) xi yi
808
A. BERESTEANU AND F. MOLINARI
= yiL I(f (xi ) < 0)f (xi ) + yiU I(f (xi ) ≥ 0)f (xi ) where I(·) is the indicator function of the event in brackets. From the arguments in Steps 1–3 it follows that for each p ∈ Sd , . Var(v(p)) = Var s(p Σ−1 Gi ) − s(p Θ) − ξp (Πi − Σ)Σ−1 p = Var yiL I(f (xi ) < 0)f (xi ) + yiU I(f (xi ) ≥ 0)f (xi ) . − ξp Πi Σ−1 p where f (xi ) is defined in (A.9). It follows from the law of iterated expectations that ) Var(v(p)) ≥ E Var yiL I(f (xi ) < 0)f (xi ) + yiU I(f (xi ) ≥ 0)f (xi ) .* − ξp Πi Σ−1 p|xi ) = E (f (xi ))2 I(f (xi ) < 0) Var(yiL |xi ) * + (f (xi ))2 I(f (xi ) ≥ 0) Var(yiU |xi ) ) * ≥ σ 2 E (f (xi ))2 I(f (xi ) < 0) + (f (xi ))2 I(f (xi ) ≥ 0) / 0 2 1 2 −1 =σ E = σ 2 p Σ−1 ΣΣ−1 p > 0 p Σ xi for each p ∈ Sd where the second inequality follows because by assumption, Var(yL |x), Var(yU |x) ≥ σ 2 > 0 P(x)-a.s., and the last inequality follows because Σ is of full rank by assumption. Hence Var(v(p)) > 0 for each p ∈ Sd , and by the same argument as in the proof of Proposition 2.1, the law of vC(Sd ) is absolutely continuous with respect to Lebesgue measure on + . Theorem 2 in Lifshits (1982) assures that the law of supp∈Sd {−v(p)}+ is absolutely continuous with respect to Lebesgue measure on ++ . Q.E.D. √ PROOF OF COROLLARY 4.4: (i) By definition, n[s(p RΘˆ n ) − s(p RΘ)] = √ n[s(R p Θˆ n ) − s(R p Θ)]. Let p˜ ≡R p A simple normalization makes p˜ an element of Sl−1 . Using the definition of Θˆ n , √ ˜ Θˆ n ) − s(p ˜ Θ)] (A.10) n[s(p √ ¯ ˜ Σˆ −1 ˜ Θ)] = n[s(p n Gn ) − s(p . √ )¯ ˜ Σˆ −1 ˜ Σˆ −1 = n s(p n Gn ) − s(p n E[G]) .* ˜ Σˆ −1 ˜ Θ) + s(p n E[G]) − s(p
809
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
√ Σ−1 ˆ −1 ¯ By Lemma A.9, n(s(· Σˆ −1 (·). Regarding the n Gn ) − s(· Σn E[G])) ⇒ z other portion of expression (A.10), using a similar argument as in the proof of Theorem 4.3, we obtain . √ ˜ Σˆ −1 ˜ Θ) n s(p n E[G]) − s(p √ ˜ Θ) − s(p ˜ Θ)) = n(s(ΣΣˆ −1 n p * √ ) ˜ Θ) − s(p ˜ Θ) ˜ Σˆ n − Σ)Σˆ −1 = n s(p−( n p d
˜ → −ξp˜ u(p) where ξ and u are defined in Theorem 4.3. The result follows. (ii) Observing that when R = [ 0 0 · · · 0 1 ], √ nH(RΘˆ n RΘ)
√ = n max |s(−R Θˆ n ) − s(−R Θ)| |s(R Θˆ n ) − s(R Θ)| √ ndH (RΘ RΘˆ n ) ) * √ = n max (s(−R Θˆ n ) − s(−R Θ)) + *
) (s(R Θˆ n ) − s(R Θ)) − Q.E.D.
the result follows easily.
PROOF OF COROLLARY 4.5: Without loss of generality, let us write, for i = 1 n, xi = [ 1 xi1 · · · xid−1 xid ] = [ xi1 xid ] and Gi = {xi yi : yi ∈ Yi }. In this proof we deviate from the notation used in the previous theorems and let X = [ X1 Xd ], X1 = [ x11 · · · xn1 ] , Xd = [ x1d · · · xnd ] . Define P = X(X X)−1 X , M = I − P, Pk = Xk (Xk Xk )−1 Xk , and Mk = n 1 I − Pk for k = 1 d. We have Σˆ n = n1 (X X) and Θˆ n = Σˆ −1 n n i=1 Gi = n −1 ˜ i = {x˜ id yi : yi ∈ Yi }. Let Θˆ d+1n = ˜ ˜ id and G (X X) i=1 Gi . Let Xd = M1 Xd = x n ˜ ˆ ˜ {θd+1 ∈ : [ θ1 θd+1 ] ∈ Θn }. Define Θd+1n = (X˜ d X˜ d )−1 i=1 G i . Note that n (X˜ d X˜ d ) = i=1 x˜ 2id is a scalar. Also note that n
˜i = G
i=1
n {x˜ id yi : yi ∈ Yi = [yiL yiU ]} i=1
=
n [min{x˜ id yiL x˜ id yiU } max{x˜ id yiL x˜ id yiU }] i=1
=
n i=1
min{x˜ id yiL x˜ id yiU }
n i=1
max{x˜ id yiL x˜ id yiU }
810
A. BERESTEANU AND F. MOLINARI
from which 1 Θ˜ d+1n = n i=1
n x˜ 2id
min{x˜ id yiL x˜ id yiU }
i=1
n
max{x˜ id yiL x˜ id yiU }
i=1
Let θˆ d ∈ Θˆ d+1n . Then there exists θˆ = [ θˆ 1 θˆ d ] ∈ Θˆ n , where θˆ 1 ∈ d . Therefore, there exists y = [ y1 · · · yn ] s.t. yi ∈ Yi , ∀i = 1 n and θˆ = (X X)−1 X y. Now by using the Frisch–Waugh–Lovell (FWL) theorem, n ˜ ˜ we can deduce that θˆ d = (X˜ d X˜ d )−1 X˜ d y ∈ (X˜ d X˜ d )−1 i=1 G i = Θd+1n . Thus ˆ ˜ Θd+1n ⊇ Θd+1n . Now take θd ∈ Θ˜ d+1n . Then we can write θd = (X˜ d X˜ d )−1 X˜ d y, where y = [ y1 · · · yn ] s.t. yi ∈ Yi , ∀i = 1 n. A simple manipulation gives
y = Py + My = X(X X)−1 X y + My = X θ¯ + My
(where θ¯ = (X X)−1 X y ∈ Θˆ n )
= X1 θ¯ 1 + Xd θ¯ d + My Now again by using FWL theorem, we get that θ¯ d = (X˜ d X˜ d )−1 X˜ d y = θd ∈ Θˆ d+1n . Hence Θ˜ d+1n ⊆ Θˆ d+1n . Q.E.D. PROOF OF PROPOSITION 4.6: Using Lemma A.9 and Theorem 2.4 in Giné and Zinn (1990), since Σˆ −1∗ is a consistent estimator of Σ−1 , we have that n √
¯∗ ˆ nH(Σˆ −1∗ n Gn Θ n ) , A √ ¯ ∗ ) − s(p Σ−1 G ¯ n )) = n sup,(s(p Σ−1 G n
rn∗ =
p∈Sd
, ˆ , ¯ + (s(p Σˆ −1∗ n Gn ) − s(p Θn )) Looking at the second portion of the above expression, we get ˆ ˆ ∗ ˆ −1∗ ˆ ˆ ˆ ¯ s(p Σˆ −1∗ n Gn ) − s(p Θn ) = s(p+(Σn − Σn )Σn p Θn ) − s(p Θn ) By Theorem 3.3.1 and the discussion on page 160 of Schneider (1993), any nonempty, convex, and compact subset of d+1 can be approximated arbitrarily accurately by a nonempty, strictly convex, and compact subset of d+1 . Hence, for any εn = op ( √1n ) we can find a sequence of nonempty, strictly convex, and compact sets Γ¯n such that H(Θˆ n Γ¯n ) < εn = op ( √1 ), which implies that n
|s(p Θˆ n ) − s(p Γ¯n )| = op ( √1n ) uniformly in p ∈ d+1 . Under the assumptions of Theorem 4.3, s(p Θ) is Fréchet differentiable at p with gradient equal to ξp .
ASYMPTOTICS FOR PARTIALLY IDENTIFIED MODELS
811
Similarly, because Γ¯n is a strictly convex set, s(p Γ¯n ) is Fréchet differentiable at p with gradient ς¯ np = Γ¯n ∩ {γ ∈ d+1 : γ p = s(p Γ¯n )}. Hence * √ ,) ˆ ˆ n, s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Θn ) − s(p Θn ) ) *, , − s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Θ) − s(p Θ) , * √ ) ˆ ˆ ≤ n, s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Θn ) − s(p Θn ) ) *, −1 ¯ ¯ , − s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Γn ) − s(p Σ Γn ) * √ ,) −1 ¯ ¯ + n, s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Γn ) − s(p Σ Γn ) ) *, , − s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Θ) − s(p Θ)
√ √ 1 ≤ nop √ + n|(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p[ς¯ np − ξp ]| + op (1) n = op (1) where the last inequality follows from the fact that s(p Γ¯n ) and s(p Θ) are Fréchet differentiable for any p ∈ d+1 \{0}. The last equality follows because √1 ((Σˆ n − Σˆ ∗n )Σˆ −1∗ n p) = Op ( n ) and because ς¯ np − ξp = op (1) due to the fact that |s(p Θ) − s(p Γ¯n )| = op (1) uniformly in p ∈ d+1 . Hence , A √ ¯ ∗ ) − s(p Σ−1 G ¯ n )) rn∗ = n sup,(s(p Σ−1 G n p∈Sd
., , + s(p+(Σˆ n − Σˆ ∗n )Σˆ −1∗ n p Θ) − s(p Θ) By Theorem 2.4 of Giné and Zinn (1990) and standard arguments, denoting by u∗n (p) ≡ (Σˆ ∗n − Σˆ n )Σˆ −1∗ n p
Σ−1 √ n1 ni=1 s(· Σ−1 G∗i ) − n1 ni=1 s(· Σ−1 Gi ) z (·) n ⇒ ∗ u(·) un (·) −1
as a sequence of processes indexed by p ∈ Sd , where ( z Σ (p) u(p) ) is the (d + 2)-dimensional normal random vector in (A.5). Therefore an application of the delta method for the bootstrap (Van der Vaart and Wellner (2000, Theorem 3.9.11)) implies that * √ ) ¯ ∗ ) − s(· Σ−1 G ¯ n )) + (s(· Σˆ −1∗ G ¯ n ) − s(· Θˆ n )) n (s(· Σ−1 G n n d
−1
→ z Σ (·) − ξp u(·)
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as a sequence of processes indexed by p ∈ Sd . By the continuous mapping d
−1
theorem rn∗ → vC(Sd ) , where v(p) ≡ z Σ (p) − ξp u(p) for each p ∈ Sd . It then follows by the same argument as in the proof of Proposition 2.1 that since Var(v(p)) > 0 for each p ∈ Sd , the critical values of the simulated distribution consistently estimate the critical values of vC(Sd ) , that is, cˆαn = Q.E.D. cα + op (1). REFERENCES ANDREWS, D. W. K. (1997): “A Conditional Kolmogorov Test,” Econometrica, 65, 1097–1128. [796] ANDREWS, D. W. K., S. BERRY, AND P. JIA (2004): “Confidence Regions for Parameters in Discrete Games With Multiple Equilibria, With an Application to Discount Chain Store Location,” Discussion Paper, Yale University. [768] ARTSTEIN, Z. (1974): “On the Calculus of Closed Set-Valued Functions,” Indiana University Mathematics Journal, 24, 433–441. [771] (1983): “Distributions of Random Sets and Random Selections,” Israel Journal of Mathematics, 46, 313–324. [801] ARTSTEIN, Z., AND R. A. VITALE (1975): “A Strong Law of Large Numbers for Random Compact Sets,” The Annals of Probability, 3, 879–882. [772,793] AUMANN, R. J. (1965): “Integrals of Set Valued Functions,” Journal of Mathematical Analysis and Applications, 12, 1–12. [764,771] BERESTEANU, A., AND F. MOLINARI (2006): “Asymptotic Properties for a Class of Partially Identified Models,” Working Paper CWP10/06, CeMMAP. [769,786,788] CHERNOZHUKOV, V., J. HAHN, AND W. NEWEY (2005): “Bound Analysis in Panel Models With Correlated Random Effects,” Discussion Paper, UCLA and MIT. [768] CHERNOZHUKOV, V., H. HONG, AND E. TAMER (2002): “Parameter Set Inference in a Class of Econometric Models,” Discussion Paper, MIT, Duke University, and Northwestern University. [768,780,788] (2007): “Estimation and Confidence Regions for Parameter Sets in Econometric Models,” Econometrica, 75, 1243–1284. [764,767,768,778,780,792] DEBLASI, F. S., AND F. IERVOLINO (1969): “Equazioni Differenziali con Soluzioni a Valore Compatto Convesso,” Bollettino della Unione Matematica Italiana, 4, 491–501. (In Italian.) [774] DOMINITZ, J., AND R. P. SHERMAN (2006): “Identification and Estimation of Bounds on School Performance Measures: A Nonparametric Analysis of a Mixture Model With Verification,” Journal of Applied Econometrics, 21, 1295–1326. [779] GALICHON, A., AND M. HENRY (2006): “Inference in Incomplete Models,” Discussion Paper, Harvard University and Columbia University. [769] GINÉ, E., AND J. ZINN (1990): “Bootstrapping General Empirical Measures,” The Annals of Probability, 18, 851–869. [773,794,810,811] GINÉ, E., M. G. HAHN, AND J. ZINN (1983): “Limit Theorems for Random Sets: Application of Probability in Banach Space Results,” in Probability in Banach Spaces, Vol. IV, Lecture Notes in Mathematics, Vol. 990. Berlin: Springer, 112–135. [793] GINTHER, D. (2000): “Alternative Estimates of the Effect of Schooling on Earnings,” Review of Economics and Statistics, 82, 103–116. [779] GONZALEZ-LUNA, M. L. (2005): “Nonparametric Bounds on the Returns to Language Skills,” Journal of Applied Econometrics, 20, 771–795. [779] GUGGENBERGER, P., J. HAHN, AND K. KIM (2006): “Specification Testing Under Moment Inequalities,” Discussion Paper, UCLA. [769] HAILE, P., AND E. TAMER (2003): “Inference With an Incomplete Model of English Auctions,” Journal of Political Economy, 111, 1–51. [779]
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POLITIS, D. N., J. P. ROMANO, AND M. WOLF (1999): Subsampling. New York: Springer Verlag. [794] ROCKAFELLAR, R. (1970): Convex Analysis. Princeton, NJ: Princeton University Press. [764] ROMANO, J. P., AND A. M. SHAIKH (2006): “Inference for the Identified Set in Partially Identified Econometric Models,” Discussion Paper, Stanford University. [768] ROSEN, A. (2006): “Confidence Sets for Partially Identified Parameters That Satisfy a Finite Number of Moment Inequalities,” Discussion Paper, UCL. [768] SCHARFSTEIN, D. O., C. F. MANSKI, AND J. C. ANTHONY (2004): “On the Construction of Bounds in Prospective Studies With Missing Ordinal Outcomes: Application to the Good Behavior Game Trial,” Biometrics, 60, 154–164. [779] SCHNEIDER, R. (1993): Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and Its Applications. Cambridge, U.K.: Cambridge University Press. [803,804,807,810] STARR, R. M. (1969): “Quasi-Equilibria in Markets With Non-Convex Preferences,” Econometrica, 37, 25–38. [772] STOYE, J. (2007): “Bounds on Generalized Linear Predictors With Incomplete Outcome Data,” Reliable Computing, 13, 293–302. [787] TSIREL’SON, V. S. (1975): “The Density of the Distribution of the Maximum of a Gaussian Process,” Theory of Probability and Its Applications, 24, 847–856. [794] VAN DER VAART, A. W. (2000): Asymptotic Statistics, Statistical and Probabilistic Mathematics. Cambridge, U.K.: Cambridge University Press. [807] VAN DER VAART, A. W., AND J. A. WELLNER (2000): Weak Convergence and Empirical Processes. New York: Springer Verlag. [811]
Econometrica, Vol. 76, No. 4 (July, 2008), 815–839
MEASURING INEQUITY AVERSION IN A HETEROGENEOUS POPULATION USING EXPERIMENTAL DECISIONS AND SUBJECTIVE PROBABILITIES BY CHARLES BELLEMARE, SABINE KRÖGER, AND ARTHUR VAN SOEST1 We combine choice data in the ultimatum game with the expectations of proposers elicited by subjective probability questions to estimate a structural model of decision making under uncertainty. The model, estimated using a large representative sample of subjects from the Dutch population, allows both nonlinear preferences for equity and expectations to vary across socioeconomic groups. Our results indicate that inequity aversion to one’s own disadvantage is an increasing and concave function of the payoff difference. We also find considerable heterogeneity in the population. Young and highly educated subjects have lower aversion for inequity than other groups. Moreover, the model that uses subjective data on expectations generates much better in- and out-ofsample predictions than a model which assumes that players have rational expectations. KEYWORDS: Ultimatum game, inequity aversion, subjective expectations.
1. INTRODUCTION DECISION MAKING UNDER UNCERTAINTY plays an important role in economic theory and practice. Economic models of choice under uncertainty typically assume that agents combine their subjective probability distribution over uncertain outcomes with their preferences to choose the optimal alternative. Experiments of proposal and response have been used to understand the preference structure of decision makers. In such games, the proposers’ payoffs not only depend on their own actions, but also on how responders will react, so that the proposers’ decisions will generally also depend on their expectations about the responders’ behavior. This paper shows, with the specific example of the ultimatum game (Güth, Schmittberger, and Schwarze (1982)), how the empirical content of experiments of proposal and response can be improved in several ways. First, as discussed by Manski (2002), many experimental studies rely on assumptions regarding agents’ expectations about other players’ actions (such as rational 1
A previous version of this paper was circulated under the title “Actions and Beliefs: Estimating Distribution Based Preferences Using a Large Scale Experiment With Probability Questions on Expectations.” We thank Vera Toepoel and Marcel Das and his team at CentERdata for support throughout the experiment. We are also grateful for the comments from participants at the Princeton workshop in econometrics and experimental economics, the 2005 ESA North American Meeting in Tucson, the 2006 ESEM meeting in Vienna, seminar participants at Concordia University, Montréal, Queens University, Kingston, Free University, Brussels, and Tor Vergata University, Rome, three anonymous referees, and a co-editor. The first author thanks the Canada Chair of Research in the Economics of Social Policies and Human Resources for support. The second author thanks the Economic Science Laboratory at the University of Arizona for their kind hospitality. 815
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
expectations) to identify preferences, since observed choice data are generally not rich enough to uncover both expectations (beliefs) and preferences. An exception is Nyarko and Schotter (2002), who collected data on actions and beliefs in a laboratory setting for repeated two player games with simultaneous decision making. They found that players are more likely to best respond to their stated beliefs than to beliefs inferred by the analyst from past decisions. Following Nyarko and Schotter (2002) and Manski (2004), we address the identification problem by collecting data on proposers’ subjective probability distributions over the actions of responders, in addition to the usual experimental data. Thus, we ask proposers direct questions on what they think are the probabilities that responders will make certain decisions. A second distinctive feature of our work is a rich econometric model in which preferences and beliefs vary with (“observed”) heterogeneity captured by observable background characteristics as well as “unobserved” heterogeneity not captured by the observed variables. Several papers provide point estimates of preference parameters, most of them based on experimental data collected in the lab with homogeneous samples (see, e.g., Goeree and Holt (2000)). We extend the existing literature by proposing a model that allows estimation of the entire distribution of preferences in a broad population. By using a large representative sample of the Dutch population rather than a convenience sample of students as used in much of the experimental literature, we follow the recent trend that has been established in the literature, for example, Harrison, Lau, and Williams (2002).2 This broad sample permits us to look into the issue of heterogeneity in preferences and beliefs. Unlike other empirical choice models which incorporate subjective expectation data, our model also allows for correlation between preferences and beliefs of proposers. For example, we allow for the possibility that proposers with optimistic beliefs about the actions of responders also have systematically different preferences, leading to a spurious correlation between beliefs and actions and inducing an endogeneity bias in the estimates of the preference parameters. Avoiding this bias, we can make causal inferences on the effect of beliefs on choices. Our application addresses preferences for inequity aversion, an issue that has received much attention in the recent experimental literature. Early research on social preferences using hypothetical questions suggests that nonlinear asymmetric inequity aversion is prevalent in student populations (Loewenstein, Thompson, and Bazerman (1989)). Recent theories of inequity aversion have mostly focused on asymmetric linear inequity aversion (Fehr and Schmidt (1999)). Here, we use a model allowing for both nonlinear and asymmetric inequity aversion, distinguishing aversion resulting from having a 2 Other studies using representative samples are Bellemare and Kröger (2007) and Fehr, Fischbacher, Rosenbladt, Schupp, and Wagner (2002). They find that behavior in the investment game varies across subgroups of the Dutch and German populations, respectively.
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higher payoff from aversion that results from having a lower payoff than the other player. Our model nests the preferences introduced by Fehr and Schmidt (1999), which can explain the common finding in ultimatum game experiments that many proposers make equitable offers, in contrast to the traditional subgame perfect equilibrium prediction that offers should not exceed the smallest positive amount that can be offered. The latter “traditional” prediction rests on two assumptions: proposers maximize their own expected monetary payoffs and expect responders not to reject any positive offer (for example, because proposers think responders also maximize their own monetary payoffs). Thus, both preferences that imply deviations from expected payoff maximization and the belief that not everyone accepts any positive offer may explain why experimental results are out of line with the traditional prediction. Our framework and the data on beliefs are particularly suited to disentangle these two explanations. Our sample was randomly divided into four groups: proposers and responders in an ultimatum game, and proposers and responders in a dictator game. The proposers in the ultimatum game were asked how much they wanted to offer the responder, and were also asked to state their subjective probabilities that other players would accept or reject any possible offer. Decisions of responders in the ultimatum game were elicited using the strategy method, asking responders to indicate their intended action for all possible offers that could be made. Proposers in the dictator game were asked how much they wanted to give to the other player; responders in this game had no active role. Our structural model estimates combined all the information on the proposers and responders in the ultimatum game. The proposer data in the dictator game were used to test the quality of out-of-sample predictions of this model. Our results can be summarized as follows. First, we found substantial deviations between the average subjective acceptance probabilities reported by the proposers and the actual acceptance rates in the responder data. Second, like Nyarko and Schotter (2002), we found that the model which incorporates proposers’ subjective probability distributions over the possible actions of responders fits the observed choice data better than a model which assumes that proposers have rational expectations. Third, we found substantial unobserved heterogeneity (i.e., heterogeneity in behavior and in expectations not attributable to observable characteristics) in the subjective probabilities as well as in preferences for inequity aversion, with a significant negative correlation between optimism in beliefs and inequity aversion to one’s own disadvantage, suggesting that persons with high optimism have a lower disutility from having less. Fourth, we found that the subgroup of young subjects with a high education level and not participating in the labor market has the most egoistic preferences in the sense that their predicted behavior in the experiment comes the closest to what is predicted by the traditional paradigm of maximizing one’s own payoff. This suggests that inequity aversion is much larger in the Dutch population as a whole than extrapolations based on student samples would suggest. Fifth, we found significant evidence that inequity aversion
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
to one’s own disadvantage is an increasing and concave function of the payoff difference between players. Finally, we used the preference parameter estimates to construct out-of-sample predictions of behavior in the dictator game. The predicted distribution of choices is very close to the sample distribution of actual choices of proposers in the dictator game. The rest of this paper is organized as follows. Our experimental design and procedure are introduced in Section 2. Section 3 presents the empirical model. Section 4 describes the data. The estimation results of the structural model are discussed in Section 5, where we also present simulations to assess the fit of the model and the role of heterogeneity, compare models with subjective and rational expectations, and present out-of-sample predictions. Section 6 concludes. 2. EXPERIMENT Subjects are members of the CentERpanel, an Internet survey managed by CentERdata, consisting of about 2000 households, who answer questions every weekend.3 There are many reasons why the CentERpanel is an attractive medium to conduct experiments. First, it provides access to a representative sample of a population, which is one of the key features of our study. Second, the experiment was double blind as participants were told that they would be anonymously matched and that their identities would not be revealed to the experimenters. Finally, CentERdata reimburses the costs for answering the questionnaire by crediting CentERpoints (hereafter CP; 100 CP = 1 euro) to the respondents’ bank accounts four times a year, allowing us to reimburse participants in a convenient way. We randomly assigned CentERpanel members to the ultimatum game or the dictator game. In both games, a proposer suggested to a responder a split of an amount of 1000 CP (10 euros). We discretized the choice set of the proposer to eight allocations: A ∈ {(1000 0), (850 150), (700 300), (550 450), (450 550) (0 1000)}, where the first and second amounts denote the payoffs for the proposer and the responder, respectively. We ruled out the equal split (500 500) to force proposers to commit themselves to offering either more or less than the equal split, a feature which intuitively should help to increase the efficiency of our estimates. In the dictator game, responders did not have an active role but had to accept the amount offered by the proposer. In the ultimatum game, on the other hand, responders could either accept or reject an offer. In our design, these decisions were elicited following the strategy method (Selten (1967)). Responders were asked whether they would accept or reject each of the eight allocations that 3 For a description of the recruitment, sampling methods, and past usages of the CentERpanel, see www.centerdata.nl. Computer screens from the original experiment (in Dutch) with translations appear in the online appendix (Bellemare, Kröger, and van Soest (2008)).
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could be offered. The response which corresponded to the actual decision of the proposer matched to this responder determined the payoff of both participants. The strategy method overcame the difficulty of having CentERpanel members interact in real time and provided more information, as responses to all eight possible allocations were elicited, including allocations that were never or hardly ever chosen by the actual proposers.4 After all participants had made their decisions, proposers and responders were randomly matched and payoffs were computed based on the decisions of each pair. Payoffs in the ultimatum game corresponded to the allocation chosen by the proposer if this allocation was accepted by the responder. If it was rejected, both participants received nothing. In the dictator game, players received the payoffs chosen by the proposer. The beliefs of the proposers in the ultimatum game were elicited with a series of subjective probability questions. To simplify their task, subjects were asked how many out of 100 persons would accept each offer.5 To be able to account for framing effects, proposers were randomly divided into groups that were asked for either their subjective acceptance or their subjective rejection probabilities for all offers. To avoid the possibility that belief elicitation influences behavior, these questions were asked after players had made their decisions. Subjects were not rewarded based on the accuracy of their expectations.6 The experiment was conducted in March 2004. Contacted individuals received an opening screen saying they were selected for an experiment carried out by a team of university researchers. A detailed description of the game and the payoff structure followed. Each person was informed that if they participated, they would be randomly assigned to one of the roles and would be randomly matched to another panel member playing the opposite role. The role was revealed once a panel member had agreed to participate. We contacted 1410 panel members, of whom 147 declined to participate. Of the 1263 panel members who completed the experiment, we had to exclude 50 people from the analysis because of missing information on some of their observable characteristics. In total, we analyzed the data of 377 (260) proposers and 335 (241) responders in the ultimatum (dictator) game.7 As announced before the exper4 McLeish and Oxoby (2004) found that decisions in the ultimatum game collected with the strategy method were not statistically different from decisions made immediately after having received an offer. Brandts and Charness (2000) did not find significant differences of choices in simple sequential games between these two methods. On the other hand, such differences have been found for binary ultimatum games (Güth, Huck, and Müller (2001)) and sequential bargaining games with costly punishment (Brosig, Weimann, and Yang (2003)). 5 This follows Hoffrage, Lindsey, Hertwig, and Gigerenzer (2000), who found that people are better at working with natural frequencies than with percent probabilities. 6 Several studies have found that rewarding subjects for the accuracy of their expectations using an incentive compatible scoring rule does not produce significantly different elicited expectations; see Friedman and Massaro (1998) and Sonnemans and Offerman (2001). 7 To balance the unequal numbers of players in both roles, some responders were randomly assigned twice to a proposer. As with all other participants, these responders received payments
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
iment, each participant received information on the outcome of the game and their final payoff two weeks after the experiment, and this amount was later credited to their bank accounts. 3. AN EMPIRICAL MODEL OF PREFERENCES AND BELIEFS In this section we introduce a structural econometric model to explain the behavior of proposers and responders in the ultimatum game, as well as the subjective acceptance probabilities reported by the proposers in this game. Proposer behavior in the dictator game was not used for estimation, but was used to evaluate the model’s potential for out-of-sample predictions (see Section 5). We allow for heterogeneity of both preferences and beliefs, which can vary with observed characteristics, such as age, education level, labor force status, and gender (included in a vector xi ), and with unobserved characteristics, not captured by variables in the data set. We also allow choices to vary with the person’s role, since preferences may vary with someone’s role in social interaction (Goeree and Holt (2000), Gächter and Riedel (2005)).8 This enables us to investigate how preferences vary with background characteristics and to test whether preferences are role dependent. Flexible Preferences With Inequity Aversion We assume that subjects have preferences with possibly nonlinear asymmetric inequity aversion. The utility of subject i from payoffs yself to him- or herself and yother to the other player is given by (1)
vi = yself − α1i max{yother − yself 0} − α2i max{yother − yself 0}2 − β1i max{yself − yother 0} − β2i max{yself − yother 0}2
For subjects who only care about their own payoff, α1i , β1i , α2i , and β2i would be zero. The linear inequity model of Fehr and Schmidt (1999) is a special case of equation (1), with α2i = β2i = 0.9 We use the specifications (2)
α1i = exp(xi α1 + uαi )
resulting from only one pairing (the first). The online appendix (Bellemare, Kröger, and van Soest (2008)) provides a summary table of our experimental design. 8 In an alternating offer bargaining experiment, Goeree and Holt (2000) found that proposers have a significantly higher disutility from having less than responders. In a two person bargaining experiment, Gächter and Riedel (2005) observed that the correlation of expectations about what fair divisions are and bargaining behavior varies between roles. 9 In principle, it would also be possible to make vi nonlinear in yself or include interactions of yself and yself − yother . We did not pursue this and expect that it would be hard to obtain accurate parameter estimates, since the utility function is identified only up to a monotonic transformation.
MEASURING INEQUITY AVERSION
(3)
821
β1i = exp(xi β1 + uβi ) α2i = xi α2 β2i = xi β2
where xi = [1 Responderi ] is a vector consisting of the intercept and a dummy Responderi taking a value of 1 for responders and 0 for proposers. This vector is combined with a person’s observable characteristics in the vector xi = [ xi xi ] .10 The terms uαi and uβi reflect unobserved heterogeneity, assumed to be independent of error terms and of xi with a bivariate normal distribution with means zero and an arbitrary covariance matrix. We expected a positive correlation between uαi and uβi since people with a general aversion to inequity might have large values for both. As explained in Section 2, each proposer had eight choices (j = 1 8), involving their own payoffs yself (1) yself (8). Proposers in the ultimatum game did not know whether their offer would be accepted. We assume expected utility maximization, where proposer i uses his own subjective probability Qij that offer j is accepted. Since utility is zero if the offer is rejected, the expected utility of offer j is given by Qij vij , where vij denotes person i’s utility of payoffs (yself (j) yother (j)) (cf. equation (1)) with yother (j) = 1000 − yself (j). Perfect optimization would imply that proposer i chooses the option j that maximizes Qij vij . To allow for suboptimal choices, we add idiosyncratic error terms λi ij and assume that proposer i chooses the option j that maximizes Qij vij + λi ij . We assume that the errors ij are independent of each other and of other variables in the model (i.e., (uαi uβi ) and xi ), and that the difference of any two ij across options follows a logistic distribution. Responder i has to trade off the utility of accepting or rejecting each offer. The utility of rejecting is zero, and the responder’s utility vij of accepting offer j immediately follows from equation (1). A perfectly utility maximizing responder would thus accept offer j if and only if vij > 0. Again, we assume that the responder accepts offer j if vij + λi ij > 0, where λi ij denotes idiosyncratic error terms which are assumed to follow a logistic distribution and to be independent across offers. The random effects uαi and uβi lead to correlation between the choices of the same responder for different offers. The size of the noise parameter λi drives the likelihood of suboptimal choice. We specify the noise parameter λi as λi = exp(xi λ), thus allowing the noise level to vary with background characteristics and role (since xi also includes the role dummy). 10 We also estimated the model with α2i and β2i depending on the complete vector xi , but this gave no significant improvement in the likelihood. See footnote 19 below.
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
Beliefs One way to incorporate beliefs in the model is to simply plug reported acceptance or rejection probabilities into the expected utility comparisons. This would have been justified if reported probabilities exactly equalled the probabilities that proposers use in making decisions, and were independent of erβ rors and (uαi ui ). In that case, the subjective probabilities Qij are observed exogenous variables and the preference distribution could be estimated without modelling the beliefs. This approach is not valid for two reasons. First, there appears to be a framing effect of asking either rejection or acceptance probabilities: the distribution of reported acceptance probabilities differed substantially from the distribution of acceptance probabilities implied by reported rejection probabilities (see next section). This framing effect cannot have affected the answers to the choice questions (since the “framed” questions on the beliefs were asked after the choices had been made), so that reported probabilities must first be purged of the framing bias before using them to construct the expected utilities in the choice model. Second, the acceptance probabilities may have been affected by unobserved factors that also drive the unobserved preference heterogeneity β terms uαi and ui . Early experiments in cognitive psychology (cf. Rapoport and Wallsten (1972)) already showed that subjective probabilities are correlated with utilities over outcomes. If acceptance probabilities are taken as exogenous in the choice model, this correlation would induce an endogeneity bias. Both issues are dealt with by modelling preferences and acceptance probabilities jointly. We allow these probabilities to vary with the same individual characteristics xi as used for the preference parameters. Since (true as well as reported) probabilities may well be 0 or 1, we allow for censoring at 0 and 1, as in a two-limit tobit model. First, we model the true (unobserved) probabilities Qij used in expected utility maximization, not affected by framing or other reporting errors: Qij∗ = xi δ + γj + uPi ⎧ if Qij∗ ≤ 0 ⎨ 0 ∗ Qij = Qij if 0 < Qij∗ < 1 ⎩ 1 if Qij∗ ≥ 1 The censoring guarantees that the true probabilities are between 0 and 1. The choice option effects γj are expected to increase with j for amounts below the equal split, since proposers probably realize that acceptance probabilities rise if the amount offered to the other player increases toward an equal split. Whether or not γj also increases with j beyond the equal split is not a priori clear, since acceptance probabilities and the beliefs about them can increase or decrease depending on the extent of inequity aversion. The unobserved heterogeneity term uPi reflects the proposer’s optimism. We assume that the joint
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distribution of (uαi uβi uPi ) is three-variate normal with mean zero and an arbitrary covariance matrix, independent of error terms and xi . Reported probabilities Pij can deviate from the true probabilities Qij because of framing bias or an idiosyncratic reporting error Pij . The latter are assumed to be identically and normally distributed, independent of each other and of everything else. The framing bias at offer j is modelled in a symmetric way, using a parameter φj . We assume that a positively framed question induces a bias opposite to the bias of a negatively framed question and that the expectations used by proposers in making their decisions are in between. Therefore, we define a framing variable Fi as 1 or −1 if belief questions are framed in terms of accepting or rejecting, respectively. The model for the reported acceptance probabilities Pij is Pij∗ = xi δ + γj + φj Fi + uPi + Pij ⎧ if Pij∗ ≤ 0 ⎨ 0 Pij = Pij∗ if 0 < Pij∗ < 1, ⎩ 1 if Pij∗ ≥ 1. The model for Pij is essentially a two-limit tobit model. The censoring guarantees that reported probabilities are between 0 and 1. Imposing symmetry on the framing effects is necessary to identify the framing parameters and the parameters γj in the true probabilities from reported beliefs alone (i.e., without relying on the data on choices and assumptions on preferences). 4. DESCRIPTIVE STATISTICS Table I describes the individual characteristics included in xi . About half of our participants were men. The median age in the sample was 48 years, with a range of 18 to 89. We used dummies for three age categories. Education level was captured by three dummies, comprising approximately the same numbers of participants. Similarly, we distinguished three income categories (based on a categorical income question in the survey) and four categories of occupational status. Figure 1 presents the distributions of amounts offered by proposers in the ultimatum and dictator games. These distributions exhibit two well-known features (see, e.g., Camerer (2003)): First, proposers sent positive amounts, with the mode around the equal split and with very few amounts much above that. Second, the distribution of amounts offered to the other player in the ultimatum game stochastically dominated that in the dictator game. A chisquared test rejected the null hypothesis that both distributions were the same (p-value = 0.000). Table II presents the choices of responders in the ultimatum game. Each line represents a choice sequence (obtained using the strategy method), with
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST TABLE I DEFINITIONS AND SAMPLE MEANS OF THE EXPLANATORY VARIABLESa Mean
Description
Male Age < 35 Age 35–54 Age > 54 Education Low Med
0.547 0.207 0.477 0.315
1 if man; 0 if woman 1 if below 35 years of age; 0 otherwise 1 if between 35 and 54 years of age; 0 otherwise 1 if above 54 years of age; 0 otherwise
0.312 0.344
High Income Low Med
0.343
1 if primary education or low level vocational training; 0 otherwise 1 if vocational training intermediate level or general secondary education; 0 otherwise 1 if highest vocational training or university education; 0 otherwise
High Market work House work Retired Other
0.320 0.586 0.125 0.157 0.132
0.260 0.420
1 if net household monthly income less than 1000 euro; 0 otherwise 1 if net household monthly income between 1000 and 2500 euro; 0 otherwise 1 if net household monthly income more than 2500 euro; 0 otherwise 1 if working for pay; 0 otherwise 1 if homemaker; 0 otherwise 1 if retired from labor force; 0 otherwise 1 if other labor force status (e.g., student, unemployed); 0 otherwise
a Sample of all players in the ultimatum and the dictator games.
FIGURE 1.—Distributions of amounts offered in the ultimatum game and the dictator game.
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MEASURING INEQUITY AVERSION TABLE II OBSERVED CHOICE SEQUENCES FOR RESPONDERS IN THE ULTIMATUM GAMEa 0
150
300
450
550
700
850
1000
N
1 0 0 0 0 0
1 1 0 0 0 0
1 1 1 0 0 0
Threshold Behavior (N = 177) 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0
1 1 1 1 1 0
1 1 1 1 1 1
17 28 30 89 12 1
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 1 1 0 0 1 1
Plateau Behavior (N = 145) 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1
0 0 0 1 0 0 0 1 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0
20 22 61 6 8 7 2 4 11 1 1 1 1
0.05
015
032
Aggregate Acceptance Rates 093 091 068
058
055
a The table columns present the acceptance decision (coded as 1 if accepted) for all 8 possible offers. N denotes the number of observations. There were 335 responders in the ultimatum game. The responses of 13 participants who answered in an inconsistent way are omitted from the table.
the frequencies in the final column. Choice sequences were grouped into two categories.11 The biggest group (52.8%) was the group of threshold players, who accepted any proposal above a certain amount. The second group (43.3%) was plateau players, who accepted offers in a range excluding both the minimum and maximum amounts that could be offered. The width of the plateau is informative of the degree of inequity aversion to both one’s own and the other player’s disadvantage, as subjects rejected offers giving them either much lower or much higher amounts than the proposer. Plateau response behavior in the ultimatum game has been reported by Huck (1999), Güth, Schmidt, and Sutter (2003), Tracer (2004), Bahry and Wilson (2006), and Hennig-Schmidt, Li, and 11 A small group (3.9%, 13 responders) exhibited what we call inconsistent behavior, with no systematic response pattern. A table containing their responses appears in the online appendix. These responses were left out of the empirical analysis. Estimates of the model including them gave very similar results.
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C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
FIGURE 2.—Proposers’ anticipated acceptance probabilities in the ultimatum game collected using the accept and reject framing.
Yang (2008).12 These studies indicated that broad subject pools and the presence of strong social norms are the most important reasons for nonmonotonic response behavior.13 The sizeable fraction of plateau responders had an immediate consequence for the aggregate acceptance rates, presented at the bottom of Table II. The acceptance rates increased from 5% for low offers to above 90% for proposals around the equal split, but then declined to just above 55% when the complete amount was offered to the responder. Figure 2 presents the means of the subjective acceptance probabilities for each offer separately for those who got the accept and the reject frames. Overall, proposers stated lower expected acceptance rates when asked in terms of acceptance than when asked in terms of rejection. This effect was smallest for amounts sent of 450 and 550 CP. We are not aware of studies that have found framing effects in expected behavior of others, although framing effects on expected own future decisions have been reported in several different contexts (Tversky and Kahneman (1981, 1986), Meyerowitz and Chaiken (1987)). The widely documented fact that “losses loom larger than gains” may explain the framing effect we found (see, e.g., Thaler (1991), Kahneman and Tversky 12 Andreoni and Miller (2002) reported laboratory evidence of nonmonotonic preferences in dictator games, suggesting that people dislike even favorable inequity. 13 In their video experiments, Hennig-Schmidt, Li, and Yang (2008) further found that moral concerns (e.g., losing face) and a perceived low probability of receiving high offers are other motivations for nonmonotonic response behavior.
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(2000)). From the point of view of proposers, the responders’ utility gain of accepting (say) 150 CP may be perceived to be smaller than the responders’ utility loss of rejecting 150 CP. This would be consistent with lower subjective acceptance probabilities in the acceptance frame relative to the rejection frame.14 Interestingly, many proposers in the ultimatum game anticipated the presence of plateau types in the responder population—the acceptance rates expected by proposers decline with offers in excess of an equal split. This result is all the more remarkable since proposers have not had a chance to learn the population pattern of response.15 5. ECONOMETRIC RESULTS We estimated the model in Section 3 by maximum simulated likelihood.16 To assess the impact of using the subjective probability distributions in the choice model, we estimated a second model by assuming that proposers in the ultimatum game had rational expectations about the responders’ acceptance rates. In that model, beliefs of all proposers were equal to observed aggregate acceptance rates of responders (cf. the bottom row of Table II). This model was estimated without the equation for beliefs. Model Fit Table III presents the distributions of observed and predicted offers and responses for different age and education levels. For the complete sample (columns 1 and 2), our model successfully predicted both distributions. In particular, it correctly separated the probability mass of fair offers of 450 CP and 550 CP, and correctly predicted the decline in acceptance probabilities for very advantageous offers. Comparing the decisions of young (<35 years; columns 3 and 4) and old (>54 years; columns 5 and 6) individuals confirmed the quality of the fit. For example, it reproduced the large difference between both age groups’ responses for offers above the equal split, with much more plateau behavior for the old than for the young. Similarly, the differences between high and low educated responders were quite well predicted. Out-of-Sample Predictions Because our model was estimated using only data from decisions in the ultimatum game, we could use the model for out-of-sample prediction of the offer 14
We thank Peter Wakker for the extensive discussion we had on this issue. Anticipation of nonmonotonic response behavior has also been documented by HennigSchmidt, Li, and Yang (2008) for proposers in ultimatum bargaining group experiments. 16 Results were generated using Ox version 3.40 (Doornik (2005)). See the online appendix for details on the estimation procedure. 15
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TABLE III MODEL FIT OF OFFER AND RESPONSE DISTRIBUTIONS FOR THE MODEL USING SUBJECTIVE EXPECTATIONSa All
High Education
2
3
Low Age 4
5
High Age 6
7
8
9
10
11
Dictators 12
Actual
Predicted
Actual
Predicted
Actual
Predicted
Actual
Predicted
Actual
Predicted
Actual
Predicted
Proposers 0 CP 150 CP 300 CP 450 CP 550 CP 700 CP 850 CP 1000 CP Sample size
0.005 0.005 0.042 0.393 0.531 0.013 0.000 0.011
0.004 0.009 0.039 0.399 0.513 0.017 0.008 0.009
0015 0015 0091 0500 0364 0000 0000 0015
0.002 0.018 0.090 0.537 0.325 0.012 0.006 0.007
0000 0000 0023 0282 0656 0023 0000 0015
0004 0005 0021 0313 0628 0017 0006 0005
0000 0000 0016 0398 0544 0041 0000 0000
0003 0006 0024 0346 0584 0016 0007 0010
0008 0000 0059 0378 0546 0000 0000 0008
0003 0020 0046 0392 0517 0015 0004 0002
0.073 0.015 0.108 0.511 0.250 0.019 0.008 0.015
0.069 0.036 0.107 0.442 0.311 0.004 0.002 0.026
Responders 0 CP 150 CP 300 CP 450 CP 550 CP 700 CP 850 CP 1000 CP Sample size
0.053 0.155 0.317 0.932 0.916 0.683 0.581 0.549
377
66 0.052 0.143 0.343 0.919 0.936 0.687 0.589 0.555
322
0048 0253 0446 0952 1000 0916 0892 0867
131 0.077 0.159 0.431 0.953 0.995 0.922 0.868 0.845
83
0010 0100 0210 0900 0840 0490 0340 0310
123 0027 0075 0268 0894 0847 0478 0368 0329
100
0079 0159 0261 0943 0829 0568 0477 0443
119 0073 0113 0306 0907 0891 0597 0475 0411
88
0050 0175 0358 0933 0925 0708 0600 0567
260 0062 0139 0406 0943 0934 0698 0607 0571
120
a “All” refers to overall fit; “Low Age” refers to subjects with less than 35 years of age; “High Age” refers to subjects above 54 years of age; “Low Education” refers to subjects
with either a primary on general secondary education; “High Education” refers to individuals with either university training or high vocational training.
C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
Low Education
1
MEASURING INEQUITY AVERSION
829
distribution of proposers in the dictator game and compare them with the data in that game. Decisions in the dictator game were simulated using the model with the estimated preferences (estimated using only ultimatum game data) and setting acceptance probabilities to 1. The last two columns of Table III present the actual and simulated distributions. The model correctly predicted the modal offer of 450 CP in the dictator game, although it underestimated the corresponding frequency, while overestimating the number of offers of 550 CP. Hence the model had some difficulties in allocating the exact probability mass between offers of 450 CP and 550 CP, but otherwise predicted dictator behavior quite well. Subjective versus Rational Expectations The usefulness of incorporating subjective probabilities is illustrated by comparing the results above with the fit obtained by the model assuming that proposers have rational expectations. The online appendix (Bellemare, Kröger, and van Soest (2008)) presents the fit of the latter model. Both models gave a similar fit of the acceptance probabilities of responders and of the offer distribution of older proposers in the ultimatum game. However, the model with rational expectations gave a substantially worse fit of the offer distribution of young proposers in the ultimatum game. In particular, it underpredicted offers of 450 CP and 550 CP by 7.5 and 5.5 percentage points, respectively, and overpredicted offers of 150 CP by almost 11 percentage points. The model using subjective expectations predicted this distribution much better. The two models also produced different (out-of-sample) predictions of the offer distribution in the dictator game. While the model with subjective expectations predicted the shape of this distribution quite well, the model with rational expectations placed too large of a probability (67.1%) on offers of 0 CP and almost no probability on offers of 450 CP. Hence, the model with subjective expectations fits and predicts substantially better than a model with rational expectations. Parameter Estimates Table IV presents parameter estimates of the model with subjective expectations.17 We only discuss the main significant effects. The null hypothesis that individual characteristics affect α1i and β1i in the same way was rejected (χ211 = 3358, p-value = 0.000). Surprisingly, there were hardly any significant effects of individual characteristics on inequity aversion to one’s own disadvantage (α); the only exception was that the retired respondents were more inequity averse than others. More variables were significant in the equation for inequity at the other player’s disadvantage (β1i ). Particularly, older respondents and those with less education were more inequity averse to the other 17
Estimates for the model imposing rational expectations are in the online appendix.
830
TABLE IV PARAMETER ESTIMATES OF THE STRUCTURAL MODEL WITH SUBJECTIVE EXPECTATIONSa α1
Constant
Male Education Medium High Age Medium High Income Medium High Paid work House work Retired
−0635∗ (0367) 0342 (0309) 0172∗ (0104)
0054 (0146) −0234 (0145)
−0323∗∗ (0113) −0265∗∗ (0125)
0143 (0148) 0234 (0200) −0246 (0221) −0188 (0261) 0168 (0193) −0097 (0252) 0559∗∗ (0274)
0743∗∗∗ (0157) 1093∗∗∗ (0205) −0123 (0164) −0352∗∗ (0178) 0009 (0146) −0361∗ (0204) 0368∗ (0201)
λ
3009∗∗∗ (0458) 0257 (0199) 0405∗ (0223)
P
Offer
0 CP 150 CP −0003 (0028)
300 CP
0159 (0272) −0027 (0323)
−0033 (0029) −0013 (0032)
450 CP
−0169 (0258) 0420 (0370)
−0028 (0037) −0071 (0046)
700 CP
−0200 (0300) −0413 (0285) 0104 (0274) −0525 (0362) −0414 (0353)
0009 (0034) 0011 (0040) 0053 (0039) 0055 (0043) 0073 (0048)
1000 CP
550 CP
850 CP
γ
−0160∗∗∗ (0022) −0140∗∗∗ (0031) −0110∗∗ (0028)
0543∗∗∗ (0056) 0588∗∗∗ (0056)
−0029 (0026) −0035 (0029)
0490∗∗∗ (0059) 0440∗∗∗ (0061)
−0130∗∗ (0036) −0131∗∗∗ (0032)
0339∗∗∗ (0056)
−0118∗∗∗ (0023)
a Standard errors are given in parentheses. ∗ , ∗∗ , and ∗∗∗ indicate significant at the 10%, 5%, and 1% levels, respectively.
Other Parameters
φ
0107∗ (0055) 0311∗∗∗ (0057) 0419∗∗∗ (0053)
0061∗∗ (0002) 0716∗∗∗ (0121) 0806∗∗∗ (0206)
V (Pij ) V (uαi ) β
V (ui ) V (uPi ) β
Corr(uαi ui ) Corr(uαi uPi ) β
Corr(uαi ui ) proposers
α2
responders
α2
proposers
β2
responders
β2
0034∗∗∗ (0004) 0245∗∗∗ (0048) 0921∗∗∗ (0103) −0062 (0182) −0515∗∗∗ (0082) −0277∗∗ (0123) 0221 (0184) −0289 (0277)
C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
Responders
β1
−1584∗∗ (0681) 1707∗∗ (0656) 0014 (0142)
MEASURING INEQUITY AVERSION
831
player’s disadvantage than younger and more highly educated subjects; higher income groups were less inequity averse than lower income subjects. We found no gender differences for disutility of having less and only weak differences for disutility of having more. This is in line with evidence from ultimatum games (e.g., Solnick (2001)) and dictator games (e.g., Bolten and Katok (1995), Andreoni and Vesterlund (2001)) using student subjects who did not know the gender of the other players.18 We found significant evidence of nonlinear aversion to inequity to one’s own disadvantage for both proposers and responders—the estimates of α2 are negative.19 No nonlinearities in preferences were found for inequity aversion at the other player’s disadvantage.20 Interestingly, we found no significant evidence of nonlinear aversion to inequity in the model imposing rational expectations (presented in the online appendix). Finally, role differences in the parameters α1 , β1 , α2 , and β2 were jointly significant (χ24 = 4563, p-value = 0.000), implying that responders had a slightly stronger disutility for having less than proposers. This result might be due to the fact that preferences depend on opportunities and on self-serving notions of fairness.21 Figure 3 sketches the implied population distribution of inequity aversion to one’s own and the other player’s disadvantage. They are based on simulating disutilities for each difference between one’s own and other player’s payments for all subjects in the sample, accounting for observed and unobserved heterogeneity. Each graph presents the mean and the first and third quartiles of the corresponding simulated disutilities. The disutility of having more than the other player is relatively homogeneous in the population and almost always positive and close to linear in (yself − yother ). There is much more dispersion in the disutility of having less than the other player. In line with the negative estimates of α2 , this disutility is a concave function of (yother − yself ), and can be negative for a substantial group of subjects, particularly for high amounts offered.22 This may reflect the fact that respondents were concerned about efficiency, making them more reluctant to reject any offer.23 If efficiency is an additional motivation, then its effect would be captured by the inequity aversion parameters, possibly shifting down the predicted disutilities by a constant 18 Andreoni and Vesterlund (2001) found no significant differences in dictator giving between male and female students for experimental parameters which are comparable to our design. 19 A likelihood ratio test did not reject the null hypothesis that α2 and β2 do not depend on xi 2 (χ20 = 2656, p-value = 0.1481). 20 The parameters β2 for proposers and responders were jointly insignificant (χ22 = 162, p-value = 0.105); the nonlinearity parameters α2 and β2 for proposers and responders were jointly significant (χ24 = 7804, p-value = 0.000). 21 We thank an anonymous referee for pointing this out. See also Babcock and Loewenstein (1997). 22 Loewenstein, Thompson, and Bazerman (1989) also reported evidence suggesting that inequity aversion to one’s own disadvantage is an increasing and concave function of the level of disadvantage. 23 See Engelmann and Strobel (2004) for evidence on efficiency concerns in dictator games.
832
C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
FIGURE 3.—Predicted disutility based on the model estimates. Predicted average disutilities of having less (left) and of having more (right) for the whole population, 25th percentile, mean, and 75th percentile.
for all offers. Our experimental design does not allow us to separately identify this, however. The estimates driving λi suggest that male proposers made more errors than female proposers, although the difference was significant only at the 10% level. Other differences between socioeconomic groups were insignificant and we thus found no evidence that participants’ understanding of the game varies across socioeconomic groups.24 The effects of individual characteristics on beliefs were all insignificant. They were also jointly insignificant (Wald test statistic χ210 = 7043, p-value = 0.721). This result implies that expectations, that is, the perception of the same uncertain situation, did not vary with background characteristics. As expected from the raw data, estimates of framing effect parameters φj were negative and significant except for proposals near the equal split.25 Finally, unobserved heterogeneity played a significant role, as shown by the positive and precisely estimated variances of uαi , uβi , and uPi . The share of total unexplained variation in beliefs of proposers in the ultimatum game captured 24 A joint test did not reject the null hypothesis that λi does not depend on xi (χ210 = 1179, p-value = 0.299). We also estimated a model where λi included interactions between the age and education variables. We found no significant improvement in the log-likelihood function (χ24 = 486, p-value = 0.302). 25 They were also jointly significant: the Wald test gives χ28 = 23182, p-value = 0.000.
MEASURING INEQUITY AVERSION
833
by individual attitudes was 35.7%.26 As conjectured, uαi and uβi were significantly positively correlated, indicating that individuals with a stronger dislike of inequity to their disadvantage also disliked inequity to their advantage. Interestingly, we found that the correlation of uPi with uαi was significant and negative, implying that proposers who are optimistic about the acceptance rates of responders had lower levels of inequity aversion to their own disadvantage. Predicted Marginal Disutility of Inequity The main qualitative implications of our model are graphed in Figure 4. The left graph plots three average predicted marginal disutilities from having less. The dashed line plots the predicted marginal disutility averaged over all players in the game.27 We found that the predicted marginal disutility is positive but decreases with the payoff difference, reflecting that the level of inequity aversion is an increasing and concave function of the level of one’s own disadvantage. The bold line presents the predicted marginal disutility averaged over subjects
FIGURE 4.—Predicted marginal disutilities based on the model estimates. Predicted average marginal disutilities of having less (left) and having more (right) for the whole population (dashed lines), only young (below 35 years of age) and high educated subjects (full lines), and Fehr and Schmidt (1999) predictions (dotted lines). This share is V (uPi )/(V (uPi ) + V (Pij )). The predicted marginal disutilities for player i of having less and having more are α1i + 1i + 2β 2i (yself − yother ), respectively. 2 α2i (yother − yself ) and β 26 27
834
C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
below 35 years of age with a high level of education. We found little difference with the predicted disutilities in the population. The dotted line presents the (constant) average marginal disutility calibrated by Fehr and Schmidt (1999) under the assumption that disutility to one’s own disadvantage is linear in the payoff difference.28 We found that the Fehr and Schmidt calibration is similar to our predictions for low, but not for high levels of inequity. The right graph of Figure 4 plots the corresponding average predicted marginal disutility from having more than the other player. In line with our model estimates, we found that the average predicted marginal disutilities are approximately linear in the payoff difference, suggesting no significant nonlinear relationship between inequity to other’s disadvantage and the level of inequity. On the other hand, young and highly educated subjects had a significantly lower marginal disutility to other’s disadvantage. Interestingly, their average predicted marginal disutility was very close to the average marginal disutility calibrated by Fehr and Schmidt (1999) mainly based on studies with student subjects. Simulations To better understand the interaction between preferences and expectations in subgroups of the population, we used the estimates of the model with subjective expectations to predict the choice distributions for four groups of nonworking men: below 35 years of age with either a university degree or high vocational training (group 1, close to a student sample), below 35 years of age with a primary or lower vocational degree (group 2), above 54 years of age with a university or higher vocational degree (group 3), and above 54 years of age with a primary or lower vocational degree (group 4). Predicted offer distributions in the ultimatum and dictator games are presented in Figure 5. For dictators, beliefs were irrelevant and offers directly revealed underlying preferences. In line with the estimates, the young and educated dictator–proposers made the most selfish offers. The graphs reveal that age differences had a stronger effect on dictator offers than educational differences, in line with the parameter estimates. In the ultimatum game, offers not only reflected preferences but also beliefs. The model predicted that all four groups of proposers make predominantly fair offers, that is, at the equal split. Thus, it seems that young and educated subjects made fair offers for strategic reasons, since lower offers had a smaller subjective probability of being accepted. On the other hand, older and less educated individuals made fair offers because of their preferences—they had large inequity aversion. Their calibrated multinomial distributions for α and β were α ∈ {0(03) 05(03) 1(03) 4(01)} and β ∈ {0(03) 025(03) 06(04)}, where the numbers in parentheses denote the calibrated proportions. 28
MEASURING INEQUITY AVERSION
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FIGURE 5.—Predicted distributions of amounts offered. Predicted offers by proposers in the ultimatum and dictator game for four groups of nonworking men (group 1: <35 years, high educated; group 2: <35 years, low educated; group 3: >54 years, high educated; group 4: >54 years, low educated).
Figure 6 presents the predicted acceptance probabilities of responders in the ultimatum game. All subgroups had similar acceptance probabilities of offers below 550 CP, in line with the parameter estimates which revealed no significant age or education differences in α1i . Acceptance probabilities of offers above 550 CP reflected the differences in β1i . The acceptance probabilities for young and highly educated responders remained above 80%, but were much smaller for older subjects. Again, differences were smaller across education levels than across age groups. Overall, the graphs predict that plateau behavior is a predominant response strategy for older and less educated subjects, while threshold behavior is common among the young and highly educated. 6. CONCLUSION We combined data on decisions and beliefs to formulate a structural microeconometric model to separately identify flexible preferences with nonlinear inequity aversion and subjective expectations. Our results indicated that the model which combined these data fits and predicts the decisions well and better than a model which assumes rational ex-
836
C. BELLEMARE, S. KRÖGER, AND A. VAN SOEST
FIGURE 6.—Predicted acceptance rates. Predicted acceptance rates of responders in the ultimatum game for four groups of nonworking men (group 1: <35 years, high educated; group 2: <35 years, low educated; group 3: >54 years, high educated; group 4: >54 years, low educated).
pectations, that is, that proposers’ beliefs were equal to the observed aggregate acceptance rates of responders. Contrary to the model with rational expectations, the model with subjective expectations also revealed a significant nonlinear relationship between aversion to one’s own disadvantage and the level of inequity. These results suggested that subjective probability data, although suffering from the problem of a substantial framing bias, can be useful to better predict and understand behavior in simple games of proposal and response. Our data also revealed that a large number of responders rejected offers which give them more than the proposers, suggesting strong aversion to inequity at other’s disadvantage. Interestingly, this nonmonotonicity in responder behavior appeared to have been—qualitatively at least—anticipated by proposers in the game. We further found that a substantial part of the nonmonotonicity in responders’ behavior could be explained by important subject pool effects. Inequity aversion, in particular aversion to other’s disadvantage, rises with age and falls with education level. Moreover, we found that young and highly educated participants represent one of the most selfish subgroups of the population under study. This suggests that care must be exerted before making population inferences based on convenience samples of students commonly used in laboratory experiments. It also implies that future research is
MEASURING INEQUITY AVERSION
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warranted to explain these differences (for example, is the age effect a cohort effect or a true age effect?) and to verify them in other ways, such as actual behavior in society (giving to charity, volunteer work, etc.) Our estimates imply that the average inequity aversion preferences of the young and highly educated were very similar to Fehr and Schmidt’s calibrated distribution based on lab experiments, although it seems that our sample made fewer zero offers than students in typical lab experiments. A thorough analysis comparing Internet experiments with “identical” lab experiments (see, e.g., Bellemare and Kröger (2007)), also focusing on error rates and inconsistent choices, is left for future research. Another question that can be addressed in future research is whether experience with these kinds of experiments reduces inconsistencies and plateau behavior. Finally, it has been argued that the ultimatum game is not a proper environment to test specific models of fairness (see Andreoni, Castillo, and Petrie (2003)). While our focus has been on estimating extended Fehr and Schmidt (1999) preferences, our data could possibly be used to fit equally well other models of fairness proposed in the literature (e.g., Bolton and Ockenfels (2000), Cox, Friedman, and Gjerstad (2007)). Future research should aim to collect richer data, possibly by enlarging the choice sets available to players. Such data could then be used to formally test the Fehr and Schmidt model (or some other model) by extending the approach presented here. This would allow the determination of the most relevant preferences that characterize heterogeneity in behavior across a broad population. Département d’économique, Université Laval, Pavillon J. A. De Sève and Centre Interuniversitataire sur le Risque, les Politiques Économiques et l’Emploi, Québec, Québec, G1K7P4 Canada;
[email protected], Département d’économique, Université Laval, Pavillon J. A. De Sève and Centre Interuniversitaire sur le Risque, les Politiques Économiques et l’Emploi (CIRPÉE), Québec, Québec, G1K7P4 Canada;
[email protected], and Dept. of Econometrics and Operations Research and Netspar, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands;
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Econometrica, Vol. 76, No. 4 (July, 2008), 841–907
EQUILIBRIUM IN CONTINUOUS-TIME FINANCIAL MARKETS: ENDOGENOUSLY DYNAMICALLY COMPLETE MARKETS BY ROBERT M. ANDERSON AND ROBERTO C. RAIMONDO1 We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy. KEYWORDS: Dynamic completeness, convergence of discrete-time finance models, continuous-time finance, general equilibrium theory.
1. INTRODUCTION IN AN ARROW–DEBREU MARKET, agents are allowed to shift consumption across states and times by trading a complete set of Arrow–Debreu contingent claims. Mas-Colell and Richard (1991), Dana (1993), and Bank and Riedel (2001) proved existence of equilibrium in a continuous-time Arrow–Debreu market. By contrast, in a securities market, agents are restricted to trading a prespecified set of securities. The securities market is said to be dynamically complete 1 This paper is dedicated to the memory of Gerard Debreu and Shizuo Kakutani. We are very grateful to Jonathan Berk, Theo Diasakos, Darrell Duffie, Steve Evans, Chiaki Hara, Ken Judd, Felix Kubler, John Quah, Paul Ruud, Jacob Sagi, Larry Samuelson, Karl Schmedders, Chris Shannon, Bill Zame, and four anonymous referees for very helpful discussions and comments. Versions of this paper have been presented, starting in 2005, at the National University of Singapore, the Conference in Memory of Gerard Debreu at Berkeley, Arizona State University, the 21 COE Conference at Keio University, the International Congress on Nonstandard Methods in Pisa, the Australasian Meetings of the Econometric Society, the Finance Lunch at the Haas School of Business, the Berkeley–Stanford Theory Workshop, Kyoto University, Seoul National University, and UC Davis; we are grateful to all participants for their comments. The work of both authors was supported by Grant SES-9710424, and Anderson’s work was supported Grant SES-0214164, from the U.S. National Science Foundation. Raimondo’s work was supported by Grant DP0558187 from the Australian Research Council. Anderson’s work was also generously supported by the Coleman Fung Chair in Risk Management at UC Berkeley. Anderson gratefully acknowledges the support and hospitality of the University of Melbourne and the hospitality of the Korea Securities Research Institute. Both authors gratefully acknowledge the support and hospitality of the Institute for Mathematical Sciences, National University of Singapore.
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if agents can, by rapidly trading the given set of securities, achieve all the consumption allocations they could achieve in an Arrow–Debreu market. When the uncertainty is driven by a Brownian motion, markets are said to be potentially dynamically complete if the number of securities is at least one more than the number of independent sources of uncertainty.2 A securities market which is potentially dynamically complete may or may not be dynamically complete. Existence of equilibrium in continuous-time finance models with a single agent has been established in a number of papers (Bick (1990), He and Leland (1995), Cox, Ingersoll and Ross (1985), Duffie and Skiadas (1994), Raimondo (2002, 2005)3 ). Surprisingly, the existing literature does not contain a satisfactory theory of the existence of equilibrium in continuous-time securities markets with more than one agent. With dynamic incompleteness, essentially nothing is known. While we hope that some of the techniques developed in this paper will help in addressing the dynamically incomplete case, we do not here present a result in that case. This paper and the existing literature all deal with the case in which markets are potentially dynamically complete. Every previous paper assumes, in one form or another, that the candidate equilibrium price process is dynamically complete and shows that this implies that the candidate equilibrium is, in fact, an equilibrium; see Basak and Cuoco (1998), Cuoco (1997), Dana and Jeanblanc (2002), Detemple and Karatzas (2003), Duffie (1986, 1991, 1995, 1996), Duffie and Huang (1985), Duffie and Zame (1989), Karatzas, Lehoczky, and Shreve (1990), and Riedel (2001). The form of the assumption varies. Some assume directly that the dispersion matrix of the candidate equilibrium price process is almost surely nonsingular, which is well known to be equivalent to dynamic completeness. Others assume that the dispersion matrix of the candidate equilibrium price process is uniformly positive definite; this is stronger than dynamic completeness and it fails for an open set of primitives in our model. Still others assume that the candidate equilibrium price of consumption is uniformly bounded above and uniformly bounded away from zero, but this is inconsistent with unbounded consumption (found in the Black–Scholes and most other continuous-time finance models) and standard utility functions. Finally, some construct securities which suffice to complete the markets, rather than taking the securities’ dividends as given by the model. But the candidate equilibrium price process is determined from the economic primitives of the model by a fixed point argument, which makes it impossible, except in knife-edge special cases, to determine from the primitives whether or not the dynamic completeness assumption is satisfied. Thus, if we 2 With non-Brownian processes, such as Lévy processes that allow jumps, the number of securities needed for potential dynamic completeness may be larger. 3 All of these papers except Raimondo (2002, 2005) require one or more endogenous assumption(s) that is (are) not expressed solely in terms of the primitives of the model.
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consider one of these models and choose specific utility functions, endowment, and dividend processes, except in the rare cases where we can solve explicitly for the candidate equilibrium, we cannot apply any of the previous theorems to determine whether or not an equilibrium exists. Indeed, the previous results do not rule out the possibility that equilibrium generically fails to exist; see the example in Section 3. While dynamic completeness plays a role in proving the existence of equilibrium, its main application in finance is to derivative pricing. Given a dynamically complete securities price process, options and other derivatives can be uniquely priced by arbitrage arguments and can be replicated by trading the underlying securities. With dynamic incompleteness, arbitrage considerations do not determine a unique option price and replication is not possible. The previous results provide no guarantee that equilibrium prices will support the standard theory of pricing and replicating options, which depends on dynamic completeness. In this paper, we prove that the candidate equilibrium price process is, in fact, dynamically complete and that the candidate equilibrium is, in fact, an equilibrium.4 Dynamic completeness of the candidate equilibrium price process and the existence of equilibrium follow from the way information is revealed by a Brownian motion and from a mild exogenous nondegeneracy condition on the terminal security dividends. To motivate our nondegeneracy condition, suppose we are given a market with K independent sources of uncertainty and K + 1 securities labeled j = 0 K, so the market is potentially dynamically complete. Now suppose that two of the securities are perfect clones of each other, in that they pay exactly the same dividends at all nodes. Clearly, this is the same as a market with K sources of uncertainty and K securities, so it cannot possibly be dynamically complete. Similarly, if the dividend processes of the securities were linearly dependent, we could not possibly have dynamic completeness. Thus, we need to assume that the dividend processes are not linearly dependent. In our model, the dividend of security j at the terminal date T is given as a function Gj (β(T ω)) of the terminal realization β(T ω) of the Brownian motion in state ω. In the important special case in which security 0 is a zero-coupon bond, our condition requires that G1 GK be C 1 functions on some open set V ⊂ RK and that the Jacobian determinant of (G1 GK ) be nonzero at one point x ∈ V . In particular, our assumption depends only on the securities 4 More specifically, we show that any consumption which is adapted to the Brownian filtration and has finite value at the candidate equilibrium price process can be replicated by an admissible self-financing trading strategy. We allow the filtration in the original continuous-time economy to be larger than the Brownian filtration, and the filtration in the Loeb-measure economy we construct is always larger than the Brownian filtration. This does not pose a problem for the existence of equilibrium because any consumption adapted to the larger filtration is a meanpreserving spread of a consumption adapted to the Brownian filtration, so agents’ demands are always adapted to the Brownian filtration.
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dividends at the terminal date, not on the other economic primitives; changing the utility function or endowment of an agent or the initial ownership of the securities has no effect on the existence of equilibrium. Clearly, the nondegeneracy condition is generically satisfied. Moreover, one can easily tell whether the condition is satisfied for any particular value of the economic primitives simply by checking whether a determinant is nonzero at least at one point; this contrasts with the situation in most generic results, in which one knows that the result holds except on a small set of primitives, but it is hard to tell whether the result holds for any specific value of the primitives. If there are just enough securities for potential dynamic completeness, then some form of linear independence of the securities dividends is a necessary condition for dynamic completeness of the Arrow–Debreu securities prices. Thus, some form of linear independence of the dividends is implicitly assumed in all the previous papers. We chose to place our nondegeneracy assumption on the lump terminal dividends because it is convenient to do so. Not all of the previous papers have lump terminal dividends; we believe it should be possible to place the assumption instead on the intermediate flow dividends, although the statement of the assumption would be more complex. We obtain explicit formulas for the equilibrium price process and its dispersion matrix, each trader’s equilibrium securities wealth, and the dispersion matrix for each traders’ equilibrium trading strategy in terms of the equilibrium consumptions; each trader’s equilibrium trading strategy can then be calculated using linear algebra. These formulas are expressed in terms of the equilibrium consumptions, which are not known a priori. However, since the equilibrium is Pareto optimal, there is a vector of Negishi (1960) utility weights λ such that, at each node, the equilibrium consumptions maximize the weighted sum of the utilities of the agents. Thus, the key features of the equilibrium can be calculated explicitly from knowledge of the primitives of the model (endowments and utility functions of the individuals, and the dividends of the securities) and the equilibrium utility weights. Moreover, even if the equilibrium utility weights are not known, the explicit nature of the formulas can potentially be used to establish general properties of equilibria, such as comparative statics results. We prove that all key elements of equilibria of discrete approximations converge to the corresponding elements of equilibria of the continuous-time model.5 This is important for two reasons: First, many people regard the discrete models as the appropriate models for “real” economies. In particular, financial transaction data are very high frequency, but nonetheless discrete. 5
When the continuous-time model has multiple equilibria, it is possible that the equilibria of the discrete approximations will be near one continuous-time equilibrium for some approximations and near a different continuous-time equilibrium for other approximations, so the sequence of equilibria of the discrete approximations converges to the set of equilibria of the continuoustime economy, rather than to a single equilibrium of the continuous-time economy.
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The theoretical literature has focused on continuous time because the formulas are much simpler in continuous time than in discrete time. Cox, Ross, and Rubinstein (1979) showed that one can compute the prices of options when the price of a stock is given by a geometric binomial random walk and showed that the discrete option prices converge to those given by the Black–Scholes formula. Our convergence theorems tell us that the equilibrium pricing formulas from the continuous-time model apply asymptotically to equilibria of the discrete models as the discretization gets finer. This shows that the formulas are applicable to discrete models. It also shows that the formulas can be used in the econometric analysis of data which are high frequency but nonetheless discrete.6 Second, since we show that the equilibria of the discretizations are close to equilibria of the continuous-time economy, algorithms to compute equilibria for discrete economies7 provide a means to compute equilibria of the continuous-time economy. Our starting point is a continuous-time model, on which we state our existence theorem, Theorem 2.1, and our characterization of the elements of the equilibrium, Proposition 2.2. A function is said to be real analytic if, at every point in its domain, there is a power series which converges to the function on an open set containing the point. We assume that the primitives of the economy are given by real analytic functions of time and the current value of the Brownian motion for times t ∈ (0 T ), where T is the terminal date.8 The assumption that utility functions are analytic is not problematic; most of the utility functions commonly studied are in fact analytic. Option payoffs are not analytic because of the kink when the stock price equals the exercise price of the option. However, options can be handled under certain conditions; see the two paragraphs preceding Equation (1) in Section 2. The assumption that the dividends at time t are a function of t and the value of the Brownian motion at time t is discussed in Section 6. 6
As noted above, if the continuous-time economy has multiple equilibria, equilibria of the discrete approximation economies converge to the set of continuous-time equilibria, rather than to a single equilibrium. Nonetheless, this tells us that the formulas given in Proposition 2.2 apply approximately to the equilibria of the discretizations. For example, Equation (6) gives a formula for the dispersion matrix of the securities prices in terms of the Brownian motion, the future dividends, and the endogenously determined prices of consumption. The convergence results in Theorem 4.2 imply that Equation (6) holds asymptotically over any sequence of equilibria of the discrete approximations, using the prices of consumption in the given discrete approximations. 7 See, for example, Brown, DeMarzo, and Eaves (1996) and Judd, Kubler, and Schmedders (2000, 2002, 2003). 8 We also require that the utility functions satisfy Inada conditions (ruling out Constant Absolute Risk Aversion (CARA) utility) and be differentiably strictly concave (ruling out risk neutrality). It is important to emphasize that the primitives of the model, and as we will find, the equilibrium, are analytic functions of time and the Brownian motion, not analytic functions of time alone. Brownian motion is almost surely nowhere differentiable and almost surely of unbounded variation on every interval of time. Consequently, the equilibrium prices and trading strategies are nowhere differentiable, and of unbounded variation, as functions of time.
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We discretize the continuous-time model to construct a sequence of modˆ 9 A naive els; in each, we replace the Brownian motion β by a random walk β. discretization would approximate the K-dimensional Brownian motion by a K-dimensional random walk in which each node has 2K successors, and the random walk moves independently in each direction at each node. However, in a discrete random walk in which each node has 2K successors, one needs at least 2K securities to obtain dynamic completeness; for K > 1, K + 1 < 2K and the discrete model cannot possibly be dynamically complete. So that the discrete approximations correspond closely to the continuous-time model it is critical that the discrete model have the same number of securities as the continuous-time model and that it be dynamically complete.10 Thus, we construct a K-dimensional random walk βˆ in which each nonterminal node has exactly K + 1 successors. Endowments, dividends, and utility functions are induced on the discrete economies from the specification of the continuoustime economy. These discrete economies are general equilibrium incomplete markets (GEI) models; the book by Magill and Quinzii (1996) is an excellent reference on GEI models. Endowments and dividends are perturbed as necessary to ensure the existence of a dynamically complete equilibrium under the Duffie–Shafer (1985, 1986) or Magill–Shafer (1990) theorems. We then prove our convergence theorem, Theorem 4.2, that equilibria of the discrete approximations converge to equilibria of the continuous-time economy. Since equilibrium prices are arbitrage-free, the equilibrium securities prices at the terminal date in the discrete model must be given by the exogenously specified dividends at the terminal date. It is well known that the equilibrium securities price process at time t is the conditional expectation of future dividends, valued at the Arrow–Debreu equilibrium prices of consumption; this comes from the first-order conditions for utility maximization (see, for example, Magill and Quinzii (1996) in the discrete case). We show that because of the smoothness properties of the Gaussian distribution, such conˆ ω)) ∈ ditional expectations are given by real analytic functions of (t β(t (0 T ) × RK , so the equilibrium securities price process is a real analytic funcˆ ω)). Moreover, the dispersion matrix of the securities prices tion of (t β(t ˆ ω)); this is and its associated determinant is a real analytic function of (t β(t a property of the way information is revealed by a Brownian motion and does not depend on any specific assumptions on the functional form of the primitives. 9 Zame (2001) considered a model in which time is discretized but the probability space is not, so each step of the random walk is normally distributed. In that setting, he found that the discrete-time approximation does not converge to the continuous-time limit. Raimondo (2001) studied the existence of equilibrium in models with discrete time and a continuum of states. 10 If one were to extend these methods to the analysis of dynamically incomplete continuoustime models, it would be critical to ensure that the discrete approximation has the same degree of incompleteness as the continuous-time model, so the random walk would have to be chosen carefully in that context also.
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We use nonstandard analysis to project the discrete model back into the continuous-time model and find that the dispersion matrix of the continuoustime securities prices is the projection of the dispersion matrix in the discrete model, and hence is real analytic. A real analytic function cannot be zero on a set of positive measure unless it is identically zero. Since the determinant associated with the dispersion matrix is nonzero on a set of positive measure, it must be nonzero except on a set of measure zero. This implies that the dispersion matrix has rank K except on a set of measure zero, but it is well known that this condition is equivalent to dynamic completeness; thus, we have shown that the candidate equilibrium prices in the continuous-time model are dynamically complete. We verify that the projected prices are equilibrium prices in the continuous-time model. Our proof depends heavily on nonstandard analysis and, in particular, on the nonstandard theory of stochastic processes. Nonetheless, the statements of our main results, Theorems 2.1 and 4.2 and Proposition 2.2, can be understood without any knowledge of nonstandard analysis. Nonstandard analysis provides powerful tools to move from discrete to continuous time, and from discrete distributions like the binomial to continuous distributions like the normal; in particular, it provides the ability to transfer computations back and forth between the discrete and continuous settings. Our sequence of discrete approximations extends to a hyperfinite approximation, one which is infinite but has all the formal properties of finite approximations. In particular, the hyperfinite approximation has a GEI equilibrium which is dynamically complete in the hyperfinite model. We then use nonstandard analysis to produce a candidate equilibrium in the continuous-time model, show that the equilibrium in the hyperfinite model is infinitely close to the candidate equilibrium in the continuous-time model, verify that the candidate prices are dynamically complete, and are in fact equilibrium prices. Anderson (1976) provided a construction for Brownian motion and Brownian stochastic integration using Loeb (1975) measure—a measure in the usual standard sense produced by a nonstandard construction. Anderson’s Brownian motion is a hyperfinite random walk which can simultaneously be viewed as being a standard Brownian motion in the usual sense of probability theory. While the standard stochastic integral is motivated by the idea of a Stieltjes integral, the actual standard definition of the stochastic integral is of necessity rather indirect because almost every path of Brownian motion is of unbounded variation, and Stieltjes integrals are only defined with respect to paths of bounded variation. However, a hyperfinite random walk is of hyperfinite variation, and hence a Stieltjes integral with respect to it makes perfect sense. Anderson showed that the standard stochastic integral can be obtained readily from this hyperfinite Stieltjes integral. We modify Anderson’s construction of the hyperfinite random walk to a random walk with branching number equal to K + 1 and extend the results on stochastic integration to that random walk. We show that equilibrium consumptions are nonzero at all times and states. Consequently, we can use the
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first-order conditions to characterize the equilibrium prices. Then we use the Loeb measure construction to produce a candidate equilibrium of the original continuous-time model. The central limit theorem then allows us to explicitly describe the candidate equilibrium prices as integrals with respect to a normal distribution; however, with more than one agent, the prices depend on the terminal distributions of wealth, which are not described in closed form. We show that the hyperfinite equilibrium is infinitely close to the candidate equilibrium, which implies that the equilibria of the discrete approximations converge to candidate equilibria of the continuous-time model. Finally, in a process analogous to that first used in Brown and Robinson (1975), we show that the candidate equilibrium is an equilibrium of the continuous-time economy.11 2. THE MODEL In this section we define the continuous-time model. There is a single consumption good. Trade and consumption occur over a compact time interval [0 T ], endowed with a measure ν which agrees with Lebesgue measure on [0 T ) and such that ν({T }) = 1. Consumption and dividends on [0 T ) are flows; consumption at the terminal date T is a lump. We choose this formulation to give a finite-horizon model in which the securities will have positive value at the terminal date T ; if securities paid only a flow dividend on [0 T ], they would expire worthless at time T and it would not be possible to formulate our nondegeneracy condition on their dividends. An alternative would be to take an infinite-horizon model, but this would require addressing certain additional technical problems. We think of our finite-time horizon T as a truncation of an infinite-horizon model. The lump of consumption at time T aggregates the flow consumption on the interval (T ∞) in the infinite-horizon model, conditional on the information available at time T . Our nondegeneracy assumption will be imposed on the lump dividend at the terminal date T . The uncertainty in the model is described by a standard K-dimensional Brownian motion β on a probability space (Ω F μ); the components of β are independent of each other and the variance of βk (t ·) equals t. We define I (t ω) = (t β(t ω)). The primitives of the economy—dividends, endowments and utility functions—will be described as functions of I (t ω). We are given a right-continuous filtration {Ft : t ∈ [0 T ]} on (Ω F μ) such that F0 contains all null sets and β is adapted to {Ft }, that is, for all t, β(t ·) is measurable with respect to Ft . Let J = K. There are J + 1 securities, indexed by j = 0 J; security j is in net supply ηj ∈ {0 1}. Security j pays dividends (measured in consumption units) at a flow rate Aj (t ω) = gj (I (t ω)) at times t ∈ [0 T ), and a lump 11 The argument is more complicated here because our continuous-time economy is more complicated than the economy in Brown and Robinson (1975).
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dividend Aj (T ω) = Gj (I (T ω)) at time T . We assume that g : [0 T ] × RK → RK+ is real analytic on (0 T ) × RK and that Gj is continuous almost everywhere on {T } × RK . For example, A0 could be a zero-coupon bond (g0 (t ω) = 0 for t ∈ [0 T ), g0 (T ω) is constant) or Aj (t ω) = eσj ·β(tω) , where σj is the jth row of a J × J matrix σ, for j = 1 J. There are I agents i = 1 I. Agent i has a flow rate of endowment ei (t ω) = fi (I (t ω)) at times t ∈ [0 T ), and a lump endowment ei (T ω) = Fi (I (T ω)), where fi is analytic on (0 T ) × RK and Fi is continuous almost I everywhere on {T } × RK . Let e(t ω) = i=1 ei (t ω) denote the aggregate endowment. The utility functions are von Neumann–Morgenstern utility functions, expectations of functions of the consumption and the process I which are analytic on (0 T ) × RK . More formally, given a measurable consumption function ci : [0 T ] × Ω → R++ , the utility function of the agent is T hi (ci (t ·) I (t ·)) dt + Hi (ci (T ·) I (T ·)) Ui (c) = Eμ 0
where the functions hi : R+ × ([0 T ) × RK ) → R ∪ {−∞} and Hi : R+ × ({T } × RK ) → R ∪ {−∞} are analytic on R++ × ((0 T ) × RK ) and C 2 on R++ × ({T } × RK ), respectively, and satisfy lim
∂hi =∞ ∂c
uniformly over ([0 T ] × RK )
lim
∂Hi =∞ ∂c
uniformly over {T } × RK
c→0+
c→0+
∂hi = 0 uniformly over ([0 T ] × RK ) c→∞ ∂c ∂Hi = 0 uniformly over {T } × RK lim c→∞ ∂c lim
lim hi (c (t x)) = hi (0 (t x)) uniformly over ([0 T ] × RK )
c→0+
lim Hi (c (T x)) = Hi (0 (T x)) uniformly over {T } × RK
c→0+
∂hi > 0 on R++ × ([0 T ] × RK ) ∂c ∂Hi > 0 on R++ × ({T } × RK ) ∂c ∂2 hi < 0 on R++ × ([0 T ] × RK ) ∂c 2 ∂2 Hi < 0 on R++ × ({T } × RK ) ∂c 2
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Note that these conditions are satisfied by all state-independent Constant Relative Risk Aversion (CRRA) utility functions. Note also that we allow quite general state-dependence of the utility function, as long as the statedependence enters through the process I . If the state-dependence were not measurable in the Brownian motions, there would be no hope of obtaining effective dynamic completeness with securities whose dividends are measurable with respect to the Brownian filtration. The assumption that the endowments and utility functions are analytic does not impose serious economic restrictions; for example, all conventional utility functions are in fact analytic. However, the dividend paid by an option is not analytic because of the kink that occurs when the stock price just equals the exercise price of the option on the exercise date. Moreover, shares of a limited liability corporation should not generally be thought of as analytic because they are, in effect, options to claim the corporation’s stream of earnings at an exercise price equal to the corporation’s debt.12 We require analyticity at the intermediate times t ∈ (0 T ) and not at t = T . Thus, our model allows us to include options and shares of limited liability corporations among our basic securities, provided that the exercise date is the terminal date T . In addition, as long as the basic securities pay dividends which are analytic functions of I over (0 T ), there is no problem in using the equilibrium prices derived from the basic securities to price options or other derivatives on those basic securities with exercise date t ∈ (0 T ).13 The equilibrium price of any security, analytic or not, is equal to the expected value of its future dividends, evaluated at the equilibrium consumption prices. Since we prove that the basic securities are essentially dynamically complete, any option or other derivative on the basic securities can be replicated by an admissible self-financing trading strategy on the basic securities. Thus, the equilibrium price of any security is also given by the equilibrium value of the portfolio specified by the replicating strategy at that node. For more detail on equilibrium pricing of options, see Anderson and Raimondo (2005, 2006). Note that we do not require Gj to be analytic, or even differentiable. Standard option payoffs are not differentiable at the strike price, and other derivatives need not be continuous. However, because derivatives represent contracts that need to be understood and enforced, they tend to have relatively simple structures, such as piecewise linearity in the underlying securities; they may or may not be continuous at the boundaries between the regions of linearity. Since our security dividends are analytic functions of the Brownian motion, 12 In particular, geometric Brownian motion is problematic as a model of the price of shares in a limited liability corporation. 13 We do not consider the payout of an option at times t < T as a dividend. The payout of an option is a lump, which does not lie in the consumption set at time t. Rather, if the option is in the money at its expiration date t < T , one exercises it by selling securities to raise the money for the exercise price, takes delivery of the underlying security, and then rebalances one’s portfolio.
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derivatives should be piecewise analytic functions of the Brownian motion. The assumption that the dividend at time T is continuous almost everywhere allows for the possibility that security Aj is a derivative, with exercise date T , on another security or securities, which may or may not be traded. Our formulation also allows Aj to be a stock in a limited liability corporation, since shares in limited liability corporations are in effect options to buy the earnings flow of the firm at an exercise price equal to the firm’s debt. We assume the endowments and dividends satisfy the following growth conditions (note that |x| denotes the Euclidean length of the vector x): (1)
∃r∈R ∀t∈[0T ]x∈RK
|fi (t x)| ≤ r + er|x| |Fi (T x)| ≤ r + er|x| |gj (t x)| ≤ r + er|x| |Gj (T x)| ≤ r + er|x| ∂fi (t x) r|x| ∂x ≤ r + e ∂gj (t x) r|x| ∂x ≤ r + e
These conditions are needed to show that if βˆ n is a sequence of random walks converging to Brownian motion β, the distribution of fi (t βˆ n (t ω)) converges to that of fi (t β(t ω)) and satisfies a uniform L2 integrability condition. In fact, the distribution of fi (t βˆ n (t ω)) will satisfy a uniform Lp integrability condition for all p ∈ [1 ∞). We are grateful to the referees for pointing out Dana’s (1993) treatment of consumptions that are not uniformly bounded away from zero. Let ⎧ I J ⎪ ⎪ ⎪ ⎪ f (t x) + ηj gj (t x) if t < T , i ⎪ ⎨ i=1 j=0 c(t x) = I J ⎪ ⎪ ⎪ ⎪ F (t x) + ηj Gj (t x) if t = T , i ⎪ ⎩ i=1
j=0
denote the social consumption at the node (t x). We make the following joint assumption on the utility functions and the social consumption: there exists r ∈ R such that ∂hi (2) ≤ r + er|x| ∂c c(tx)/I ∂Hi ≤ r + er|x| ∂c c(Tx)/I
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Inequality (2) says if we give each of the I agents an equal share of the social consumption, then every agent’s marginal utility of consumption is bounded above by a linear exponential of the magnitude of the Brownian motion. Note that it implies that the social endowment is strictly positive at almost all nodes. The assumption would follow without any additional assumptions on the utility functions if the social consumption were uniformly bounded away from zero. However, the assumption also allows for the social consumption to approach zero, as long as the marginal utility does not grow too quickly. For example, if each agent has state-independent CRRA utility and the social endowment is a geometric Brownian motion, the condition is satisfied. In continuous-time models, it is commonly assumed that the zeroth security is a money-market account, in other words, it is instantaneously risk-free. Since we are determining securities prices endogenously, the assumption that a security is instantaneously risk-free is an endogenous assumption. For example, if we assume that A0 is a zero-coupon bond (i.e., G0 (t ω) = 0 for t < T and G0 (T ω) = 1, so the dividends of A0 are risk-free), the Arrow–Debreu equilibrium price of A0 will not be instantaneously risk-free except in degenerate situations. However, as long as the equilibrium securities prices are positive Itô processes (and we shall show that they are), one is free to divide the securities price process and the consumption price process by the equilibrium price of the zeroth security. Under this renormalization, relative prices are preserved and the price of the zeroth security is identically 1, which is obviously instantaneously risk-free. When one does this, the set of admissible self-financing trading strategies is left unchanged; see Section 4.8 of Nielsen (1999) or Duffie (1996), as well as Nielsen (2007). Consequently, the consumption processes that lie in the budget set remain invariant, and the renormalized prices are equilibrium prices. This motivates the form of our exogenous nondegeneracy condition on the terminal dividends of the securities. We assume that there is an open set V ⊂ RK such that G0 (T x) > 0 for all x ∈ V and for j = 1 J and i = 1 I, ⎞ ⎛ ∂(G1 /G0 ) (3)
Gj Fi ∈ C (V ) and ∃x∈V 1
⎜ rank ⎝
∂β
(Tx)
∂(GJ /G0 ) ∂β
⎟ ⎠ = K
(Tx)
Note that if A0 is a bond, the rank condition is equivalent to assuming that the K × K matrix ⎞ ⎛ ∂G1 ⎜ ⎝
∂β
(Tx)
∂GJ ∂β
(Tx)
⎟ ⎠
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is nonsingular. This simply says that there is some possible terminal value of the Brownian motion so that the dividends of securities A1 AJ are locally linearly independent. Agent i is initially endowed with deterministic security holdings eiA = (eiA0 eiAJ ) ∈ RJ+1 satisfying I
eiAj = ηj
i=1
Note that the initial holdings are independent of the state ω. We require the following condition for each i = 1 I: (4)
∀(tx)∈[0T )×RK ∀x∈RK
fi (t x) + eiA g(t x) ≥ 0
Fi (T x) + eiA G(T x) ≥ 0
with strictly inequality on a set of positive measure (recall that the single time T carries measure 1). It says that if an agent never traded the securities and simply consumed his/her endowment and dividends, consumption would be nonnegative at each node and strictly positive on a set of positive measure; it guarantees that each agent has strictly positive income at every candidate equilibrium price process. We allow an agent to be endowed with a short position in securities, as long as the condition is satisfied. To define the budget set of an agent, we need to have a way of calculating the capital gain the agent receives from a given trading strategy. In other words, we need to impose conditions on prices and strategies that ensure that the stochastic integral of a trading strategy with respect to a price process is defined. The essential requirements are that the trading strategy at time t not depend on information which has not been revealed by time t, and the trading strategy times the variation in the price yields a finite integral. Specifically, a consumption price process is an Itô process pC (t ω). A securities price process is an Itô process pA = (pA0 pAJ ) : Ω × [0 T ] → RJ+1 such that the associated cumulative gains process t pC (s ω)Aj (s ω) ds γj (t ω) = pAj (t ω) + 0
is a martingale. Securities are priced cum dividend at time T . Given a securities price process pA , an admissible trading strategy for agent i is a row vector process zi which is Itô integrable with respect to γ (written zi ∈ L2 (γ))14 and 14 Formally, zi ∈ L2 (γ) if zi : [0 T ) × Ω → RJ+1 , zi (t ·) is Ft -measurable for all t ∈ [0 T ), and zi is measurable on the product [0 T ) × Ω. If the Itô process γ is given by dγ = a dt + σ dβ, then since γ is a martingale, a must be zero almost surely. Itô integrability with respect to γ
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R. M. ANDERSON AND R. C. RAIMONDO
such that zi dγ is a martingale.15 Given a securities price process pA and a consumption price process pC , the budget set for agent i is the set of all consumption plans ci such that there exists an admissible trading strategy so that ci and ti satisfy the budget constraint pA (t ω) · zi (t ω)
t
= pA (0 ω) · eiA (ω) +
zi dγ 0
t
+
pC (s ω)(ei (s ω) − ci (s ω)) ds 0
for almost all ω and all t ∈ [0 T )
T
0 = pA (0 ω) · eiA (0 ω) +
zi dγ 0
T
+
pC (s ω)(ei (s ω) − ci (s ω)) ds 0
+ pC (T ω)(ei (T ω) − ci (T ω))
for almost all ω
Given a price process p, the demand of the agent is a consumption plan and an admissible trading strategy which satisfy the budget constraint and such that the consumption plan maximizes utility over the budget set. An equilibrium for the economy is a securities price process pA , a consumption price process pC , and a profile of admissible trading strategies z1 zI and consumption plans c1 cI which lies in the demand set so that the securities and goods markets clear, that is, for almost all ω, I
ziAj (t ω) = ηj
for j = 0 J and almost all (t ω)
i=1 I i=1
ci (t ω) =
I i=1
ei (t ω) +
J
ηj Aj (t ω)
for almost all (t ω)
j=0
requires two conditions: that zi (· ω)·a(· ω) ∈ L1 ([0 T ]) almost surely, which is trivially satisfied, and that zi (· ω)σ(· ω) ∈ L2 ([0 T ]) almost surely. A stronger condition is that zi ∈ H2 (γ); the definition of H2 (γ) is the same as that of L2 (γ), except that we strengthen the condition that zi (· ω)σ(· ω) ∈ L2 ([0 T ]) almost surely to zi σ ∈ L2 ([0 T ] × Ω). We also define L2 = L2 (β) and H2 = H2 (β). For more details, see Nielsen (1999). 15 The requirement that zi dγ be a martingale is the standard “admissibility” condition ruling out arbitrage strategies such as the doubling strategy of Harrison and Kreps (1979).
EQUILIBRIUM IN FINANCIAL MARKETS
855
We say that an equilibrium is effectively dynamically complete if every consumption process c which is adapted to the Brownian filtration and which satisfies T pC (t ω)c(t ω) dt < ∞ E pC (T ω)c(T ω) + 0
can be financed by an admissible trading strategy. The following theorem and proposition are proved in Appendix D. THEOREM 2.1: The continuous-time finance model just described has an equilibrium, which is Pareto optimal. The equilibrium pricing process is effectively dynamically complete and the admissible replicating strategies are unique. PROPOSITION 2.2: Let pA , pc , ci , and zi denote the equilibrium securities prices, consumption prices, consumptions, and trading strategies. Let Σ = ∂p∂βA denote the dispersion matrix of the securities prices, let Σi = zi Σ denote the dispersion (row) vector of agent i’s trading strategy, and let Wi = zi · pA denote agent i’s securities wealth. Then (5) pA (t β) = E pC (T β(T ))A(T β(T ))
T
+
pC (s β(s))A(s β(s)) dsβ(t) = β
t
pC (T β(T ))A(T β(T ))(β(T ) − β) T −t T pC (s β)A(s β(s))(β(s) − β) + β(t) = β s−t t I There exists a unique vector λ ∈ RI+ of Negishi utility weights with i=1 λi = 1 such that for all (t β), c1 (t β) cI (t β) solve the problem I I I J (7) λi hi (ci ) : ci = ei (t β) + ηj Aj (t β) max (6)
Σ(t β) = E
i=1
i=1
i=1
j=1
for t ∈ [0 T ) and the analogous problem with Hi substituted for hi for t = T . Each Σi and Wi has a continuous version (still denoted Σi and Wi ): pC (T β(T ))(ci (T β(T )) − ei (T β(T )))(β(T ) − β) (8) Σi (t β) = E T −t T + pC (s β(s)) ci (s β(s)) − ei (s β(s)) t
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R. M. ANDERSON AND R. C. RAIMONDO
× (β(s) − β) (s − t) dsβ(t) = β
−1
(9)
Wi (t β) = E pC (T β(T )) ci (T β(T )) − ei (T β(T ))
T
+
pC (s β(s)) ci (s β(s)) − ei (s β(s)) ds
s=t
β(t) = β There is a open set B of full measure in (0 T ) × RK and analytic functions Z1 ZI with domain B such that zi (t ω) = Zi (I (t ω)) whenever I (t ω) ∈ B; Zi is uniquely determined on B by the linear equations Σi = Zi Σ
Wi = Zi · pA
pA pC c1 cI W1 WI ∈ L2 ; Σ Σ1 ΣI ∈ H2 ; z1 zI ∈ H2 (γ); pA pC W1 WI are functions of I (t ω) which are continuous on [0 T ] × RK and analytic on (0 T ) × RK ; Σ Σ1 ΣI are functions of I (t ω) which are continuous on [0 T ] × V and analytic on (0 T ) × RK ; pc and ci are given separately over [0 T ) × Rk and {T } × RK as functions of I (t ω); the functions over [0 T ) × RK are analytic on (0 T ) × RK . REMARK 2.3: Equation (6) says that the (j k) entry of the dispersion matrix of the securities prices is the integral, over future times s (including s = T ), of the regression coefficient of dividends of the jth security at time s, valued at the equilibrium consumption prices, on the change in the kth component of the Brownian motion up to time s. Equation (8) says that the kth element of Σi is the integral, over future times s (including s = T ), of the regression coefficient of i’s consumption minus i’s endowment at time s, valued at the equilibrium consumption prices, on the change in the kth component of the Brownian motion up to time s. 3. EXAMPLE To appreciate how our existence result compares to the previous literature, it is useful to consider the following parametric family of continuous-time securities markets. It is derived from the Merton (1973, 1990) model, except that our securities are described by their dividend processes, rather than by their price processes. Agent i is endowed with a flow rate of consumption αi for t ∈ [0 T ) and a lump of consumption αi at time T (in Merton, this is described
EQUILIBRIUM IN FINANCIAL MARKETS
857
as a bequest), where αi ∈ R++ . Agent i’s utility for a stream of consumption ci is T hi (ci (s)) ds E hi (ci (T )) + 0
where hi is a state-independent CRRA utility function with coefficient of relative risk aversion γi . There are K + 1 securities, which pay no dividends at times t ∈ [0 T ) and which pay lump dividends at time T . The zeroth security is a zero-coupon bond which pays a lump dividend of one unit of consumption at time T ; the dividends of the other securities are given by eσβ(T ) , where σ is a constant K ×K matrix, so the dividends are terminal values of a K-dimensional geometric Brownian motion. The zeroth security is in zero net supply, while the remaining securities are in net supply one. At time zero, agent 1 has an initial holding of δ1 ∈ RK+1 units of the securities; agent’s 2’s initial holding is (0 1 1) − δ1 . The utility functions and dividend processes are standard benchmarks in continuous-time finance. For what set of parameters does this securities market have an equilibrium? If an equilibrium exists, is it dynamically complete? If α1 = α2 , γ1 = γ2 , and δ1 = δ2 , this is effectively a single agent economy, and it is known that the economy has an equilibrium, but it is not known whether it is dynamically complete for K > 1. If γ1 = γ2 , the candidate equilibrium is Pareto optimal, so there are Negishi weights λ1 and λ2 such that the candidate equilibrium consumptions maximize h(c) = λ1 h1 (c1 ) + λ2 h2 (c2 ) subject to c1 + c2 = c at each node. But the weights λ cannot be computed in closed form and h is not CRRA. Diasakos (2007) recently showed in the case K = 1 that the candidate equilibrium price is dynamically complete.16 His proof does not require computing the candidate equilibrium price in closed form, but it is a difficult calculation. For the case K > 1, the only known way to show the existence of equilibrium has been to compute the candidate equilibrium price process explicitly and check that it is dynamically complete. However, the candidate equilibrium prices can be computed explicitly only for a very small set of parameter values. Thus, none of the previous results rules out the possibility that the set of parameters (α1 α2 γ1 γ2 δ1 σ T ) for which an equilibrium exists has Lebesgue measure zero. Theorem 2.1 implies that if σ is nonsingular, then an equilibrium exists and is dynamically complete for every value of the other parameters, that is, every (α1 α2 γ1 γ2 δ1 T ) ∈ R++ × R++ × R+ × R+ × ({0} × [0 1] × [0 1]) × R++ 16 Diasakos considered a single agent economy, but the Negishi utility h satisfies his assumptions on the utility function.
858
R. M. ANDERSON AND R. C. RAIMONDO
4. DISCRETE APPROXIMATIONS In this section, we describe our process for discretizing a continuous-time model. We then state a theorem indicating that the equilibria of the discretized economy are close to those of the continuous-time economy. This result has two important consequences. First, it provides an effective computational method for computing equilibria of the continuous-time economy. All one has to do is compute, using standard algorithms, an equilibrium of a sufficiently fine discretization. Second, actual securities markets are discrete in a number of important ways. Prices are restricted to lie in a grid, trades are carried out in integral numbers of shares, and trades take a certain amount of time to execute. Continuous-time models are useful because pricing formulas can be expressed more cleanly in continuous time than in discrete time. However, to know that the formulas obtained from continuous-time models are applicable to real markets, we need to know that the behavior of large discrete models is close to the behavior of large continuous-time models. Here is the formal description of our discretization procedure: , where x denotes the greatest Choose n ∈ N. For t ∈ [0 T ], define tˆ = nt n integer less than or equal to x; in particular, Tˆ = nTn . Define T = n1 and T = {0 T 2T Tˆ }. Define a measure νˆ on T by νˆ ({t}) = T if t < Tˆ and νˆ ({Tˆ }) = 1. LEMMA 4.1: We can choose K + 1 vectors v0 vK ∈ RK such that K if j = k, vj · vk = −1 if j = k, K (vk )i (vk )j = δij (K + 1) k=0 K
vk = 0
k=0
where δij = 1 if i = j and 0 otherwise. See Appendix A for the proof. To ensure that the discrete model is potentially dynamically complete, it is critical that every nonterminal node in the random walk have exactly K + 1 successor nodes; see the comments on this in the Introduction. Now, we deˆ filtration {Ft }, and random scribe how we construct our probability space Ω, ˆ Let Ωˆ = {ω : T \ {0} → {0 1 2 K}}. The measure μˆ on Ωˆ is given walk β. ˆ ˆ by μ(A) ˆ = |A| ˆ for every A ∈ F , the algebra of all subsets of Ω; here, |A| de|Ω| notes the cardinality of A. For t ∈ T , Fˆ t is the algebra of all subsets of Ωˆ that respect the equivalence relation ω ∼t ω ⇔ ω(s) = ω (s) for all s ≤ t.
EQUILIBRIUM IN FINANCIAL MARKETS
859
For s ∈ T , define the random variable vs (ω) = vω(s) , where v0 vk are the vectors chosen in Lemma 4.1. If s = t, the random variables vs and vt are independent. Moreover, for each ∈ {1 K}, the random variable (vs ) has mean 0 and standard deviation 1. Define βˆ : T × Ωˆ → RK by
ˆ ω) = β(t
√ vs (ω) T
0<s≤ts∈T
ˆ ·) has variance–covariance βˆ is a K-dimensional random walk. Note that β(t matrix tI, where I is the K × K identity matrix. Define ˆ ω)) Iˆ (t ω) = (t β(t Given a consumption plan cˆ : T × Ωˆ → R+ , the agent’s utility is ˆ = Eμˆ Uˆ i (c)
ˆ ˆ ω) I (t ω)) T hi (c(t s∈T s
ˆ Tˆ ω)) ˆ Tˆ ω) β( + Hi (c( Our discrete economy is a GEI economy, the social endowment (including dividends) is strictly positive at every node, and our utility functions are completely well behaved, so the discrete economy satisfies the assumptions of the Duffie–Shafer (1985, 1986) and Magill–Shafer theorems (1990). These theorems state that there is an arbitrarily small perturbation of the endowments and security dividends such that the perturbed economy has an equilibrium. Let eˆ i (t ω) ≥ fi (Iˆ (t ω)), |eˆ i (t ω) − fi (Iˆ (t ω))| ≤ (T )2 (t < Tˆ ), and eˆ i (Tˆ ω) ≥ Fi (Tˆ ω), |eˆ i (Tˆ ω) − Fi (Iˆ (Tˆ ω))| ≤ (T )2 denote the perˆ let A(t ˆ ω) denote the perturbed diviturbed endowments. For all ω ∈ Ω, ˆ ω) − g(Iˆ (t ω))| ≤ (T )2 for all t < Tˆ , and ˆ ω) ≥ g(Iˆ (t ω)), |A(t dends A(t ˆ Tˆ ω) ≥ G(Iˆ (Tˆ ω)), |A( ˆ Tˆ ω) − G(Iˆ (Tˆ ω))| ≤ (T )2 . Notice that the soA( ˆ ω) + Jj=0 ηj Aˆ j (t ω) > 0. cial consumption e(t Recall there are J + 1 securities indexed by j = 0 J. A securities price process is a function pˆ A : T × Ωˆ → RJ+1 which is adapted with respect to {Fˆ t }t∈Tˆ . We will price securities ex dividend for t < Tˆ ; it will be convenient to price securities cum dividend at t = Tˆ . A consumption price process is a function pˆ C : T × Ωˆ → R+ which is adapted with respect to {Fˆ t }t∈Tˆ .
860
R. M. ANDERSON AND R. C. RAIMONDO
Given a securities price process pˆ A and a consumption price process pˆ C , the associated total gains process γˆ for the securities is defined to be ⎧ ˆ ω) if t < Tˆ , ⎪ ˆ p (t ω) + (T ) pˆ C (s ω)A(s ⎪ A ⎪ ⎨ s∈T s≤t γ(t ˆ ω) = ⎪ ˆ ω) if t = Tˆ . ⎪ pˆ A (t ω) + (T ) pˆ C (s ω)A(s ⎪ ⎩ s∈T s
Note that since securities are priced ex dividend at times t < Tˆ , the dividend at time t is included in the sum and not in the securities price pˆ A ; since securities are priced cum dividend at time Tˆ , the dividend at time Tˆ is included in the securities price pˆ A and not in the sum. A trading strategy17 for agent i is a row vector function zˆ i : (T ∪ {−T }) × Ωˆ → RJ+1 which is adapted with respect to {Fˆ t }t∈T such that zˆ i (−T ω) = eiA (this ensures that each agent arrives at time 0 holding the correct initial securities endowment; there is no trade in goods or securities at time −T ) and zˆ i (Tˆ ω) = zˆ i (Tˆ − T ω). A consumption plan for agent i is a function cˆi : T × Ωˆ → R+ . The budget set for agent i is the set of all consumption plans cˆi such that there exists a trading strategy zˆ i for which cˆi satisfies the budget constraint ˆ ω) · pˆ C (s ω) cˆi (s ω) − eˆ i (s ω) − zˆ i (s − T ω) · A(s = (ˆzi (s ω) − zˆ i (s − T ω)) · pˆ A (s ω) ˆ The budget equation says that the value of the for all s ∈ T and all ω ∈ Ω. agent’s consumption at every node equals the value of the agent’s endowment at that node plus the value of the dividends generated by the agent’s portfolio at that node plus the value of the securities sold by the agent at that node.18 Note that since zˆ is required to be adapted to {Fˆ t }t∈Tˆ , it follows that cˆi is adapted to {Fˆ t }t∈Tˆ . Given a security price pˆ and a consumption price pˆ C , the demand of the agent is a consumption plan and a trading strategy which satisfy the budget constraint, and such that the consumption plan maximizes utility over the budget set. ˆ a consumption An equilibrium for the economy is a security price process p, ˆ ˆ price process pC , trading strategies zi , and consumption plans cˆi which lies in 17 In the discrete model, stochastic integrals with zero drift are automatically martingales, so we do not need to require admissibility as a separate assumption. 18 We take zˆ i (−T ω) = eiA (0), so that the agent enters period 0 holding the securities with which s/he is endowed. Since securities are priced cum dividend at t = Tˆ , we require that zˆ i (Tˆ ω) = zˆ i (Tˆ − T ω).
EQUILIBRIUM IN FINANCIAL MARKETS
861
the demand sets of the agents so that the securities and goods markets clear, ˆ that is, for all t ∈ T and all ω ∈ Ω, I
zˆ i (t ω) = (η0 ηJ )
i=1 I i=1
cˆi (t ω) =
I
eˆ i (t ω) +
i=1
J
ηj Aˆ j (ω t)
j=0
Since short selling is not restricted, the first-order conditions imply that the total gains process γˆ is a vector martingale (pages 230–231 of Magill and Quinzii (1996)). Now we identify the equilibrium trading strategies and the discrete analogue σˆ of the dispersion matrix of the equilibrium securities price process and the dispersion matrix σˆ i of agent i’s securities wealth. We have the equilibrium securities prices pˆ A (t ω) and at the K + 1 successor nodes to (t ω). Because ˆ + T ω) − β(t ˆ ω) span RK , and pˆ A and γˆ the K + 1 possible values of β(t 19 ˆ ω) are both martingales, there is a unique (K + 1) × K matrix process σ(t adapted to Fˆ t , satisfying pˆ A (t + T ω) + pˆ C (t + T ω)A(t + T ω) − pˆ A (t ω) ˆ + T ω) − β(t ˆ ω)) = σ(t ˆ ω)(β(t Because the equilibrium is dynamically complete, σ(t ˆ ω) has rank K for all nodes (t ω). For each consumer i = 1 I, define ˆ i (t ω) = E pˆ C (Tˆ ·)(cˆi (Tˆ ·) − eˆ i (Tˆ ·)) D Tˆ −T
+ T
pˆ C (s ·)(cˆi (s ·) − eˆ i (s ·)) (t ω)
s=0
ˆ i is a martingale; it is the conditional expectation of the From the definition, D difference in value, at the prices of consumption, between i’s consumption and i’s endowment. For each i, there is20 a unique (K + 1) × K matrix process σˆ i (t ω) adapted to Fˆ t such that ˆ i (t ω) = σˆ i (t ω)(β(t ˆ i (t + T ω) − D ˆ + T ω) − β(t ˆ ω)) D 19 20
For details, see Appendix A. For details, see Appendix A.
862
R. M. ANDERSON AND R. C. RAIMONDO
Since the securities wealth of agent i must exactly equal the conditional expectation of the difference in value between i’s future consumption and i’s endowment, zˆ i (t ω) must be the unique K × 1 row vector process adapted to Fˆ t which satisfies the budget constraint and the equation σˆ i (t ω) = zˆ i (t ω)σ(t ˆ ω) Thus, zˆ i is determined uniquely from pˆ A and cˆi by linear algebra. In the following theorem, whose proof is given in Appendix D, we show that equilibria of the discretized economies are close to equilibria of the continuous-time economy. If the continuous-time economy has a unique equilibrium, then pnA , pnC , cni , and λn are given by the unique equilibrium, and thus are independent of n. If the continuous-time economy has multiple equilibria, the equilibrium corresponding to the nth discretization may well cycle among the multiple equilibria. We could, for example, have the discrete-time equilibria close to one continuous-time equilibrium for n odd and close to a different continuous-time equilibrium for n even, so our continuous-time equilibrium will in general have to be chosen differently for different discretizations. THEOREM 4.2: Given a continuous-time economy satisfying the assumptions of Theorem 2.1, let pˆ nA , pˆ nC , cˆni , and zˆ ni denote any equilibrium securities prices, consumption prices, consumptions, and trading strategies of the discretized sequence of economies just described. Let λˆ n denote the vector of Negishi utility weights maximized at that equilibrium. Then there are equilibria of the continuoustime economy with securities prices pnA , consumption prices pnC , consumptions cni , trading strategies zni , and utility weights λn satisfying the conclusions of Theorem 2.1 and Proposition 2.2 such that (10)
|λˆ n − λn | → 0
(11)
pˆ nA − pnA ◦ Iˆ 2 → 0 and in probability
(12)
pˆ nc − pnc ◦ Iˆ 2 → 0
and
in probability (13)
t∈Tn
maxpˆ nc (t ·) − pnc (Iˆ (t ·)) → 0 t∈Tn
and
maxσˆ n (t ·) − Σn (Iˆ (t ·)) → 0
cˆni − cni ◦ Iˆ 2 → 0 and
maxcˆni (t ·) − cni (Iˆ (t ·)) → 0
σˆ n − Σn ◦ Iˆ 2 → 0 in probability
(14)
maxpˆ nA (t ·) − pnA (Iˆ (t ·)) → 0
in probability
t∈Tn
t∈Tn
EQUILIBRIUM IN FINANCIAL MARKETS
(15)
|Uˆ ni (cˆni ) − Uni (cni )| → 0
(16)
|ˆzni − zni ◦ Iˆ | → 0 in probability σˆ ni − Σni ◦ Iˆ 2 → 0 and maxσˆ ni (t ·) − Σni (Iˆ (t ·)) → 0
(17)
863
t∈Tn
in probability (18)
ˆzni · pˆ An − Wni ◦ Iˆ 2 → 0 and maxzˆ ni (t ·) · pˆ An − Wni (Iˆ (t ·)) → 0 t∈Tn
in probability
5. DISCUSSION Nonstandard analysis is a conservative extension of conventional analysis. In other words, the existence of nonstandard models follows from the conventional axioms of set theory and analysis, so the theorems presented here do not depend on any additional set theoretic or analytic axioms. In particular, the proofs presented here can be mechanically translated into standard proofs using only the usual axioms, but the resulting standard proofs would be exceedingly long and unintelligible. We believe it would be extremely difficult to provide tractable standard proofs of the convergence theorems. We think it is likely possible to use the analytic function techniques developed here to prove that, in the settings of the previous literature,21 the candidate equilibrium prices are dynamically complete. This would allow one to remove the assumption of dynamic completeness of the candidate equilibrium price process from the papers cited, and would rely on the techniques used in those papers and our analytic function techniques, but would not require nonstandard analysis. We have chosen not to proceed in this direction for the following three reasons: First, our ultimate goal is to extend these methods to obtain the existence of equilibria in continuous-time financial markets which are dynamically incomplete. The approach in the previous literature begins with an Arrow–Debreu complete markets equilibrium. Since there is no known theorem asserting the existence of equilibrium with more than one agent, an infinite-dimensional commodity space, and incomplete markets, there seems to be no hope of extending the approach in the previous literature to incomplete markets. By contrast, our approach begins with a GEI equilibrium in a discrete setting. Many steps in our argument work just as well for the dynamically incomplete case as for the dynamically complete case. While critical problems remain to be solved, our methods at least provide a way of attacking the dynamically incomplete case. Second, we feel the convergence theorem is at least as important as the existence theorem. As is often the case in nonstandard analysis, our convergence theorem is essentially an immediate corollary 21
See the references listed in the fifth paragraph of Section 1.
864
R. M. ANDERSON AND R. C. RAIMONDO
of the nonstandard existence proof; however, we doubt that a tractable standard proof of the convergence theorem can be given. Third, while our proof makes extensive use of nonstandard analysis and stochastic processes, it is completely independent of knowledge of functional analysis. Since most who work in continuous-time finance have a background in stochastic processes but not in functional analysis, it is desirable to have a proof that does not depend on functional analysis. Anderson’s construction of Brownian motion has been used to answer a number of questions in stochastic processes, and we are able to make do in this paper with slight extensions of it. However, we anticipate that extending this work to the dynamically incomplete case, or the case of dividends with jumps, will require using subsequent work in nonstandard stochastic analysis.22 6. MEAN REVERSION We assumed throughout that the primitives of the model are given as functions of time and the current value of the Brownian motion. It would be desirable to allow the primitives to be given by more general Itô processes. Significant technical problems would have to be overcome, so we are leaving this problem for future work. Most models of interest rate determination involve mean reversion, so in this section we briefly sketch how to incorporate mean reversion into our model. 22 The work of Keisler (1984) on stochastic differential equations with respect to Brownian motion, and the work by Albeverio and Herzberg (2006), Hoover and Perkins (1983a, 1983b), Lindstrøm (1980a, 1980b, 1980c, 1980d, 2004, 2008), and Ng (2008) on stochastic integration with respect to more general martingales, including Lévy processes, is likely to be of particular relevance. The nonstandard theory of stochastic integration has previously been applied to option pricing by Cutland, Kopp, and Willinger (1991a, 1991b, 1991c, 1993, 1995a, 1995b) and Cutland, Kopp, Willinger, and Wyman (1997). Those papers, which are not in an equilibrium context, primarily concern convergence of discrete versions of options to continuous-time versions, and their methods can likely be used to establish convergence results for the equilibrium option pricing formulas developed by Anderson and Raimondo (2005, 2006). Anderson and Rashid (1978) provided a nonstandard characterization of weak convergence. Cutland, Kopp, and Willinger (1993, 1995b) and Cutland (2000) proposed a convergence notion for stochastic processes, D2 -convergence, which is stronger than the topology of weak convergence. The statement of the convergence notion is nonstandard, and to the best of our knowledge no one has identified a standard topology on stochastic processes that corresponds to D2 -convergence. The equilibria of our discrete approximations D2 -converge to the set of equilibria of the continuous-time economy. Ng (2003) provided a treatment of mathematical finance based on hyperfinite random walks, and in particular presented the Cox–Ross–Rubinstein (1979) binomial model in a nonstandard context. Khan and Sun (1997, 2001) related the capital asset pricing model and arbitrage pricing theory in a single-period setting. Malliavin calculus provides an approach to continuous-time finance with asymmetric information; nonstandard treatments of Malliavin calculus were given by Cutland and Ng (1995) and Osswald (2006). Keisler (1996) presented a nonstandard stochastic process model of nontatonnement price adjustment; agents are randomly chosen to come to the market, and their trades move the market price rapidly toward equilibrium.
EQUILIBRIUM IN FINANCIAL MARKETS
865
The canonical mean-reverting process is the Ornstein–Uhlenbeck process, which is by definition a solution φ of the stochastic differential equation (19)
dφ = a(b − φ) dt + σ dβ
where a, b, and σ are real constants with a > 0. The Vasicek (1977) model assumes that the short-run risk-free rate of return r is an Ornstein–Uhlenbeck process; one can then derive the dynamics of bonds of various maturities. From an equilibrium perspective, the short-run risk-free interest rate should be determined in equilibrium, not given as a primitive of the model. When markets are dynamically complete, the securities market equilibrium is the Arrow– Debreu equilibrium, in which the price of consumption is the marginal utility of consumption. In the proof of Theorem 2.1, we show that one can construct a riskless security denominated in utils, and that the risk-free interest rate in this normalization is zero. Observable interest rates are typically expressed in terms of monetary units, but ours, like most general equilibrium models, is not a monetary model. However, there are securities, such as treasury inflationprotected securities (TIPS), which are automatically adjusted for inflation, and it is reasonable to think of TIPS as being denominated in units of consumption. If we denominate prices in units of consumption, the equilibrium real risk-free interest rate is the negative of the drift term of the price of consumption: r=−
∂pC ∂t
+
1 ∂2 pC 2 ∂β2
pC
r will be an Ornstein–Uhlenbeck process only for a knife-edge set of primitives of our equilibrium model. Our model can easily generate mean reversion of the risk-free rate. For example, suppose that the social consumption is a periodic function of β such as 2 + sin β(t ω) and that the utility functions are state- and time-independent (in particular, there is no time discounting). When sin β(t ω) > 0, social consumption has a downward drift, so pC (t ω) has an upward drift, so r(t ω) < 0; one can show that r(t ω) has an upward drift. Similarly, when sin β(t ω) < 0, social consumption has an upward drift, so pC (t ω) has a downward drift, so r(t ω) > 0 and has a downward drift. Adding impatience and expressing the interest rate in nominal rather than real terms will produce a risk-free rate which is positive and mean reverting. Suppose φ is an Ornstein–Uhlenbeck process satisfying Equation (19). It is not possible to write φ(t ω) as a function of t and β(t ω) alone, so our theorem as stated does not cover the case in which the social consumption equals
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R. M. ANDERSON AND R. C. RAIMONDO
φ(t ω). However, our proof can be extended to the case in which the primitives are analytic functions of t and φ(t ω). They key is that φ is a Markov process, and the conditional distribution of φ(T ), conditional on (t φ(t)), is well behaved in the same ways that the conditional distribution of β(T ), conditional on (t β(t)), is well behaved. Dept. of Economics, University of California at Berkeley, 508-1 Evans Hall 3880, Berkeley, CA 94720-3880, U.S.A.;
[email protected] and Dept. of Economics, University of Melbourne, Victoria 3010, Australia;
[email protected]. Manuscript received September, 2006; final revision received December, 2007.
APPENDIX A: DETAILS ON THE DISCRETE MODEL In this appendix, we provide more detail on the analysis of the discrete model. PROOF OF LEMMA 4.1: Suppose first K = 1; let v0 = (1) and v1 = (−1). Then v0 · v0 = v1 · v1 = 1 and v0 · v1 = −1, (v0 )1 (v0 )1 + (v1 )1 (v1 )1 = 1 + 1 = 2, and (v0 )1 + (v1 )1 = 1 − 1 = 0. Now suppose we have chosen v0 vK ∈ RK such that vj · vk = −1 if j = k K K and vk · vk = K; k=0 (vk )i (vk )j = δij (K + 1) and k=0 vk = 0. Let ⎧ K+2 K+2 1 ⎪ ⎪ ⎪ (vk )1 (vk )K − √ ⎪ ⎪ K+1 K+1 K+1 ⎨ v˜ k = for k = 0 K, ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎩ (0 0 K + 1) for k = K + 1. For j k ∈ {0 K}, K+2 1 v j · vk + K+1 K+1 ⎧ K + 2 1 ⎪ ⎪ if j = k, ⎨ K + 1 (−1) + K + 1 = −1 = ⎪ K+2 1 K 2 + 2K + 1 ⎪ ⎩ K+ = = K + 1 if j = k, K+1 K+1 K+1 v˜ j · v˜ K+1 = −1
v˜ j · v˜ k =
v˜ K+1 · v˜ K+1 = K + 1
EQUILIBRIUM IN FINANCIAL MARKETS
867
For i j ∈ {1 K}, K+1 K (v˜ k )i (v˜ k )j = (v˜ k )i (v˜ k )j + 0 k=0
k=0
K+2 (vk )i (vk )j K + 1 k=2 K
=
K+2 (K + 1)δij K+1 = δij (K + 2) =
For i ∈ {1 K} and j = K + 1, K+1 K K+2 1 (vk )i − √ (v˜ k )i (v˜ k )j = +0· K+1 K+1 K+1 k=0 k=0 √
=−
K+2 (vk )i + 0 K + 1 k=0 K
=0 = δij (K + 2) For i = K + 1 and j ∈ {1 K}, note that the formula is invariant under switching i and j. Finally, for i = j = K + 1, K+1 (v˜ k )i (v˜ k )j = (K + 1) k=0
1 + (K + 1) K+1
= 1 + (K + 1) =K+2 = δij (K + 2) For j = 1 K, K+1 (v˜ k )j = k=0
K+2 (vk )j = K + 1 k=0 K
K+2 · 0 = 0 K+1
Moreover, K+1 K+1 + K + 1 = 0 (v˜ k )K+1 = − √ K+1 k=0
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R. M. ANDERSON AND R. C. RAIMONDO
K+1 This shows that k=0 v˜ k = 0. Thus, by induction, we can choose vectors v0 vK ∈ RK with the specified properties. Q.E.D. Determination of σˆ and σˆ i Here, as promised in footnotes 19 and 20, we provide more detail on the determination of σˆ and σˆ i . Recall that vk is a K × 1 column vector, so vk is a ˆ i is a scalar. Let M 1 × K row vector; γˆ is a (K + 1) × 1 column vector, and D be the K × K matrix, let Γ be the (K + 1) × K matrix, and let Γi (i = 1 I) be the 1 × K row vectors: ⎛ ⎞ v0 √ M = T ⎝ ⎠ vK−1 (γ(t ˆ + T ω) − γ(t ˆ ω)) · · · (γ(t ˆ + T ω) − γ(t ˆ ω)) Γ = when ω(t + T ) = 0 · · · when ω(t + T ) = K − 1 ˆ i (t + T ω) − D ˆ i (t ω)) · · · (D ˆ i (t + T ω) − D ˆ i (t ω)) ( D Γi = when ω(t + T ) = 0 ··· when ω(t + T ) = K − 1 ˆ + T ω) − Thus, the kth row (k = 0 K − 1) of M is the value of β(t ˆ ω) when ω(t + T ) = k, the kth column (k = 0 K − 1) of Γ is the β(t value of γ(t ˆ + T ω) − γ(t ˆ ω) when ω(t + T ) = k, and the kth entry (k = ˆ i (t +T ω)− D ˆ i (t ω) when ω(t +T ) = k; 0 K −1) of Γi is the value of D note that Γi is neither a row nor a column of Γ . Since v0 vK−1 are linearly independent, M is nonsingular. Let σ(t ˆ ω) = Γ M −1 σˆ i (t ω) = Γi M −1 σ(t ˆ ω) is the unique (K + 1) × K matrix and σˆ 1 (t ω) σˆ I (t ω) are the unique 1 × K row vectors such that (20)
ˆ + T ω) − β(t ˆ ω)) γ(t ˆ + T ω) − γ(t ˆ ω) = σ(t ˆ ω)(β(t ˆ 1 (t ω) = σˆ 1 (t ω)(β(t ˆ 1 (t + T ω) − D ˆ + T ω) − β(t ˆ ω)) D ˆ I (t + T ω) − D ˆ I (t ω) = σˆ I (t ω)(β(t ˆ + T ω) − β(t ˆ ω)) D
ˆ i D ˆ I are martinˆ γ, whenever ω(t + T ) ∈ {0 1 K − 1}. Since β, ˆ and D ˆ the change gales, the sum (over the K + 1 successor nodes) of the change in β,
EQUILIBRIUM IN FINANCIAL MARKETS
869
ˆ i is zero, so the fact that Equation (20) holds for in γ, ˆ and the change in D ω(t + T ) ∈ {0 1 K − 1} implies it holds also for ω(t + T ) = K. Thus, the processes σˆ and σˆ 1 σˆ I are adapted. APPENDIX B: REAL ANALYTIC FUNCTIONS OF SEVERAL VARIABLES In this appendix, we summarize the results on real analytic functions of several variables used in our proof. DEFINITION B.1: Let U ⊂ Rn be open. A function F : U → Rm is real analytic if, for every x0 ∈ U, there is a power series Gx0 (x) centered at x0 with a positive radius of convergence δx0 such that F(x) = Gx0 (x) whenever x ∈ U and |x − x0 | < δx0 . THEOREM B.2—The Analytic Implicit Function Theorem: Suppose X ⊂ Rn Y ⊂ Rm are open, F : X × Y → Rn is real analytic, x0 ∈ X, y0 ∈ Y , F(x0 y0 ) = 0, and det
∂F = 0 ∂x
at (x0 y0 ). Then there are neighborhoods U of x0 , and V of y0 and a real analytic function φ : V → U such that for all y ∈ V , F(φ(y) y) = 0 Moreover, if x ∈ U, y ∈ V , and F(x y) = 0, then x = φ(y). PROOF: The conventional statement of the implicit function theorem establishes all the claims except the claim that the implicit function φ is real analytic. For this, see Theorem 2.3.5 of Krantz and Parks (2002). Q.E.D. THEOREM B.3: Let U ⊂ Rn be open and convex,23 F : U → R real analytic. If {x ∈ U : F(x) = 0} has positive Lebesgue measure, then F is identically zero on U. PROOF: Lojasiewicz’s structure theorem for varieties (Theorem 6.3.3 of Krantz and Parks (2002)) states that if U is an open set in Rn and F : U → R is analytic, then for every x0 ∈ U, there exists a neighborhood Vx0 of x0 such that either F(x) = 0 for all x ∈ Vx0 or {x ∈ Vx0 : F(x) = 0} is a finite union of real algebraic varieties of dimension less than n. If F(x) = 0 for all x ∈ Vx0 and y ∈ U, there is a ray that passes through Vx0 and through y; the restriction of F 23 It is sufficient to assume that U is connected and open. We prove the easier case when U is convex, since that is the case we use.
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R. M. ANDERSON AND R. C. RAIMONDO
to the ray is an analytic function of a single variable, and it vanishes on an interval (the intersection of the ray with the set V0 ); since it is well known that an analytic function of one variable that vanishes on an interval is identically zero, we must have F(y) = 0 for all y ∈ U and we are done. On the other hand, if {x ∈ V : F(x) = 0} is a finite union of algebraic varieties of dimension less than n, {x ∈ Vx0 : F(x) = 0} has Lebesgue measure zero. There is a countable col lection {xn : n ∈ N} such that n∈N Vxn ⊃ U, so {x ∈ U : F(x) = 0} has Lebesgue measure zero. Q.E.D. Finally, we show that the conditional expectation of a function of G(β(T )), conditional on (t β(t)), is, under mild hypotheses on G, an analytic function of (t β(t)). This fact is widely known among probabilists, but we were unable to find a specific reference that shows that the analyticity is joint in (t β(t)), and we need the joint analyticity. Therefore, we provide a proof. THEOREM B.4: Suppose F is measurable on RK and there exists r ∈ R such that |F(x)| ≤ r + er|x| Let β be a standard K-dimensional Brownian motion and let G(t β) = E F(β(T ))|β(t) = β Then G(t β) is an analytic function of (t β) ∈ (0 T ) × RK . PROOF: Fix t < T . Then (21)
1 G(t β) = (2π(T − t))K/2 1 = (2π(T − t))K/2 2
=
e−|β| /2(T −t) (2π(T − t))K/2 2
e−|β| /2(T −t) = (2π(T − t))K/2
RK
F(β + x)e−|x|
2 /2(T −t)
2 /2(T −t)
F(y)e−|y−β|
dx
dy
RK
eβ·y/(T −t) F(y)e−|y| RK
∞ RK
k=0
2 /2(T −t)
dy
β·y k T −t
k!
F(y)e−|y|
2 /2(T −t)
dy
2 ∞ e−|β| /2(T −t) 1 = K/2 k (2π(T − t)) RK k=0 (T − t) k! k × (β1 y1 )k1 · · · (βK yK )kK k1 · · · kK k1 +···+kK =k
× F(y)e−|y|
2 /2(T −t)
dy
EQUILIBRIUM IN FINANCIAL MARKETS 2
e−|β| /2(T −t) = (2π(T − t))K/2 ×
k1 +···+kK
∞ RK
k=0
871
1 (T − t)k
(β1 y1 ) · · · (βK yK )kK 2 F(y)e−|y| /2(T −t) dy k1 ! · · · kK ! =k k1
We view the two sums as an integral over {(k k1 kK ) : k k1 kK ∈ N ∪ {0} k = k1 + · · · + kK } where each element has mass 1, so the expression becomes the integral of a product measure space. We want to use Fubini’s theorem to interchange the order of the integral and the sums. To do this, we need to show that the integrand is L1 on the product. In the following calculation, Equation (22) folk k lows because the integral of |y1 1 · · · yKK | over any orthant equals the integral of k1 kK y1 · · · yk over the positive orthant, and there are 2K orthants. Equation (24) follows because the component normal random variables y1 yK are independent, and from the formula for the ki th moment of the absolute value of a normal random variable. In Equation (25), the integrand is greater than or equal to the integrand in Equation (23) on the set {y ∈ RK+ : |y| ≤ 4r(T − t)}. In Equation (26), the integrand is greater than or equal to the integrand in Equation (23) on the set {y ∈ RK+ : |y| > 4r(T − t)}; integrating over all of RK+ makes the integral still larger. In Equation (27), the second term comes from the fact that Equation (25) is the integral of a constant over {y ∈ RK+ : |y| ≤ 4r(T − t)} times the density of a probability distribution over RK ; the third term comes from the formula for the ki th moment of the absolute value of a normal random variable: k1 kK 1 y · · · y 2 /2(T −t) K 1 −|y| dy (2π(T − t))K/2 K k ! · · · k ! F(y)e 1 K R k k 1 (r + er|y| )|y1 1 · · · yKK | −|y|2 /2(T −t) ≤ e dy (2π(T − t))K/2 RK k1 ! · · · kK ! k k (r + er|y| )y1 1 · · · yKK −|y|2 /2(T −t) 2K e (22) dy = (2π(T − t))K/2 RK+ k1 ! · · · kK ! k k 2K r y1 1 · · · yKK −|y|2 /2(T −t) = e (23) dy (2π(T − t))K/2 RK+ k1 ! · · · kK ! k k 2K y1 1 · · · yKK er|y| −|y|2 /2(T −t) + e dy (2π(T − t))K/2 RK+ k1 ! · · · kK ! √ √ k1 !(T − t)k1 /2 · · · kK !(T − t)kK /2 (24) ≤r k1 ! · · · kK !
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R. M. ANDERSON AND R. C. RAIMONDO
+
(25)
2K (2π(T − t))K/2
(4(T − t))k e4r(T −t) −|y|2 /2(T −t) e dy k1 ! · · · kK ! {y∈RK + :|y|≤4r(T −t)} k k 2K y1 1 · · · yKK −|y|2 /4(T −t) + e dy (2π(T − t))K/2 RK+ k1 ! · · · kK ! ×
(26) (27)
r(T − t)k/2 (4(T − t))k e4r(T −t) r(2(T − t))k/2 + √ ≤√ + k1 ! · · · kK ! k1 ! · · · kK ! k1 ! · · · kK !
Consider the power series (28)
∞ k=0
1 (T − t)k
×
k
k1 +···+kK =k
k1 1
RK
k
β1 1 · · · βKK
1 (2π(T − t))K/2
k
y · · · yKK 2 F(y)e−|y| /2(T −t) dy k1 ! · · · kK ! k
y
By Equation (27), the absolute value of the coefficient of β1 1 · · · βKK is bounded by r(T − t)k/2 (4(T − t))k e4r(T −t) r(2(T − t))k/2 1 + + √ √ (T − t)k k1 ! · · · kK ! k1 ! · · · kK ! k1 ! · · · kK ! 3r(T − t)−k/2 4k e4r(T −t) ≤ √ + k1 ! · · · kK ! k1 ! · · · kK ! 3r(T − t)−k/2 + 4k e4r(T −t) √ k1 ! · · · kK ! ! "k (3r + e4r(T −t) ) max T 1−t 16 ≤ √ k1 ! · · · kK ! ≤
which shows that the power series in Equation (28) converges absolutely within a positive radius of convergence, by Proposition 2.2.10 of Krantz and Parks (2002). Thus, we can apply Fubini’s theorem to obtain e−|β|
2 /2(T −t)
∞ k=0
×
1 (T − t)k
1 (2π(T − t))K/2
k
k
β1 1 · · · βKK
k1 +···+kK =k
k
RK
k
y1 1 · · · yKK 2 F(y)e−|y| /2(T −t) dy k1 ! · · · kK !
EQUILIBRIUM IN FINANCIAL MARKETS 2
e−|β| /2(T −t) = (2π(T − t))K/2
∞ RK k=0
−|y|2 /2(T −t)
× F(y)e
1 (T − t)k
k1 +···+kK
873
(β1 y1 )k1 · · · (βK yK )kK k1 ! · · · kK ! =k
dy
= G(t β) by Equation (21). Therefore, for fixed t < T , G(t β) is a product of two analytic functions, hence is an analytic function of β ∈ RK . For any s ∈ (0 t), we have 1 2 G(t β + x)e−|x| /2(t−s) dx G(s β) = K/2 (2π(t − s)) RK The right side is an integral with respect to x of an analytic function of (s β x), and hence G is analytic on (0 t) × RK by Proposition 2.2.3 of Krantz and Parks (2002). Since this is true for every t < T , G is analytic on (0 T ) × RK . Q.E.D. APPENDIX C: NONSTANDARD STOCHASTIC INTEGRATION Up to now, all of our definitions and results have been stated without any reference to nonstandard analysis. Our proof makes extensive use of nonstandard analysis, in particular Anderson’s (1976) construction of Brownian motion and the Itô integral. It is beyond the scope of this paper to develop these methods; see Anderson (2000) and Hurd and Loeb (1985) for references to nonstandard analysis. To show that the equilibrium of the hyperfinite economy generates an equilibrium of the standard continuous-time economy, one needs to show that capital gains are the same in the two settings. Capital gains are given by Stieltjes integrals with respect to securities prices in the hyperfinite setting and by Itô integrals with respect to securities prices in the continuous-time setting. Anderson (1976) showed that the Itô integral with respect to Brownian motion is the standard part of a Stieltjes integral with respect to a hyperfinite random walk. Anderson’s theorem covers hyperfinite random walks which move √ independently in each component by an amount ±1/ n. In that random walk, each node in the tree has 2K successor nodes. As discussed above, to obtain dynamic completeness in the hyperfinite model, we need to use a random walk in which each node has K + 1 successor nodes. Thus, Anderson’s theorem does not cover the case considered here. Lindstrøm (1980a, 1980b, 1980c, 1980d) showed that the stochastic integral with respect to a square integrable martingale is the standard part of a Stieltjes integral with respect to a hyperfinite SL2 martingale. Lindstrøm’s theorem is limited to one-dimensional martingales. Because the components of a vector Brownian motion are uncorrelated, a process is Itô integrable with respect to a
874
R. M. ANDERSON AND R. C. RAIMONDO
vector Brownian motion if and only if it is integrable with respect to each component. However, the components of a vector martingale can be correlated and, consequently, a process can be integrable with respect to a vector martingale even if it is not integrable with respect to the individual components. This fact has economic significance. If two components of the vector martingale are instantaneously nearly perfectly correlated at some point, then the equilibrium trading strategy may well require taking a nonstandard infinite long position in one security and a nonstandard infinite short position in the other. In both the hyperfinite and continuous-time models, the capital gain is well defined and finite when computed with respect to the vector martingale. However, the hyperfinite capital gain may be a positive nonstandard infinite number in one component and a negative nonstandard infinite number in the other components; they add up to a well defined finite integral when both components are considered. The continuous-time capital gain may be undefined with respect to the two components when considered separately, but well defined and finite when the integral is computed with respect to the vector martingale. Thus, we need to extend either Anderson’s theorem or Lindstrøm’s theorem. The more general approach would be to extend Lindstrøm’s theorem to vector martingales; such an extension is probably needed to tackle the dynamically incomplete markets case. However, it is considerably easier, and sufficient for our purposes in this paper, to extend Anderson’s theorem to the particular kind of random walk considered here. This is the approach we follow. Anderson (1976) proved that β is a standard Brownian motion provided that βˆ moves up or down √1n independently in each coordinate at every node; this would require each node in the tree for βˆ to have 2K successor nodes, precluding dynamic completeness in the hyperfinite model for K > 1. Neither Anderson nor Keisler (1984) quite covers the random walk βˆ considered here because the coordinates of βˆ are uncorrelated but not independent. For definitions of standard terms in stochastic integration (such as H2 and 2 L ), see Nielsen (1999). For definitions of nonstandard terms such as SL2 lifting, see Anderson (1976). THEOREM C.1: β is a standard Brownian motion. PROOF: We claim that the coordinates of β are independent. To see this, fix x ∈ RK . Then {x · vs : s ∈ T } is a family of independent and identically distributed (IID) random variables with standard distribution, mean zero, and finite variance σx , so Anderson’s Theorem 21 implies that x · β(t ·) is Normal mean zero variance tσx . Since x · β(t ·) is Normal for all x ∈ RK , it is well known that β(t ·) is system Normal (see, for example, Bryc (1995, Theorem 2.2.4)). Since β(t ·) is system Normal with variance–covariance matrix tI, where I is the K × K identity matrix, the components of β are independent. Each component of the random walk βˆ is a hypermartingale (martingale with respect to
EQUILIBRIUM IN FINANCIAL MARKETS
875
the hyperfinite filtration) and so satisfies the S-continuity property by Keisler’s continuity theorem for hypermartingales, and it follows from Anderson’s proof that β is almost surely continuous. Anderson’s proof that β has independent increments goes through without change in the present setting. Anderson used a slightly smaller filtration than the one considered here, while Keisler used the filtration considered here. Q.E.D. THEOREM C.2: Let βˆ be the hyperfinite random walk defined above, and let β = ◦ βˆ be the standard Brownian motion it generates. Suppose Z ∈ H2 . Then there is an SL2 lifting Zˆ of Z. Given any SL2 lifting Zˆ of Z, for every t ∈ T , we have t ◦t ◦ Z dβ Zˆ d βˆ = 0
0
PROOF: Lemma 31 in Anderson (1976) proves the existence of an SL2 lifting; the hyperfinite probability space is slightly different from the one considered here, but the proof goes through without change. Theorem 33 in Anderson (1976) shows that with respect to the hyperfinite random walk and Brownian motion considered there, the Itô integral is the standard part of the hyperfinite Stieltjes integral. The proof of Theorem 33 depends on the specific form of the hyperfinite random walk only in establishing the Itô isometry, so we show that the Itô isometry holds for the random walk ˆ Zˆ may be a 1 × 1 scalar process, a 1 × K vector process, or a (J + 1) × K β. matrix process. The proofs in the vector and matrix cases are virtually identical apart from notation, while the proof in the scalar case is easier, so we assume that Zˆ is a 1 × K vector process with kth component Zˆ k , # # t # # # Zˆ d βˆ # # # 0
2
# t # # # # ˆ ˆ = # Zk d βk # # 0 2 # K # # √ # # # =# Zˆ k (s ·)(vs+T )k T # # # k=1 s∈T s
=
K
2
√ Zˆ k (s ·)(vs+T )k T 2
k=1 s∈T s
(because the terms Zˆ k (s ω)vs+T are uncorrelated across s k) =
K k=1 s∈T s
T Zˆ k (s ·)2
876
R. M. ANDERSON AND R. C. RAIMONDO
=
K
Zˆ k |{s∈T :s
k=1
ˆ {s∈T :s
◦
t
◦ 0
Zˆ d βˆ =
◦t
Z dβ 0
PROOF: Let f (ω) = Z(· ω)2 and find f¯ internal such that ◦ f¯(ω) = f (ω) L(μ)-almost ˆ surely. Let Z¯ be an internal nonanticipating process such that ◦ ¯ ˆ everywhere, so for L(μ)-almost ˆ all Z(t ω) = Z(◦ t ω) L(ˆν ) × L(μ)-almost ◦ ¯ ◦ ∗ ˆ ω, Z(· ω) = Z( t ω) for ν-almost all t ∈ T \ {T }. For m ∈ N, let ! " (Z¯ m )ij (t ω) = max −m min{m Z¯ ij (t ω)} f¯m (ω) = Z¯ m (· ω)2 For all ω, we have f¯m+1 (ω) ≥ f¯m (ω). For L(μ)-almost ˆ all ω, we have ◦ ¯ limm∈Nn→∞ fm (ω) = f (ω) < ∞. Therefore, there exists m0 ∈ N such that for all m ∈ N, m ≥ m0 , $ 1 1 ¯ ¯ μˆ ω : |fm (ω) − f (ω)| < >1− m m so we may find m ∈ ∗ N \ N such that $ 1 1 ¯ ¯ >1− μˆ ω : |fm (ω) − f (ω)| < m m so ◦ f¯m (ω) = f¯(ω) L(μ)-almost ˆ surely; for any such ω, Z¯ m (· ω) ∈ SL2 (Anderson (1976, Theorem 11)), so if we define Zˆ = Z¯ m , Zˆ is an SL2 lifting of Z.
EQUILIBRIUM IN FINANCIAL MARKETS
877
Now, suppose Zˆ is any SL2 lifting of Z. For m ∈ N, define the internal stopping time $ t 2 ˆ Z(s ω)2 d νˆ ≤ m τˆ m (ω) = max t ∈ T : 0
and define
ˆ ω) if t ≤ τm (ω), Z(t ˆ Zm (t ω) = 0 if t > τm (ω).
ˆ all ω, there exists m(ω) such that Let Zm (t ω) = ◦ Zˆ m (tˆ ω). For L(μ)-almost ˆ ˆ ˆ τm(ω) = T , in which case Zm (· ω) = Z(· ω) and therefore t t ˆ ˆ ω) d β ˆ ˆ Zm d β = Z(· 0
0
By the definition of the standard stochastic integral (see, for example, page 96 of Steele (2001)) and Theorem C.2, ◦t ◦t Z dβ = lim Zm dβ 0
m→∞
= lim
m→∞
t
=◦ 0
0
◦
t
Zˆ m d βˆ
0
ˆ Zˆ d β
Q.E.D.
APPENDIX D: PROOFS OF THEOREMS 2.1 AND 4.2 AND PROPOSITION 2.2 We construct our probability space, filtration, and Brownian motion following Anderson’s (1976) construction. Choose n ∈ ∗ N \ N. Using this hyperfinite ˆ β, ˆ Fˆ , Fˆ t , Uˆ i , and so on exactly as they were defined in ˆ Ω, n, define T , tˆ, Tˆ , νˆ , μ, Section 4; note that this involves perturbing endowments and dividends to ensure the existence of a Pareto optimal equilibrium, and that the measures νˆ and μˆ are defined by internal cardinalities. By the transfer principle, the economy has an equilibrium, and the equilibrium is Pareto optimal. Let (T L(ˆν )) denote the complete Loeb measure generated by νˆ on T ; note that the Loeb measure is a standard countably additive measure on the same underlying point set T as ν. For B ⊂ [0 T ], let st−1 (B) = {t ∈ T : ◦ t ∈ B}. For any Lebesgue measurable set B ⊂ [0 T ], st−1 (B) is Loeb measurable and L(ˆν )(st−1 (B)) = ν(B) (Anderson (1976)). ˆ Fˆ μ) ˆ be the (complete) Loeb measure generated by (Ω ˆ Let (Ω F L(μ)) ˆ is generated by a nonstandard con(Loeb (1975)). Although (Ω F L(μ)) struction, Loeb showed that it is a probability space in the usual standard sense.
878 Let
R. M. ANDERSON AND R. C. RAIMONDO
! Ft = B ∈ F : L(μ)(BC) ˆ = 0 for some C which respects
" the equivalence relation ω ∼ ω ⇔ ω ∼s ω for all s t
ˆ tˆ ω)). β is a KLet β : [0 T ] × Ω → RK be defined by β(t ω) = ◦ (β( dimensional Brownian motion in the usual standard sense and β(t ·) = E(β(T ·)|Ft ). Let I : [0 T ] × Ω → RK+1 be defined by
I (t ω) = (t β(t ω)) Since the equilibrium is Pareto optimal, the marginal utility of consumption is infinite at zero, the aggregate consumption is strictly positive at each node, and all agents have strictly positive income by Equation (4), the equilibrium consumptions of all agents are strictly positive at each node; they may be infinitesimal. Let Δ be the open (I − 1)-dimensional simplex in RI++ . Pareto optimality and strict positivity of the equilibrium consumptions imply that there exists λˆ = (λˆ 1 λˆ I ) ∈ ∗ Δ such that at each node (t ω), the consumptions maximize I
λˆ i ∗ hi (cˆi (t ω) Iˆ (t ω))
and
i=1
I
λˆ i ∗ Hi (cˆi (Tˆ ω) Iˆ (Tˆ ω))
i=1
Since each cˆi (t ω) > 0, each λˆ i > 0, so there is a positive constant ιˆ(t ω) such that ∂h ∂h 1 I ∗ ∗ = · · · = λˆ I = ιˆ(t ω) λˆ 1 ∂c1 (cˆ1 (tω)Iˆ (tω)) ∂cI (cˆI (tω)Iˆ (tω)) for t < Tˆ ∂H1 λˆ 1 ∗ ∂c ˆ 1
I
(cˆ1 (T ω)Iˆ (Tˆ ω))
= · · · = λˆ I ∗
∂HI = ιˆ(Tˆ ω) ∂cI (cˆI (Tˆ ω)Iˆ (Tˆ ω))
ˆ ω) = i=1 cˆi (t ω). Let c(t We claim that ιˆ(· ·) ∈ SLp for each p ∈ [1 ∞), and that ιˆ(t ω) 0 for Loeb almost all (t ω). For every node (t ω) there is an agent i(t ω) such ˆ . The following calculation assumes t < Tˆ ; if t = Tˆ , subthat cˆi(tω) (t ω) ≥ c(tω) I stitute Hi for hi . Since λˆ i ≤ 1 for each i, ∂hi(tω) ∗ ˆ ιˆ(t ω) = λi(tω) ∂ci(tω) (cˆi(tω) (tω)Iˆ (tω)) ∂hi(tω) ∗ ≤ ∂ci(tω) ((c(tω))/I ˆ Iˆ (tω))
879
EQUILIBRIUM IN FINANCIAL MARKETS
∂hi(tω) ∗ ≤ ∂c
i(tω) ((
I
∗
ˆ i=1 fi (I (tω))+
J
∗
ˆ ˆ j=0 ηj gj (I (tω)))/II (tω))
ˆ
≤ r + er|β(tω)| ˆ by inequality (2). Therefore, (ˆι(t ω))p ≤ r¯ + er¯|β(tω)| for some r¯ ∈ R, so (ˆι)p ∈ I 1 SL by Raimondo (2002, 2005, Proposition 3.1), so ιˆ ∈ SLp . Since i=1 λˆ i = 1, we can assume without loss of generality that λ1 0. Then ∂hi ιˆ(t ω) = λˆ 1 ∗ ∂ci (cˆi (tω)Iˆ (tω)) ∂hi ∗ ˆ ≥ λ1 ∂ci (c(tω) ˆ Iˆ (tω)) ∂hi ≥ λˆ 1 ∗ ∂ci (Ii=1 ∗ fi (Iˆ(tω))+J ηj ∗ gj (I (tω)+O((T )2 )) j=0
0 whenever the social consumption is finite, that is, for Loeb almost every (t ω). If we set pˆ C (t ω) = ιˆ(t ω) for t ∈ T , pˆ C gives the Arrow–Debreu prices of consumption. We claim that λˆ i 0 for each i = 1 I. If not, we may assume without loss of generality that λˆ I 0. For every node for which ιˆ(t ω) 0, we have ∂hI ιˆ(t ω) ∗ = ∞ or ∂cI (cˆI (tω)I(tω)) ˆ λˆ I ∂HI ιˆ(Tˆ ω) ∗ = ∞ ∂cI (cˆI (Tˆ ω)I(ˆ Tˆ ω)) λˆ I as appropriate, so cˆI (t ω) 0, which shows that cˆI (t ω) 0 Loeb almost ˆ ˆ ω) ≤ r + er|β(tω)| + O((T )2 ), then as above, everywhere. Since cˆI (t ω) ≤ c(t p cˆI ∈ SL for all p ∈ [1 ∞) by Raimondo (2002, 2005, Proposition 3.1), and hence pˆ C × cˆI ∈ SLp for all p ∈ [1 ∞) by the Cauchy–Schwarz inequality and Anderson (1976); Tˆ −T ◦ ˆ ˆ E pˆ C (T ω)cˆI (T ω) + T pˆ c (t ω)cˆI (t ω) =E
◦
t=0
pˆ C (Tˆ ω) cˆI (Tˆ ω) +
T
◦
0
= 0
◦
◦
pˆ c (t ω) cˆI (t ω) dt
880
R. M. ANDERSON AND R. C. RAIMONDO
However, since the hyperfinite market is dynamically complete, agent I can afford any bundle satisfying the Arrow–Debreu budget constraint. In the folˆ lowing calculation, Equation (29) follows because eˆ I (t ω) ≤ r + er|β(tω)| + ˆ p 2 r|β(tω)| 2 ˆ ˆ O((T ) ) and Aj (t ω) ≤ r + e + O((T ) ), so eˆ I A ∈ SL for all p ∈ [1 ∞), so pˆ c eˆ I pˆ c Aˆ ∈ SLp for all p ∈ [1 ∞). Agent I’s income is eˆ IA · pˆ A (0 0) + E pˆ C (Tˆ ·)eˆ I (Tˆ ·) + T
Tˆ −T
pˆ C (t ·)eˆ I (t ·)
t=0
= E pˆ C (Tˆ ·)(eˆ IA A(Tˆ ·) + eˆ I (Tˆ ·))
Tˆ −T
+ T E
pˆ C (t ·)(eˆ IA A(t ·) + eˆ I (t ·))
t=0
(29)
◦
pˆ C (Tˆ ·)(eIA A(Tˆ ·) + eI (Tˆ ·))
+
T
◦
pC (t ·)(eIA A(t ·) + eI (t ·))
0
= 0 since the expectation is of a function which is nonnegative and strictly positive on a set of positive measure by inequality (4). Thus, cˆI lies strictly inside the Arrow–Debreu budget set; since preferences are strictly monotonic, it cannot be the demand of agent I, a contradiction which shows that λˆ i 0 for each i = 1 I. We now show that the equilibrium consumptions and consumption prices are given by standard analytic functions. Consider the standard analytic function ρ : (0 ∞)I × R++ × Δ × R++ × ((0 T ) × RK ) → RI+1 defined by ρ((c1 cI ) ι λ c (t β)) ∂h1 ∂hI = λ1 − ι λI ∂c ∂c 1 (c1 tβ)
− ι c1 + · · · + cI − c
I (cI tβ)
ρ encodes the first-order conditions for the equilibrium consumptions. The determinant of the Jacobian matrix of partial derivatives of ρ with respect to
881
EQUILIBRIUM IN FINANCIAL MARKETS
((c1 cI ) ι) is given by ∂ρ ∂((c1 cI ) ι) ⎛ ∂2 h 0 λ1 21 ⎜ ∂c1 2 ⎜ 0 λ2 ∂∂ch22 ⎜ 2 ⎜ ⎜ = det ⎜ ⎜ ⎜ ⎜ 0 0 ⎝
det
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ = det ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 ∂2 h1 1 ∂c 2 1
λ
= λ1 · · · λI
···
0
0
···
0
···
0
λI ∂∂ch2I
1
1
···
1
0
0
···
0
−1 −1
2
−1
2
I
λ2 ∂∂ch22 2
0
···
0
0
0
···
0
λI ∂∂ch2I
0
···
0
⎟ −1 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −1 ⎟ ⎠ 0
0
0
⎞
0
2
I
0
⎞
−1 I
2 I ∂2 h1 ∂ hi ∂2 h I % 1 λi 2 · · · ∂c12 ∂cI2 i=1 ∂ci
i=1
1 λi
∂2 hi ∂c 2 i
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
= 0 since λ ∈ Δ and
∂2 hi ∂ci2
< 0. By the analytic implicit function theorem (Theo-
rem B.2 in Appendix B) and the transfer principle, there exist standard analytic functions π, ˆ ψˆ i : Δ × R++ × ((0 T ) × RK ) → R for i = 1 I such that ˆ c(t ˆ ω) Iˆ (t ω)) for t < Tˆ ιˆ(t ω) = ∗ π( ˆ λ ˆ c(t ˆ ω) Iˆ (t ω)) cˆi (t ω) = ∗ ψˆ i (λ
for t < Tˆ
Using a similar calculation at t = Tˆ and the ordinary implicit function theorem, ˆ Ψˆ i : Δ × R++ × ({T } × by Equation (3), there are standard C 1 functions Π, K R ) → R such that ˆ λ ˆ c( ˆ Tˆ ω) Iˆ (Tˆ ω)) ιˆ(Tˆ ω) = ∗ Π( ˆ c( ˆ Tˆ ω) Iˆ (Tˆ ω)) cˆi (Tˆ ω) = ∗ Ψˆ i (λ
882
R. M. ANDERSON AND R. C. RAIMONDO
Let λ ∈ Δ, π : R++ × RK × [0 T ) → R, Π : R++ × RK → R, ψi : R++ × RK × [0 T ) → R, and Ψi : R++ × RK → R be defined by ˆ λ = ◦ λ π(c I ) = π(λ ˆ c I ) ˆ Π(c β) = Π(λ c β) ψi (c I ) = ψˆ i (λ c I ) Ψi (c β) = Ψˆ i (λ c β) Because πˆ and ψˆ are standard analytic functions, π and Ψ are standard analytic functions; because Πˆ and Ψˆ are standard C 1 functions, Π and Ψ are stanˆ ω) and ci (t ω) = ◦ cˆi (t ω). Since c(t ω) dard C 1 functions. Let c(t ω) = ◦ c(t and ψi (c β t) are analytic functions for t ∈ (0 T ), each ci (t ω) is an analytic function of I (t ω). Each ci (T ω) is a C 1 function of c(T ω); however, since c(T ω) is a continuous function of I (T ω) almost everywhere, ci (T ω) is a continuous function of I (T ω) almost everywhere. Let pC (t ω) = ◦ pˆ C (t ω) then pC (t ω) = ◦ pˆ C (t ω) ˆ c(t ˆ ω) Iˆ (t ω)) = ◦ ∗ π( ˆ λ ˆ ◦ c(t ˆ ω) ◦ Iˆ (t ω)) = π( ˆ ◦ λ = π(λ ˆ c(t ω) I (t ω)) = π(c(t ω) I (t ω)) is an analytic function of I (t ω) on (0 T ) × RK , since π is analytic and c(t ω) is an analytic function of I (t ω). Since c(T ω) = Π(c(T ω) β(T ω)) is a C 1 function of β(T ω) ∈ V , pC (T ·) is a C 1 function of β(T ω) ∈ V . In the following calculation, Equation (30) holds because the first-order conditions for demand imply the price of a security is the expected value of future dividends times the weighted marginal utility of consumption, which equals the expected value of the dividend times the price of consumption. Equation (31) follows because pˆ A (Tˆ ·) = pˆ C (Tˆ ω)A(Tˆ ω), pˆ C ∈ SLp for all p ∈ [1 ∞), ˆ + O((T )2 ), so as above pˆ A (Tˆ ·) ∈ SLp for all and Aj (t ω) ≤ r + er|β(tω)| ˆ ·) ∈ SLp p ∈ [1 ∞), by Raimondo (2002, 2005 Proposition 3.1), so pˆ C (· ·)A(· for all p ∈ [1 ∞) (Anderson (1976)); and the internal integrands are Scontinuous functions of Iˆ (t ω) whenever ◦ Iˆ (t ω) is at a point at which c and
883
EQUILIBRIUM IN FINANCIAL MARKETS
G are continuous. Equation (32) follows because Iˆ (· ω) is almost surely Scontinuous, and for any such ω and any s ∈ [◦ t T ], the conditional distribution of I (s ω) is the same given (◦ t ω) as it is given (t ω). ◦
pˆ A (t ω) ˆ Tˆ ·) + T pˆ C (Tˆ ·)A(
◦
= E (30)
Tˆ −T
ˆ ·) (t ω) pˆ C (s ·)A(s
s=t+T
∗
ˆ λ ˆ Tˆ ·)|(t ω) ˆ c( ˆ Tˆ ·) Iˆ (Tˆ ·))A( = E Π( Tˆ −T ◦ ∗ ˆ ·) (t ω) ˆ c(s ˆ ·) Iˆ (s ·))A(s + E T π( ˆ λ ◦
s=t+T
∗
ˆ λ ˆ c( ˆ Tˆ ·) Iˆ (Tˆ ·))∗ G(Iˆ (Tˆ ·)) + O((T )2 )|(t ω) = ◦ E Π( Tˆ −T ◦ ∗ ˆ c(s ˆ ·) Iˆ (s ·)) π( ˆ λ + E T s=t+T
× g(Iˆ (s ·)) + O((T )2 ) (t ω) ∗
(31)
◦t
= E Π(c(T ·) I (T ·))G(I (T ·)) +
Let
= E Π(c(T ·) I (T ·))G(I (T ·))|(t ω) T +E π(c(s ·) I (s ·))g(I (s ·)) ds(t ω)
(32)
T ◦t
◦ π(c(s ·) I (s ·)) g(I (s ·)) ds ( t ω)
pA (t ω) = E Π(c(T ·) I (T ·))G(I (T ·)) +
T
◦t
π(c(s ·) I (s ·)) g(I (s ·)) ds (t ω)
ˆ ω)) ∈ We have shown that ◦ pˆ A (t ω) is an S-continuous function of (t β(t −1 K ◦ ˆ ω) is st ([0 T ) × R ) and that pˆ A (t ω) = pA (t ω) whenever the path β(· S-continuous, that is, Loeb-almost everywhere. For almost all ω, pˆ A (· ω) is an S-continuous function of t ∈ T . Moreover, pA is an analytic function of
884
R. M. ANDERSON AND R. C. RAIMONDO
(t β(t ω)) ∈ (0 T ) × RK by Theorem B.4 and Proposition 2.2.3 of Krantz and Parks (2002). In the following calculation of γ(t ˆ + T ω) − γ(t ˆ ω), note that we do not calculate how the future dividend values are affected by the information reˆ + T ω) − β(t ˆ ω). Instead, we calculate how the probability vealed by β(t of each possible future dividend is affected by the information revealed by √ ˆ + T ω) − β(t ˆ ω) = T vω(t+T ) . The calculation uses the fact that the β(t distribution of the random walk is multinomial. In order to know the value of ˆ ω) − β(t ˆ ω), it is enough to count how many of the s−t steps of the ranβ(s T dom walk in this time interval lie in each of the possible directions v0 vK ; it does not depend on the order in √which the steps occur. In Equation (33), we K ˆ ˆ write pˆ C A(s β(t ω) + i=0 ki vi T ) as an abbreviation for K K √ √ ˆ ˆ ˆ ki vi T A s β(t ω) + ki vi T pˆ C s β(t ω) + i=0
i=0
Equation (34) holds because for each s > t s ∈ T , including s = Tˆ , and for k0 kK ∈ ∗ (N ∪ {0}) satisfying k0 + · · · + kK = (s − t)/T , √ √ K T T vω(t+T ) i=0 ki vi s−t K T ki vi · vω(t+T ) = s − t i=0 T Kkω(t+T ) + (−1) = ki s−t i =ω(t+T ) s−t T Kkω(t+T ) − − kω(t+T ) = s−t T T s−t = (K + 1)kω(t+T ) − s−t T
=
(K + 1)kω(t+T ) − 1 (s − t)/T
ˆ ˆ ω) ≤ r + er|β(tω)| Equation (36) follows from the inequalities A(t + O((T )2 ) ˆ ˆ ˆ ˆ Aˆ βˆ ∈ ˆ ≤ |β|(r ˆ + er|β| ) ≤ (r|β| ˆ + e(r+1)|β| ) ≤ 2e(r+1)|β| , so as above A and |Aˆ β| p SL for all p ∈ [1 ∞) by Raimondo (2002, 2005 Proposition 3.1), and we previously showed pˆ C ∈ SLp for all p ∈ [0 ∞); it follows that pˆ A (Tˆ ·) = ˆ Tˆ ·) ∈ SLp and pˆ C Aˆ βˆ ∈ SLp for all p ∈ [1 ∞) (Anderson (1976)); pˆ C (Tˆ ·)A( and the internal integrands are S-continuous functions of Iˆ (t ω) whenever
885
EQUILIBRIUM IN FINANCIAL MARKETS
Iˆ (t ω) is at a point at which c and G are continuous. Equation (37) follows because Iˆ (· ω) is almost surely S-continuous, and for any such ω and any s ∈ [◦ t T ], the conditional distribution of I (s ω) is the same given (◦ t ω) as it is given (t ω): ◦
γ(t ˆ + T ω) − γ(t ˆ ω)
= pˆ A (t + T ω) + (T )
ˆ ω) pˆ C (s ω)A(s
s∈T s≤t+T
− pˆ A (t ω) + (T )
ˆ ω) pˆ C (s ω)A(s
s∈T s≤t
ˆ + T ω) − pˆ A (t ω) = pˆ A (t + T ω) + pˆ c (t + T ω)A(t = E (pˆ A (Tˆ ·))|(t + T ω) − E (pˆ A (Tˆ ·))|(t ω) Tˆ −T
+ T
ˆ ·))|(t + T ω) E (pˆ C (s ·)A(s
s=t+T
ˆ ·))|(t ω) − E (pˆ C (s ·)A(s =
1 (K + 1)(Tˆ −t)/T
ˆ ω) + pˆ A Tˆ β(t
k0 +···+kK =(Tˆ −t)/T
K
√ ki vi T
i=0
(K + 1)(((Tˆ − t)/T ) − 1)! ((Tˆ − t)/T )! − × k0 !k1 !(kω(t+T ) − 1)! · · · kK ! k0 !k1 ! · · · kK !
Tˆ −T
+ T
(33)
×
1 (K + 1)(s−t)/T s=t+T K √ ˆ ˆ ki vi T pˆ C A s β(t ω) +
k0 +···+kK =(s−t)/T
i=0
(K + 1)(((s − t)/T ) − 1)! ((s − t)/T )! − k0 !k1 !(kω(t+T ) − 1)! · · · kK ! k0 !k1 ! · · · kK ! K √ 1 ˆ ω) + = ki vi T pˆ A Tˆ β(t (K + 1)(Tˆ −t)/T i=0 ˆ
×
k0 +···+kK =(T −t)/T
((Tˆ − t)/T )! × k0 !k1 ! · · · kK !
(K + 1)kω(t+T ) −1 ((Tˆ − t)/T )
886
R. M. ANDERSON AND R. C. RAIMONDO Tˆ −T
+ T
×
1 (s−t)/T (K + 1) s=t+T K √ ˆ ω) + ki vi T pˆ C Aˆ s β(t
k0 +···+kK =(s−t)/T
((s − t)/T )! × k0 !k1 ! · · · kK ! =
i=0
(K + 1)kω(t+T ) −1 ((s − t)/T )
1 (K + 1)(Tˆ −t)/T
×
k0 +···+kK =(Tˆ −t)/T
ˆ ω) + pˆ A Tˆ β(t
K
√ ki vi T
i=0
√ √ K ˆ T ((T − t)/T )! i=0 ki vi T · vω(t+T ) × k0 !k1 ! · · · kK ! Tˆ − t Tˆ −T
+ T
×
1 (s−t)/T (K + 1) s=t+T K √ ˆ ω) + ki vi T pˆ C Aˆ s β(t
k0 +···+kK =(s−t)/T
(34)
(35)
(36)
i=0
√ √ K T ((s − t)/T )! i=0 ki vi T · vω(t+T ) × k0 !k1 ! · · · kK ! s−t ˆ Tˆ ·))(β( ˆ Tˆ ·) − β(t ˆ ·)) pˆ A (Tˆ β( =E Tˆ − t Tˆ −T ˆ β(s ˆ ·))(β(s ˆ ·) − β(t ˆ ·)) T s=t+T pˆ C A(s + (t ω) s−t √ × vω(t+T ) T pA (T β(T ·))(β(T ·) − β(t ·)) =E T −t T pC (s β(s ·))A(s β(s ·))(β(s ·) − β(t ·)) ds(t ω) + s − ◦t ◦t √ ˆ + T ω) − β(t ˆ ω)) + o( T ) × (β(t
EQUILIBRIUM IN FINANCIAL MARKETS
(37)
887
σ(t ˆ ω) pA (T β(T ·))(β(T ·) − β(t ·)) E T −t T pC (s β(s ·))A(s β(s ·))(β(s ·) − β(t ·)) ds(t ω) + s − ◦t ◦t pC (T β(T ·))A(T β(T ·))(β(T ·) − β(t ·)) =E T −t T pC (s β(s ·))A(s β(s ·))(β(s ·) − β(t ·)) ◦ ds ( t ω) + s − ◦t ◦t
ˆ ∈ st−1 ([0 T ) × RK ). This shows that σˆ is an S-continuous function of (t β) Note that the integrand is the product of the (K + 1) × 1 column vector A and the 1 × K row vector β , so it is a (K + 1) × K matrix. Letting Σ : [0 T ) × RK → R(K+1)×K be given by pC (T β(T ))A(T β(T ))(β(T ) − β(t)) Σ(t β) = E T −t T pC (s β(s))A(s β(s))(β(s) − β(t)) dsβ(t) = β + s−t t we have ◦
σ(t ˆ ω) = Σ(I (t ω))
for every (t ω) such that Iˆ (t ω) is finite. Since σˆ is SL2 and adapted to {Fˆ t }, and is an SL2 lifting of Σ ◦ I , σ ◦ I is L2 and adapted to {Ft } (Anderson (1976)). Therefore, by Theorem C.2, we have pA (t ω) = ◦ pˆ A (tˆ ω) tˆ−T ◦
= pˆ A (0 ω) +
◦
ˆ + T ω) − β(s ˆ ω)) σ(s ˆ ω)(β(s
s=0
ˆ ω) − (T )pˆ C (s ω)A(s t Σ(I (s ω)) dβ(s ω) = pA (0 ω) + −
0 t
pC (s ω)A(s ω) ds 0
γ(t ω) = ◦ γ( ˆ tˆ ω)
888
R. M. ANDERSON AND R. C. RAIMONDO tˆ−T
= ◦ pˆ A (0 ω) + ◦
ˆ + T ω) − β(s ˆ ω)) σ(s ˆ ω)(β(s
s=0
t
= pA (0 ω) +
Σ(I (s ω)) dβ(s ω)
0
Thus, γ is the total gains process of pA and is a vector martingale. Since pA is an analytic (hence C 2 ) function of (t β), by Itô’s lemma and the uniqueness of Itô coefficients, Σ(t β) =
∂pA (t β) ∂β
is a partial derivative of an analytic function, so Σ is analytic on (0 T ) × RK . Extend Σ to {T } × V by Σ(T β) =
∂pA (T β) ∂β
for β ∈ V ; note that here, we are using β ∈ RK to denote a particular value that the Brownian motion might take. We claim that Σ is a continuous function of (t β) ∈ [0 T ] × V . Notice that the second term in the definition of Σ tends to zero as t → T , uniformly over β ranging over compact subsets of RK , so we can restrict attention to the first term. Suppose β0 ∈ V . Fix ε > 0. Since A(T ·) is C 1 on V and pC (T ·) is C 1 on V , pA (T ·) = A(T ·)pC (T ·) is C 1 on {T } × V . Since V is open, we may find δ > 0 such that B(β0 2δ) ⊂ V and # # # ε # ∂pA ∂p A #< β y ∈ B(β0 2δ) ⇒ # − # ∂β ∂β (Tβ) # 3 (Ty) For any β ∈ B(β0 δ) and y ∈ B(β δ), we have y ∈ B(β0 2δ). Using the mean value theorem one component at a time, we find that ε|y − β| pA (T y) − pA (T β) − ∂pA (y − β) < ∂β (Tβ) 3 so
# # # ε|y − β|2 # # # pA (T y) − pA (T β) − ∂pA (y − β) (y − β) #< # ∂β (Tβ) 3
Since pC ≤ r + er|β| and |Aj (T y)| ≤ r + er|y| , and ∂pA /∂β|(Tβ) is uniformly bounded over β ∈ B(β0 δ), there is a constant r¯ ∈ R such that for every β ∈ B(β0 δ), for all y ∈ RK , # # # # # r¯|y| # pA (T y) − pA (T β) − ∂pA (y − β) (y − β) # ≤ r¯ + e # ∂β (Tβ)
EQUILIBRIUM IN FINANCIAL MARKETS
889
Find t0 < T such that for all t ∈ [t0 T ), 1 (2π(T − t))K/2
RK \B(βδ)
ε r¯ + er¯|y| −|y−β|2 /2(T −t) e dy < T −t 3
If t ∈ [t0 T ) and β ∈ B(β0 δ), # # # E(pA (T β(T ·))(β(T ·) − β(t ·)) |β(t ω) = β) ∂pA # # # − # T −t ∂β (Tβ0 ) # √ √ # # 1 pA (T β + T − tx) T − tx −|x|2 /2 # e =# dx (2π)K/2 RK T −t # # ∂pA # − ∂β (Tβ0 ) # # # 1 pA (T y)(y − β) −|y−β|2 /(2(T −t)) =# e dy # (2π(T − t))K/2 K T −t R # # ∂pA # − ∂β (Tβ0 ) # # ∂pA (y − β)(y − β) # 1 2 ∂β (Tβ) # ≤# e−|y−β| /(2(T −t)) dy (2π(T − t))K/2 RK T −t # # ∂pA # − ∂β (Tβ0 ) # # # # # 1 p (t β)(y − β) 2 /(2(T −t)) A −|y−β| +# e dy # # (2π(T − t))K/2 K # T −t R # # # # 1 ε(y − β)(y − β) 2 /(2(T −t)) −|y−β| +# e dy # # (2π(T − t))K/2 # 3(T − t) B(βδ) 1 r¯ + er¯|y| −|y−β|2 /(2(T −t)) + dy e (2π(T − t))K/2 RK \B(βδ) T − t # # # ∂pA # ∂pA # #+0+ ε + ε =# − ∂β (Tβ) ∂β (Tβ0 ) # 3 3 <
ε ε ε + 0 + + = ε 3 3 3
which shows that Σ is continuous on [0 T ] × V . Now, we show that the pricing process is dynamically complete. Let Σj (j = ¯ β) be the K × K matrix whose 0 J) denote the jth row of Σ and let Σ(t
890
R. M. ANDERSON AND R. C. RAIMONDO
jth row (j = 1 J) is pA0 (t β)Σj (t β) − pAj (t β)Σ0 (t β) Σ¯ j (t β) = (pA0 (t β))2 =
∂(pAj /pA0 ) ∂β
Notice that (38)
¯ β) ≤ rank Σ(t β) rank Σ(t
Let ¯ I ) = 0} B = {I ∈ (0 T ) × RK : det Σ( ¯ I ) = 0 for every Suppose that B has positive Lebesgue measure. Then det Σ( I ∈ B. The determinant is a polynomial function of the entries of the matrix, ¯ I ) must be identically hence is an analytic function of I ∈ RK × (0 T ), so det Σ( 24 zero on (0 T ); since it is continuous on [0 T ] × V , it is identically zero on {T } × V . Using the nondegeneracy assumption (Equation (3)), choose ω0 such ˆ Tˆ ω0 ) ∈ V , and ˆ ω0 ) is finite, ◦ β( that β(· ⎞ ⎛ ∂(G1 /G0 ) ⎜ det ⎝
∂β
I (Tω0 )
∂(GJ /G0 ) ∂β
⎟ ⎠ = 0
I (Tω0 )
Since the securities prices are equilibrium prices, they must be arbitrage-free, ˆ Tˆ ω) ∈ V , so we have for j = 1 J and any ω such that ◦ β( pAj (T ω) pA0 (T ω)
24
pˆ Aj (Tˆ ω) pˆ A (Tˆ ω) 0
=
Aˆ j (Tˆ ω) Aˆ 0 (Tˆ ω)
=
ˆ Tˆ ω)) + O((T )2 ) Gj (Tˆ β( ∗ ˆ Tˆ ω)) + O((T )2 ) G0 (Tˆ β(
ˆ Tˆ ω)) Gj (Tˆ β( ∗ ˆ Tˆ ω)) G0 (Tˆ β(
Gj (T β(T ω)) G0 (T β(T ω))
∗
∗
See Theorem B.3 in Appendix B.
EQUILIBRIUM IN FINANCIAL MARKETS
891
so ¯ det Σ(T β(T ω0 )) ⎛ ∂(pA /pA ) ⎞ ⎞ ⎛ ∂(G1 /G0 ) 1 0 ∂β I (Tω0 ) ∂β I (Tω0 ) ⎜ ⎟ ⎟ ⎟ = det ⎜ = det ⎜ ⎠ = 0 ⎝ ⎝ ⎠ ∂(pA /pA ) ∂(G /G ) J 0 J 0 ∂β
I (Tω0 )
∂β
I (Tω0 )
a contradiction which proves that B is a set of measure zero. If we let Bt = {β ∈ RK : (t β) ∈ B} denote the t section of B, then by Fubini’s theorem, λ({t : Bt has positive Lebesgue measure}) = 0. Since the distribution of β(t ·) is absolutely continuous with respect to Lebesgue measure, " ! (λ × L(μ)) ˆ (t ω) : (I (t ω)) ∈ B ! " = L(μ) ˆ ω : (I (t ω)) ∈ B dλ =
[0T ]
[0T ]
L(μ) ˆ {ω : β(t ω) ∈ Bt } dλ
= 0 ¯ β(t ω)) = K, and hence rank Σ(t β(t Thus, we have shown that rank Σ(t ω)) = K, except on a set of measure zero. Now, we construct a money-market account: an admissible self-financing trading strategy which is instantaneously riskless. We first construct a moneymarket account in the hyperfinite model. Because the hyperfinite economy is internally dynamically complete, for every node (t ω), σ(t ˆ ω) has rank K. We claim that pˆ A (t ω) does not lie in the span of the columns of σ(t ˆ ω). If it did, note that dynamic completeness and the absence of arbitrage imply that there is a security holding whose value (ex dividend) at (t ω) is nonzero and whose value (cum dividend) is nonzero and the same at each of the successor nodes, so z · σ(t ˆ ω) = 0. Since pˆ A (t ω) lies in the span of the columns of σ(t ˆ ω), ˆ then z · pA (t ω) = 0, a contradiction which establishes the claim. Let σˆ (t ω) be the (K + 1) × (K + 1) matrix whose first column is pˆ A (t ω) and whose remaining columns are the columns of σ(t ˆ ω); we have just shown that σˆ (t ω) is nonsingular for every (t ω), so there is a unique 1 × (K + 1) row vector zˆ (t ω) satisfying the equation ⎛ ⎞ 1 0⎟ ⎜ ⎟ zˆ (t ω)Σˆ (t ω) = ⎜ ⎝ ⎠ 0
892
R. M. ANDERSON AND R. C. RAIMONDO
ˆ ω)pˆ A (t β(t ω)) = 1 and z(t ˆ ω)σ(t so z(t ˆ β(t ω)) = 0. Since γˆ is an internal martingale, zˆ d γˆ is an internal martingale; since zˆ σˆ = 0, zˆ d γˆ is identically zero, so zˆ is self-financing. ¯ β(t ω)) = K. If pA (t ω) lies in the span of Fix (t ω) such that rank Σ(t the columns of Σ(t β(t ω)), then there is a K × 1 column vector x such that pA (t ω) = Σ(t β(t ω))x, so the directional derivative of pA in the direction x/|x| is parallel to pA , so the directional derivative of the normal¯ so rank Σ¯ ≤ ized prices pA /pA0 in the direction x/|x| is zero, so x ∈ ker Σ, ◦ ˆ ω) is finite. Thus, z(t ˆ ω) K − 1, contradiction. Thus, σˆ is nonsingular, so z(t is finite almost everywhere, so we define a trading strategy almost everyˆ ω)σ(t ˆ ω)) = 0 and where by z(t ω) = ◦ zˆ (t ω). z(t ω)Σ(t β(t ω)) = ◦ (z(t 2 ◦ ˆ ˆ = (ˆ z (t ω) p (t ω)) = 1. Trivially, z σ ˆ is an SL lifting of zΣ, z(tω)pA (t ω) A ◦ zˆ d γˆ by Theorem C.4, so z dγ is identically zero. Thus, z so z dγ = is self-financing, admissible, and instantaneously risk-free, so it is a moneymarket account. Note that the risk-free rate of interest, expressed in the equilibrium prices, is zero.25 Now let c be any consumption bundle adapted to the Brownian filtration with finite value at the consumption price pC . Recall that γ is the total gains process associated with the equilibrium securities prices pA . Let V (ω) = pC (T ω)c(T ω) +
T
pC (t ω)c(t ω) dt
0
Since γ is a martingale, the process which is identically one is a state price process26 for γ. The dispersion matrix of γ is Σ, the dispersion matrix of pA ; we have seen it has rank K almost surely. Thus, the assumptions of Theorem 5.6 of Nielsen (1999) are satisfied. (Alternatively, since γ is a martingale, the true probability is a martingale measure for γ, so the assumptions of Duffie (1996, Proposition 6.I) are satisfied.) Since E(V ) < ∞, there is an ad27 missible self-financing trading t strategy zV which replicates V (ω) at time T . ¯ ω) for Let zc (t ω) = zV (t ω) − (( 0 pC (s ω)c(s ω) ds)/z¯ (t ω)pA (t ω))z(t t ∈ [0 T ). Then z is an admissible trading strategy which finances the consumption c with respect to the total gains process γ; it is, of course, not selffinancing unless c is identically zero for t < T . z is unique by Nielsen (1999, Proposition 5.5). Therefore, pA is effectively dynamically complete. We now derive the formulas for the hyperfinite equilibrium trading strategies and show that they are sufficiently regular to extract a candidate trading 25
Recall that the prices of consumption are marginal utility, so the units are utils. Equilibrium requires that a riskless security that costs 1 util today be worth 1 util tomorrow. 26 See Nielsen (1999, Sec. 4.3, p. 130) for the definition of a state price process. 27 Nielsen established admissibility, while Duffie did not require it in the definition of dynamic completeness and did not prove it here.
EQUILIBRIUM IN FINANCIAL MARKETS
893
strategy in continuous time. Let ˆ Tˆ )) cˆi (Tˆ β( ˆ Tˆ )) − eˆ i (Tˆ β( ˆ Tˆ )) Wˆ i (t ω) = E pˆ C (Tˆ β( Tˆ −T
+ T
ˆ ˆ ˆ cˆi (s β(s)) − eˆ i (s β(s)) pˆ C (s β(s)) (t ω)
s=t
Since zˆ i finances cˆi , we must have zˆ i (t ω) · pˆ A (t ω) = Wˆ i (t ω) ˆ we have By the same arguments we used to derive the formulas for pˆ A and σ, ˆ for every ω such that β(· ω) is S-continuous, for all t ∈ T , (39)
Wˆ i (t ω)
E pC (T β(T )) ci (T β(T )) − ei (T β(T )) +
T ◦t
◦ pC (s β(s)) ci (s β(s)) − ei (s β(s)) ds( t ω)
ˆ i (t + T ω) − D ˆ i (t ω) D pC (T β(T ))(ci (T β(T )) − ei (T β(T )))(β(T ) − β(t)) =E T −t T + pC (s β(s)) ci (s β(s)) − ei (s β(s)) (β(s) − β(t)) ◦t
× (s − ◦ t)−1 ds(◦ t ω) √ ˆ + T ω) − β(t ˆ ω)) + o( T ) × (β(t (40)
σˆ i (t ω) pC (T β(T ))(ci (T β(T )) − ei (T β(T )))(β(T ) − β(t)) E T − ◦t T + pC (s β(s)) ci (s β(s)) − ei (s β(s)) (β(s) − β(t)) ◦t
◦ × (s − t) ds( t ω) ◦
−1
894
R. M. ANDERSON AND R. C. RAIMONDO
Define standard functions Σ1 ΣI : [0 T ) × RK → RK and W1 WI : [0 T ] × RK → RK by Σi (t β) pC (T β(T ))(ci (T β(T )) − ei (T β(T )))(β(T ) − β(t)) =E T −t T + pC (s β(s)) ci (s β(s)) − ei (s β(s)) (β(s) − β(t)) t
× (s − t)−1 dsβ(t) = β Wi (t β) = E pC (T β(T )) ci (T β(T )) − ei (T β(T ))
T
+
pC (s β(s)) ci (s β(s)) − ei (s β(s)) dsβ(t) = β
t
Σ1 ΣI are continuous on [0 T ) and analytic on (0 T ); W1 WI are continuous on [0 T ] and analytic on (0 T ). Equations (39) and (40) show that σˆ i (t ω) Σi (I (◦ t ω)) zˆ i (t ω) · pˆ A (t ω) Wi (I (◦ t ω)) ˆ ω) is S-continuous. Summing up, we have for every ω such that β(· (41)
pˆ A (t ω) pA (◦ t ω) pˆ c (t ω) pc (◦ t ω) σ(t ˆ ω) Σ(I (◦ t ω)) cˆi (t ω) ci (◦ t ω) zˆ i (t ω) · σ(t ˆ ω) = σˆ i (t ω) Σi (I (◦ t ω)) zˆ i (t ω) · pˆ A (t ω) Wi (I (◦ t ω))
ˆ ω) is S-continuous. Thus, for every ω for all t ∈ T , for every ω such that β(· ˆ ω) is S-continuous, we have for every t ∈ T ,28 such that β(· (42)
pˆ A (t ω) pA (◦ t ω) ∗ pA (Iˆ (t ω))
28 In Equation (42), when we write ∗ pA (Iˆ (t ω)), we mean pA is a standard function of I (t ω) ∈ [0 T ] × RK ; take the nonstandard extension of this standard function defined on [0 T ] × RK and evaluate it at Iˆ (t ω). The analogous definition is used for ∗ pc ∗ Σ ∗ Σ1 ∗ ΣI ∗ W1 ∗ WI .
895
EQUILIBRIUM IN FINANCIAL MARKETS
pˆ c (t ω) pc (◦ t ω) ∗ pc (Iˆ (t ω)) σ(t ˆ ω) Σ(I (◦ t ω)) ∗ Σ(Iˆ (t ω)) cˆi (t ω) ci (I (◦ t ω)) ∗ ci (Iˆ (t ω)) ˆ ω) = σˆ i (t ω) Σi (I (◦ t ω)) ∗ Σi (Iˆ (t ω)) zˆ i (t ω) · σ(t zˆ i (t ω) · pˆ A (t ω) Wi (I (◦ t ω)) ∗ Wi (Iˆ (t ω)) ˆ ω) is S-continuous, Given ω such that β(· maxpˆ A (t ω) − ∗ pA (Iˆ (t ω)) 0 t∈T
since T is hyperfinite, and every hyperfinite set contains its maximum by the ˆ ω) is S-continuous (see the transfer principle. But for L(μ)-almost ˆ all ω, β(t proof of Theorem C.1), so for L(μ)-almost ˆ all ω, maxpˆ A (t ω) − ∗ pA (Iˆ (t ω)) 0 t∈T
Fix ε ∈ R++ . ' & ω : maxpˆ A (t ω) − ∗ pA (Iˆ (t ω)) < ε t∈T
is an internal set which contains a set of Loeb measure 1, so (& ') μˆ ω : maxpˆ A (t ω) − ∗ pA (Iˆ (t ω)) < ε > 1 − ε (43) t∈T
We have already noted that pˆ C pˆ A ∈ SLp for all p ∈ [1 ∞). Equation (35) and the growth conditions (Equation (1)) imply that, regardless of whether ˆ ˆ Iˆ (t ω) is finite, |σ(t ˆ ω)| ≤ r + er|β(tω)| and |σˆ i (t ω)| ≤ r + er|β(tω)| , which implies as above by Raimondo (2002, 2005, Proposition 3.1) that σ ˆ σˆ 1 σˆ I ∈ SLp for all p ∈ [1 ∞), in particular, for p = 2. zˆ i · pˆ A is bounded below by minus the value (at pˆ c ) of i’s future endowment and bounded above by the value of total market future consumption; hence there exists r¯¯ ∈ R such that ˆ ) , so zˆ i · pˆ A is SLp for all p ∈ [1 ∞). cˆi (t ω) ≤ zˆ i · pˆ A (t ω) ≤ r¯¯ + er¯¯(|β(tω)|+T ˆ p r + er|β(tω)| , so cˆi is SL for all p ∈ [1 ∞): ˆ
pˆ Aj (t ω) ≤ r + er(|β(tω)|+T )
⇒
pAj (I (t ω)) ≤ r + er(|β(tω)|+T )
⇒
∗
ˆ pAj (Iˆ (t ω)) ≤ r + er(|β(tω)|+T )
⇒
∗
pAj ◦ Iˆ ∈ SLp
(1 ≤ p < ∞)
896
R. M. ANDERSON AND R. C. RAIMONDO
Similarly, ∗ pc ◦ Iˆ ∗ Σ ◦ Iˆ ∗ ci ◦ Iˆ ∗ Σi ◦ Iˆ ∗ Wi ◦ Iˆ ∈ SLp for all p ∈ [1 ∞). Since pˆ A and ∗ pA ◦ I are SLp for all p ∈ [1 ∞) and ◦ (pˆ A (· ·) − ∗ pA (Iˆ (· ·))) is zero on a set of full Loeb measure, # # (44) pˆ A − ∗ pA ◦ Iˆ 2 #◦ (pˆ A − ∗ pA ◦ Iˆ )#2 = 0 ˆ ω) Now we consider the form of the equilibrium trading strategies zˆ i . β(t ¯ is finite and det Σ(t β(t ω)) = 0 at (Loeb) almost every node; fix such a node ˆ 0 + T ω0 ) is also finite. From the growth condition on div(t0 ω0 ). Then β(t ˆ 0 + T ω0 ) is finite. Let ω1 ωK be elements of idends, it follows that A(t Ω such that ωk (s) = ω0 (s) for s ≤ t0 and ωk (t0 + T ) = k; we can assume without loss of generality that ω0 (t0 + T√) = 0 (recall that ω(t0 + T ) = k ˆ 0 ω) = vk T ). Since zˆ i finances cˆi , we have, ˆ 0 + T ω) − β(t means that β(t for k = 0 K, ˆ 0 + T ωk ) − β(t ˆ 0 ω0 )) σˆ i (t0 ω0 )(β(t ˆ i (t0 ω0 ) ˆ i (t0 + T ωk ) − D =D ˆ 0 + T ωk ) − γ(t ˆ 0 ω0 )) = zˆ i (t0 ω0 ) · (γ(t ˆ 0 + T ωk ) − β(t ˆ 0 ω0 )) = zˆ i (t0 ω0 ) · σ(t ˆ 0 ω0 )(β(t ˆ 0 + T ωk ) − β(t ˆ 0 ω0 )) ˆ 0 ω0 )) (β(t = zˆ i (t0 ω0 )(σ(t Since {v0 vK } spans RK , we have ˆ 0 ω0 ) = σˆ i (t0 ω0 ) zˆ i (t0 ω0 )σ(t In addition, we have zˆ i (t0 ω0 ) · pˆ A (t0 ω0 ) = Wˆ i (t0 ω0 ) Let σ(t ˜ 0 ω0 ) = (σ(t ˆ 0 ω0 )|pˆ A (t0 ω0 )) σ˜ i (t0 ω0 ) = (σˆ i (t0 ω0 )|Wˆ i (t0 ω0 )) denote the matrix whose first K columns are the columns of σ(t ˆ 0 ω0 ) and whose (K + 1)st column is pˆ A (t0 ω0 ), and the row vector whose first K entries are the entries of σˆ i (t0 ω0 ) and whose (K + 1)st entry is Wˆ i (t0 ω0 ). Then ˜ 0 ω0 ) = σ˜ i (t0 ω0 ) zˆ i (t0 ω0 )σ(t and σ(t ˜ 0 ω0 ) is (K + 1) × (K + 1).
EQUILIBRIUM IN FINANCIAL MARKETS
897
˜ 0 ω0 ) is nonsingular. ◦ σ(t ˆ 0 ω0 ) = Σ(I (◦ t ω)) is (K + We claim that ◦ σ(t 1) × K and has rank K by Equation (38). If pA (◦ t0 ω0 ) = ◦ pˆ A (t0 ω0 ) lies in the span of the columns of Σ(I (◦ t0 ω0 )), there is a vector x ∈ RK such that Σ(I (◦ t0 ω0 ))x = pA (◦ t0 ω0 ). Since Σ(I (◦ t0 ω0 )) = ∂pA /∂β, we have for α ∈ R, pA (◦ t0 β(◦ t0 ω0 ) + αx) = pA (I (◦ t0 ω0 )) + αΣ(I (◦ t0 ω0 ))x + o(α) = pA (I (◦ t0 ω0 )) + αpA (I (◦ t0 ω0 )) + o(α) = (1 + α)pA (I (◦ t0 ω0 )) + o(α) and
1 pAj (◦ t0 β(◦ t0 ω0 ) + αx) pAj (I (◦ t0 ω0 )) − α pA0 (◦ t0 β(◦ t0 ω0 ) + αx) pA0 (I (◦ t0 ω0 )) 1 (1 + α)pAj (I (◦ t0 ω0 )) + o(α) pAj (I (◦ t0 ω0 )) − = α (1 + α)pA0 (I (◦ t0 ω0 )) + o(α) pA0 (I (◦ t0 ω0 )) pAj (I (◦ t0 ω0 )) 1 pAj (I (◦ t0 ω0 )) = + o(α) − α pA0 (I (◦ t0 ω0 )) pA0 (I (◦ t0 ω0 )) = o(1) as
α → 0
so the directional derivative of pAj /pA0 in the direction x is zero. Since ¯ I (◦ t0 ω0 )) is the Jacobian of (pA pA )/pA , Σ( J 1 0 ¯ I (◦ t0 ω0 ))x = 0 Σ( ¯ I (◦ t0 ω0 )) = 0. Accordingly, ◦ σ(t contradicting the fact that det Σ( ˜ 0 ω0 ) has rank K + 1 and thus is invertible. Since the formula for the inverse of a matrix is a polynomial function of the matrix components, the coefficients of the inverse ˜ 0 ω0 ) is also matrix are given by a standard analytic function of I (◦ t0 ω0 ). σ(t invertible and ◦
(σ(t ˜ 0 ω0 )−1 ) = (◦ σ(t ˜ 0 ω0 ))−1
Then ˜ 0 ω0 ))−1 zˆ i (t0 ω0 ) = σ˜ i (t0 ω0 )(σ(t so there are standard analytic functions29 Z1 ZI : (((0 T ) × RK ) \ B) → RK+1 such that zˆ i (t0 ω) Zi (I (◦ t0 ω)) ∗ Zi (Iˆ (t0 ω)) 29
¯ I ) = 0} is a set of measure zero. Recall that B = {I ∈ (0 T ) × RK : det Σ(
898
R. M. ANDERSON AND R. C. RAIMONDO
whenever
I (◦ t0 ω0 ) ∈ ((0 T ) × RK ) \ B Define zi (t ω) = ◦ zˆ i (tˆ ω); zˆ i σˆ = σˆ i is an SL2 lifting of zi σ, which belongs to H2 . We now show that (pA pC (c1 cI ) (Z1 ZI )) is an equilibrium of the Loeb continuous-time economy with utility weights λ and induces an equilibrium with utility weights λ for the original continuous-time economy. For Loeb almost all (t ω), I
ci (t ω) =
i=1
I
◦
cˆi (tˆ ω)
i=1
=◦ =◦
I
cˆi (tˆ ω)
i=1 I
eˆ i (tˆ ω) +
J
i=1
eˆ i (tˆ ω) +
i=1
=
i=1
zi (t ω) =
I
J
ei (t ω) +
ηj Aj (t ω)
j=0
I
◦
zˆ i (tˆ ω)
i=1
=◦
ηj ◦ Aˆ j (tˆ ω)
j=0
i=1 I
J
◦
ηj Aˆ j (tˆ ω)
j=0
I
=
I
zˆ i (tˆ ω)
i=1
= ◦ (η0 ηJ ) Thus, the consumptions c1 cI clear the goods market and the trading strategies z1 zI clear the securities market. By Theorem C.4, for all t ∈ [0 T ], pA (t ω) = ◦ pˆ A (tˆ ω)
899
EQUILIBRIUM IN FINANCIAL MARKETS
= ◦ γ( ˆ tˆ ω) − T
ˆ ω) pˆ C (s ω)A(s
s≤tˆs
tˆ ˆ ω) ˆ ω) + = ◦ γ(0 σˆ d βˆ − T pˆ C (s ω)A(s 0
= γ(0 ω) +
s≤tˆs
t
t
σ dβ − 0
pC (s ω)A(s ω) ds 0
Given any hyperfinite trading strategy zˆ i and t ∈ T , we have t ˆ + T ω) − γ(s ˆ ω)) zˆ i d γˆ = zˆ i (s ω) · (γ(s 0
s∈T s
=
ˆ + T ω) − β(s ˆ ω)) (ˆzi (s ω)σ(s ˆ ω)) · (β(s
s∈T s
t
=
ˆ (ˆzi σ) ˆ d β
0
Since cˆ1 cˆI z1 zI are adapted to the hyperfinite filtration, cˆ1 cˆI z1 zI are adapted to the Loeb filtration; since cˆ1 cI ∈ SLp for all p ∈ [1 ∞), c1 cI ∈ Lp for all p ∈ [1 ∞) (Anderson (1976)). For all t ∈ [0 T ), pA (t ω) · zi (t ω) − pA (0 ω) · eiA (ω) t t zi dγ − pC (s ω)(ei (s ω) − ci (s ω)) ds − 0
0
= pA (t ω) · zi (t ω) − pA (0 ω) · eiA (ω) t t − zi σ dβ − pC (s ω)(ei (s ω) − ci (s ω)) ds 0
0
pˆ A (tˆ ω) · zˆ i (t ω) − pˆ A (0 ω) · eˆ iA (ω) tˆ tˆ − zˆ i σˆ d βˆ − pˆ C (s ω)(eˆ i (s ω) − cˆi (s ω)) ds 0
0
= pˆ A (tˆ ω) · zˆ i (tˆ ω) − pˆ A (0 ω) · eˆ iA (ω) tˆ tˆ − zˆ i d γˆ − pˆ C (s ω)(eˆ i (s ω) − cˆi (s ω)) ds 0
0
= 0 since the trading strategy zˆ i finances the consumption cˆi . Thus, the trading strategy zi finances the consumption ci .
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For all nodes (t ω) in the hyperfinite economy, λˆ i ∗ hi (ci (t ω) I (t ω)) = pˆ C (t ω) for each i. Recalling that λˆ i 0, so λi = 0, and ◦ pˆ C ∈ (0 ∞) Loeb almost everywhere, ∂hi ◦ ˆ ◦ ∗ ∂hi λi = ( λ ) i ∂ci (ci (tω)I (tω)) ∂ci (cˆi (tω)Iˆ (tω)) ◦ ˆ ∗ ∂hi = λi ∂ci (cˆi (tω)Iˆ (tω)) = ◦ pˆ C (Iˆ (t ω)) = pC (I (t ω)) holds for almost all nodes in the Loeb continuous-time economy. This shows that for almost all nodes, the consumptions c1 (t ω) cI (t ω) satisfy the necessary conditions for the Negishi weight problem (maximize I I i=1 λi hi (ci (t ω) I (t ω)) subject to i=1 ci (t ω) = c(t ω)), as well as the necessary conditions for demand. Since preferences are strictly concave, the necessary conditions are in fact sufficient. Here, we present the details for the demand; the argument for the Negishi weight problem is similar. Suppose that ci lies in the budget set. Then E pC (T )ci (T ) +
T
i
pC (t)c (t) dt 0
T
≤ eiA · pA (0) + E
pC (t)ei (t) dt 0
= E pC (T )ci (T ) +
T
pC (t)ci (t) dt
0
Since hi and Hi are concave, hi (ci (t β) (t β))
∂hi (c (t β) − ci (t β)) ≤ hi (ci (t β) (t β)) + ∂ci (ci (tβ)(tβ)) i
Hi (ci (T β) (T β))
∂Hi (c (T β) − ci (T β)) ≤ Hi (ci (T β) (T β)) + ∂ci (ci (Tβ)(Tβ)) i
EQUILIBRIUM IN FINANCIAL MARKETS
901
so U(ci ) − U(ci ) t hi (ci (t)) − hi (ci (t)) dt = E Hi (ci (T )) − Hi (ci (T )) + ∂Hi ≤E ∂c
0
i
(ci (T ) − ci (T )) (ci (T )T(β))
∂hi (c (t) − c (t)) dt i i 0 ∂ci (ci (t)(tβ)) t 1 pC (t)(ci (t) − ci (t)) dt = E pC (T )(ci (T ) − ci (T )) + λi 0 T 1 E pC (T )ci (T ) + pC (t)ci (t) dt = λi 0 T pC (t)ci (t) dt − E pC (T )ci (T ) +
t
+
0
≤ 0 so ci lies in agent i’s demand set. This shows that we have an equilibrium for the Loeb continuous-time economy and that λ is a vector of Negishi utility weights for the Loeb continuous-time economy. Now, consider the original continuous-time economy specified in the statement of Theorem 2.1. To distinguish the original economy from the Loeb economy, we will denote the Brownian motion in the original economy by β¯ and let ¯ ω)). The equilibrium prices, consumptions, and (except for I¯ (t ω) = (t β(t a set of measure zero) trading strategies of the Loeb economy are given by functions of I (t ω): pA (t ω) = PA (I (t ω)) pC (t ω) = PC (I (t ω)) ci (t ω) = Ci (I (t ω)) zi (t ω) = Zi (I (t ω)) Using these functions, define a candidate equilibrium for the original continuous-time economy: p¯ A (t ω) = PA (I¯ (t ω)) p¯ C (t ω) = PC (I¯ (t ω))
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c¯i (t ω) = Ci (I¯ (t ω)) z¯ i (t ω) = Zi (I¯ (t ω)) ¯ ω ¯ 0 ): Fix ω ¯ 0 ∈ Ω¯ and suppose that ω0 ∈ Ω satisfies β(t ω0 ) = β(t ¯ ω ¯ ω ¯ 0 )p¯ A (t ω ¯ 0 ) = Zi (t β(t ¯ 0 ))PA (t β(t ¯ 0 )) z¯ i (t ω = Zi (t β(t ω0 ))PA (t β(t ω0 )) = Wi (t β(t ω0 )) ¯ ω0 )) = Wi (t β(t where Wi is an analytic function of (t β) ∈ (0 T ) × RK . By Itô’s lemma, ¯ Wi (t β(t)) must satisfy the same stochastic differential equation with respect ¯ to β as Wi (t β(t)) satisfies with respect to β, and the coefficients of the sto¯ and (t β), respectively: chastic differential equation are functions of (t β) d Wi (t β(t)) = a(t β(t)) dt + b(t β(t)) dβ ¯ ¯ ¯ ¯ d Wi (t β(t)) = a(t β(t)) dt + b(t β(t)) d β Since zi finances ci , we have a(t β) = (Ci (t β) − fi (t β)) b(t β) =
∂Di (t β) ∂β
for all (t β) ∈ (0 T ) × RK , so ¯ − fi (t β)) ¯ ¯ = (Ci (t β) a(t β) ¯ ¯ = ∂Di (t β) b(t β) ∂β¯ so z¯ i finances c¯i . Since zi is admissible, z¯ i is admissible and E p¯ C (T )c¯i (T ) +
T
p¯ C (t)c¯i (t) dt 0
T
= e¯ iA · p¯ A (t) + E 0
I
i=1
I
c¯i (t β) =
i=1
ci (t β)
p¯ C (t)e¯ i (t) dt
EQUILIBRIUM IN FINANCIAL MARKETS
=
I
ei (t β) +
i=1
=
z¯ i (t β) =
i=1
ηj Aj (t β)
j=0
I
e¯ i (t β) +
i=1 I
J
903
J
ηj A¯ j (t β)
j=0
I
zi (t β)
i=1
=
J
ηj
j=0
Thus, the markets for consumption and securities clear at almost all nodes in the original continuous-time economy. For almost all nodes in the original continuous time economy, ∂hi ∂hi = λi λi ∂ci (c¯i (tβ)(t ∂ci (ci (tβ)(tβ)) ¯ ¯ β)) = pC (I (t ω)) = p¯ C (I¯ (t ω)) This shows that for almost all nodes, the consumptions c1 (t ω) cI (t ω) satisfy the necessary conditions for the Negishi weight problem maximize I I i=1 λi hi (ci (t ω) I (t ω)) subject to i=1 ci (t ω) = c(t ω), as well as the necessary conditions to be the demand. Since preferences are strictly concave, the necessary conditions are in fact sufficient; the argument is exactly the same as the argument above for the Loeb continuous-time economy. This shows that we have an equilibrium for the original continuous-time economy, and that λ is a vector of Negishi utility weights for the original continuous-time economy. This completes the proofs of Theorem 2.1 and Proposition 2.2. Let E denote the set of all equilibria of the original continuous-time economy. Recall that we have been working in a discretization indexed by a particular n ∈ ∗ N \ N, but suppressing the index n. Putting the index back into the notation, we have shown that for n ∈ ∗ N \ N, (λn pnA pnC (cn1 cnI ) (Zn1 ZnI )) ∈ E so30 (λn ∗ pnA ∗ pnC (∗ cn1 ∗ cnI ) (∗ Zn1 ∗ ZnI )) ∈ ∗ E 30
Since λn ∈ R, ∗ λn = λn .
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Fix ε ∈ R++ . Equation (43) tells us that ∃m∈∗ N ∀n≥m ∃(λn pnA pnC (Zn1 ZnI )(cn1 cnI ))∈∗ E (& ') μˆ ω : maxpˆ nA (t ω) − ∗ pnA (Iˆ (t ω)) < ε > 1 − ε t∈Tn
(any m ∈ ∗ N \ N will do). Since the construction of the hyperfinite economy and the price process pˆ nA is just the transfer of the construction for finite n, the transfer principle tells us that ∃m∈N ∀n≥m ∃(λn pnA pnC (Zn1 ZnI )(cn1 cnI ))∈E ') (& μˆ ω : maxpˆ nA (t ω) − pnA (Iˆ (t ω)) < ε > 1 − ε t∈Tn
which is the convergence in probability part of Equation (11); Equation (16) and the convergence in probability parts of Equations (12), (13), (14), (17), and (18) in the statement of Theorem 4.2 follow in exactly the same way. Equation (44) tells us that ∃m∈∗ N ∀n≥m ∃(λn pnA pnC (Zn1 ZnI )(cn1 cnI ))∈∗ E
pˆ nA − ∗ pnA ◦ Iˆ 2 < ε
As before, the transfer principle tells us that ∃m∈N ∀n≥m ∃(λn pnA pnC (Zn1 ZnI )(cn1 cnI ))∈E
pˆ nA − pnA ◦ Iˆ 2 < ε
This is the convergence in norm part of Equations (11); Equations (10) and (15), and the convergence in norm parts of Equations (12), (13), (14), (17), and (18) in the statement of Theorem 4.2 follow in exactly the same way. This completes the proof of Theorem 4.2. Q.E.D. REFERENCES ALBEVERIO, S., AND F. S. HERZBERG (2006): “A Combinatorial Infinitesimal Representation of Lévy Processes and Possible Applications,” Stochastics, 78, 301–325. [864] ANDERSON, R. M. (1976): “A Nonstandard Representation of Brownian Motion and Itô Integration,” Israel Journal of Mathematics, 25, 15–46. [847,873-877,879,882,884,887,899] (2000): “Infinitesimal Methods in Mathematical Economics,” Preprint, http://www. econ.berkeley.edu/~anderson/Book.pdf. [873] ANDERSON, R. M., AND R. C. RAIMONDO (2005): “Market Clearing and Derivative Pricing,” Economic Theory, 25, 21–34. [850,864] (2006): “Equilibrium Pricing of Derivative Securities in Dynamically Incomplete Markets,” in Institutions, Equilibria and Efficiency: Essays in Honor of Birgit Grodal, ed. by C. Schultz and K. Vind. New York: Springer, 27–48. [850,864] ANDERSON, R. M., AND S. RASHID (1978): “A Nonstandard Characterization of Weak Convergence,” Proceedings of the American Mathematical Society, 69, 327–332. [864] BANK, P., AND F. RIEDEL (2001): “Existence and Structure of Stochastic Equilibria With Intertemporal Substitution,” Finance and Stochastics, 5, 487–509. [841]
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BASAK, S., AND D. CUOCO (1998): “An Equilibrium Model With Restricted Stock Market Participation,” Review of Financial Studies, 11, 309–341. [842] BICK, A. (1990): “On Viable Diffusion Price Processes of the Market Portfolio,” Journal of Finance, 45, 673–689. [842] BROWN, D. J., AND A. ROBINSON (1975): “Nonstandard Exchange Economies,” Econometrica, 43, 41–55. [848] BROWN, D. J., P. M. DEMARZO, AND B. C. EAVES (1996): “Computing Equilibria When Asset Markets Are Incomplete,” Econometrica, 64, 1–27. [845] BRYC, W. (1995): The Normal Distribution: Characterizations With Applications, Lecture Notes in Statistics, Vol. 100. New York: Springer-Verlag. [874] COX, J. C., J. E. INGERSOLL JR., AND S. A. ROSS (1985): “An Intertemporal General Equilibrium Model of Asset Prices,” Econometrica, 53, 363–384. [842] COX, J. C., S. A. ROSS, AND M. RUBINSTEIN (1979): “Option Pricing: A Simplified Approach,” Journal of Financial Economics, 7, 229–263. [845,864] CUOCO, D. (1997): “Optimal Consumption and Equilibrium Prices With Portfolio Constraints and Stochastic Income,” Journal of Economic Theory, 72, 33–73. [842] CUTLAND, N. J. (2000): Loeb Measures in Practice: Recent Advances, Lecture Notes in Mathematics, Vol. 1751. Berlin: Springer-Verlag. [864] CUTLAND, N. J., AND S.-A. NG (1995): “A Nonstandard Approach to the Malliavin Calculus,” in Advances in Analysis, Probability and Mathematical Physics (Blaubeuren 1992), ed. by S. Albeverio, M. P. Wolff, and W. A. J. Luxemburg, Mathematics and Applications. Dodrecht: Kluwer, 149–170. [864] CUTLAND, N. J., P. E. KOPP, AND W. WILLINGER (1991a): “A Nonstandard Approach to Option Pricing,” Mathematical Finance, 1, 1–38. [864] (1991b): “Nonstandard Methods in Option Pricing,” in Proceedings of the 30th IEEE Conference in Decision and Control. New York: IEEE, 1293–1298. [864] (1991c): “Convergence of Cox–Ross–Rubinstein to Black–Scholes,” Unpublished Manuscript, University of York, U.K. [864] (1993): “From Discrete to Continuous Financial Models: New Convergence Results for Option Pricing,” Mathematical Finance, 3, 101–123. [864] (1995a): “Stock Price Returns and the Joseph Effect: A Fractional Version of the Black– Scholes Model,” Progress in Probability, 36, 327–351. [864] (1995b): “From Discrete to Continuous Stochastic Calculus,” Stochastics and Stochastics Reports, 52, 173–192. [864] CUTLAND, N. J., P. E. KOPP, W. WILLINGER, AND M. C. WYMAN (1997): “Convergence of Snell Envelopes and Critical Prices in the American Put,” in Mathematics for Derivative Securities, Isaac Newton Institute Proceedings Volume, ed. by M. H. A. Demmpster and S. R. Pliska. Cambridge, U.K.: Cambridge University Press, 126–140. [864] DANA, R. A. (1993): “Existence and Uniqueness of Equilibria When Preferences Are Additively Separable,” Econometrica, 61, 953–957. [841,851] DANA, R. A., AND M. JEANBLANC (2002): Financial Markets in Continuous Time. Berlin: Springer-Verlag. [842] DETEMPLE, J. B., AND I. KARATZAS (2003): “Non-Addictive Habits: Optimal Consumption– Portfolio Policies,” Journal of Economic Theory, 113, 265–285. [842] DIASAKOS, T. (2007): “Comparative Statics of General Equilibrium Asset-Pricing,” Ph.D. Dissertation, Department of Economics, University of California, Berkeley. [857] DUFFIE, D. (1986): “Stochastic Equilibria: Existence, Spanning Number and the “No Expected Financial Gains From Trade” Hypothesis,” Econometrica, 54, 1161–1184. [842] (1991): “The Theory of Value in Security Markets,” in Handbook of Mathematical Economics, Vol. IV, ed. by W. Hildenbrand and H. Sonnenschein. Amsterdam: Elsevier Science Publishers, Chapter 31, 1615–1682. [842] (1995): Security Markets, Stochastic Models. Boston, MA: Academic Press. [842]
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(1996): Dynamic Asset Pricing Theory (Second Ed.). Princeton, NJ: Princeton University Press. [842,852,892] DUFFIE, D., AND C.-F. HUANG (1985): “Implementing Arrow–Debreu Equilibria by Continuous Trading of Long-Lived Securities,” Econometrica, 53, 1337–1356. [842] DUFFIE, D., AND W. SHAFER (1985): “Equilibrium in Incomplete Markets I: Basic Model of Generic Existence,” Journal of Mathematical Economics, 14, 285–300. [846,859] (1986): “Equilibrium in Incomplete Markets II: Generic Existence in Stochastic Economies,” Journal of Mathematical Economics, 15, 199–216. [846,859] DUFFIE, D., AND C. SKIADAS (1994): “Continuous-Time Security Pricing: A Utility Gradient Approach,” Journal of Mathematical Economics, 23, 107–131. [842] DUFFIE, D., AND W. R. ZAME (1989): “The Consumption-Based Capital Asset Pricing Model,” Econometrica, 57, 1279–1297. [842] HARRISON, J. M., AND D. M. KREPS (1979): “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, 20, 381–408. [854] HE, H., AND H. LELAND (1995): “On Equilibrium Asset Price Processes,” Review of Financial Studies, 6, 593–617. [842] HOOVER, D. N., AND E. A. PERKINS (1983a): “Nonstandard Construction of the Stochastic Integral and Applications to Stochastic Differential Equations I,” Transactions of the American Mathematical Society, 275, 1–36. [864] (1983b): “Nonstandard Construction of the Stochastic Integral and Applications to Stochastic Differential Equations II,” Transactions of the American Mathematical Society, 275, 37–58. [864] HURD, A. E., AND P. A. LOEB (1985): An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press. [873] JUDD, K. L., F. KUBLER, AND K. SCHMEDDERS (2000): “Computing Equilibria in InfiniteHorizon Finance Economies: The Case of One Agent,” Journal of Economic Dynamics and Control, 24, 1047–1078. [845] (2002): “A Solution Method for Incomplete Asset Markets With Heterogeneous Agents,” Preprint, Stanford University, http://bucky.stanford.edu/papers/jks22k2.pdf. [845] (2003): “Computational Methods for Incomplete Asset Markets With Heterogeneous Agents,” in Advances in Economics and Econometrics, ed. by M. Dewatripont, L. P. Hansen, and S. Turnovsky. Cambridge, U.K.: Cambridge University Press, 243–290. [845] KARATZAS, I., J. P. LEHOCZKY, AND S. E. SHREVE (1990): “Existence and Uniqueness of MultiAgent Equilibrium in a Stochastic, Dynamic Consumption/Investment Model,” Mathematics of Operations Research, 15, 80–128. [842] KEISLER, H. J. (1984): An Infinitesimal Approach to Stochastic Analysis, Memoirs of the American Mathematical Society, Vol. 48 (297). [864,874] (1996): “Getting to a Competitive Equilibrium,” Econometrica, 64, 29–49. [864] KHAN, M. A., AND Y. SUN (1997): “The Capital–Asset-Pricing Model and Arbitrage Pricing Theory: A Unification,” Proceedings of the National Academy of Sciences of the United States of America, 94, 4229–4232. [864] (2001): “Asymptotic Arbitrage and the APT With or Without Measure-Theoretic Structures,” Journal of Economic Theory, 101, 222–251. [864] KRANTZ, S. G., AND H. R. PARKS (2002): A Primer of Real Analytic Functions (Second Ed.). Boston, MA: Birkhäuser. [869,872,873,884] LINDSTRØM, T. L. (1980a): “Hyperfinite Stochastic Integration I: The Nonstandard Theory,” Mathematica Scandinavica, 46, 265–292. [864,873] (1980b): “Hyperfinite Stochastic Integration II: Comparison With the Standard Theory,” Mathematica Scandinavica, 46, 293–314. [864,873] (1980c): “Hyperfinite Stochastic Integration III: Hyperfinite Representations of Standard Martingales,” Mathematica Scandinavica, 46, 315–331. [864,873] (1980d): “Addendum to Hyperfinite Stochastic Integration III,” Mathematica Scandinavica, 46, 332–333. [864,873]
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(2004): “Hyperfinite Lévy Processes,” Stochastics and Stochastics Reports, 76, 517–548. [864] (2008): “Nonlinear Stochastic Integrals for Hyperfinite Lévy Processes,” Logic and Analysis (forthcoming). [864] LOEB, P. A. (1975): “Conversion From Nonstandard to Standard Measure Spaces and Applications in Potential Theory,” Transactions of the American Mathematical Society, 211, 113–122. [847,877] MAGILL, M., AND M. QUINZII (1996): Theory of Incomplete Markets, Vol. 1. Cambridge, MA: MIT Press. [846,861] MAGILL, M., AND W. SHAFER (1990): “Characterization of Generically Complete Real Asset Structures,” Journal of Mathematical Economics, 19, 167–194. [846,859] MAS-COLELL, A., AND S. RICHARD (1991): “A New Approach to the Existence of Equilibrium in Vector Lattices,” Journal of Economic Theory, 53, 1–11. [841] MERTON, R. C. (1973): “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41, 867–887. [856] (1990): “A Complete-Markets General Equilibrium Theory of Finance in Continuous Time,” in Continuous-Time Finance, ed. by R. C. Merton. Cambridge, MA: Basil Blackwell, Chapter 16, 524–576. [856] NEGISHI, T. (1960): “Welfare Economics and Existence of an Equilibrium for a Competitive Economy,” Metroeconomica, 12, 92–97. [844] NG, S.-A. (2003): Hypermodels in Mathematical Finance. River Edge, NJ: World Scientific. [864] (2008): “A Nonstandard Lévy–Khintchine Formula and Lévy Processes,” Acta Mathematica Sinica, 24, 241–252. [864] NIELSEN, L. T. (1999): Pricing and Hedging of Derivative Securities. Oxford, U.K.: Oxford University Press. [852,854,874,892] (2007): “Dividends in the Theory of Derivative Security Pricing,” Economic Theory, 31, 447–471. [852] OSSWALD, H. (2006): “A Discrete Approach to Malliavin Calculus for Lévy Processes,” Preprint, Department of Mathematics, University of Munich. [864] RAIMONDO, R. C. (2001): “Incomplete Markets With a Continuum of States,” Preprint, http: //www.economics.unimelb.edu.au/rraimondo/raimondo.htm. [846] (2002): “Essays in Incomplete Markets Theory,” Ph.D. Dissertation, Department of Economics, University of California, Berkeley. [842,879,882,884,895] (2005): “Market Clearing, Utility Functions, and Securities Prices,” Economic Theory, 25, 265–285. [842,879,882,884,895] RIEDEL, F. (2001): “Existence of Arrow–Radner Equilibrium With Endogenously Complete Markets With Incomplete Information,” Journal of Economic Theory, 97, 109–122. [842] STEELE, J. M. (2001): “Stochastic Calculus and Financial Applications,” in Applications of Mathematics: Stochastic Modelling and Applied Probability, Vol. 45, ed. by I. Karatzas and M. Yor. New York: Springer-Verlag. [877] VASICEK, O. (1977): “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5, 177–188. [865] ZAME, W. R. (2001): “Continuous Trading: A Cautionary Tale,” Preprint, Department of Economics, UCLA. [846]
Econometrica, Vol. 76, No. 4 (July, 2008), 909–933
COMMON LEARNING BY MARTIN W. CRIPPS, JEFFREY C. ELY, GEORGE J. MAILATH, AND LARRY SAMUELSON1 Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. The signals are independent and identically distributed across time but not necessarily across agents. We show that when each agent’s signal space is finite, the agents will commonly learn the value of the parameter, that is, that the true value of the parameter will become approximate common knowledge. The essential step in this argument is to express the expectation of one agent’s signals, conditional on those of the other agent, in terms of a Markov chain. This allows us to invoke a contraction mapping principle ensuring that if one agent’s signals are close to those expected under a particular value of the parameter, then that agent expects the other agent’s signals to be even closer to those expected under the parameter value. In contrast, if the agents’ observations come from a countably infinite signal space, then this contraction mapping property fails. We show by example that common learning can fail in this case. KEYWORDS: Common learning, common belief, private signals, private beliefs.
1. INTRODUCTION CONSIDER TWO AGENTS who observe sequences of private signals sufficiently rich that they almost surely learn the value of an underlying parameter. The signals are independent and identically distributed across time but not necessarily across agents. Does it follow that the agents will commonly learn the parameter value, that is, that the true value will become approximate common knowledge? We show that the answer is affirmative when each agent’s signal space is finite and show by example that common learning can fail when observations come from a countably infinite signal space. Common learning is precisely what is needed to make efficient outcomes possible in coordination problems in which agents privately learn the appropriate course of action. For example, suppose that in every period t = 0 1 each agent receives a signal bearing information about a parameter θ. The agent can then choose action A, action B, or to wait (W ) until the next period. Simultaneous choices of A when the parameter is θA or B when it is θB bring payoffs of 1 each. Lone choices of A or B or joint choices that do not match the parameter bring a payoff of −c < 0 and cause the joint opportunity to disappear. Waiting is costless. Figure 1 summarizes these payoffs. Under what circumstances do there exist equilibria of this investment game in which the agents do not always wait? Choosing action A is optimal for an c agent in some period t only if the agent attaches probability at least c+1 ≡q 1 Cripps thanks the Economic and Social Research Council (U.K.) via ELSE, and Mailath and Samuelson thank the National Science Foundation (Grants SES-0350969 and SES-0549946, respectively) for financial support. We thank Stephen Morris, the editor, and three referees for helpful comments.
909
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FIGURE 1.—Payoffs from a potential joint opportunity, with actions A, B, or wait (W ) available to each agent in each period.
to the joint event that the parameter is θA and the other agent chooses A. Now consider the set of histories A at which both agents choose A. At any such history, each agent must assign probability at least q to A, that is, A must be q-evident (Monderer and Samet (1989)). Furthermore, at any history in A, each agent must assign probability at least q to the parameter θA . But this pair of conditions is equivalent to the statement that θA is common q-belief. The existence of histories at which there is common q-belief is thus a necessary condition for eventual coordination in this game. Conversely, the possibility of common q-belief is sufficient for a nontrivial equilibrium, as it is an equilibrium for each agent to choose A on the q-evident event on which θA is common q-belief. Now suppose that various forms of this opportunity arise, characterized by different values of the miscoordination penalty c. What does it take to ensure that all of these opportunities can be exploited? It suffices that the information process be such that the parameter eventually becomes arbitrarily close to common 1-belief. More generally, common learning is a potentially important tool in the analysis of dynamic games with incomplete information. Examples include reputation models such as Cripps, Mailath, and Samuelson (2007), where one player learns the “type” of the other, and experimentation models such as Wiseman (2005), where players learn about their joint payoffs in an attempt to coordinate on some (enforceable) target outcome. Characterizing equilibria in these games requires analyzing not only each player’s beliefs about payoffs, but also higher-order beliefs about the beliefs of others. Existing studies of these models have imposed strong assumptions on the information structure to keep the analysis tractable. We view our research as potentially leading to some general tools for studying common learning in dynamic games. Passing from individual to common learning requires showing that agent eventually attaches high probability to the true parameter value as well as to ˆ attaching high probability to the true parameter value. events such as agent ’s This will obviously hold in the special case of public signals, where beliefs are
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identical. At the opposite extreme, suppose agents’ signals are stochastically independent, and so (conditional on the parameter) each learns nothing about the other’s beliefs. Signals typical of parameter value θ then lead agent to place high probability on parameter value θ and—knowing that agent ˆ is facing informative signals—a high probability on ˆ similarly having seen signals typical of θ. Moreover, conditional on the parameter θ, the probability that assigns to ˆ seeing typical signals is independent of ’s signals. This uniformity in ’s beliefs plays a vital role in establishing common learning (Proposition 2). However, when signals are private but not independent, agent may observe signals that are highly likely under some parameter θ but which lead to believe, conditional on θ, that agent ˆ has observed signals less typical of θ, disrupting the straightforward common-learning argument that suffices for independent signals. The key observation in our general argument is that when the set of signals is finite, the distribution of one agent’s signals, conditional on the other agent’s signals, has a Markov chain interpretation.2 This allows us to appeal to a contraction mapping principle that effectively replaces the uniformity in beliefs of the independent-signals case, ensuring that if agent ’s signals are close to those that would be expected under some parameter value θ, then believes ˆ signals are even closer to what would be expected under θ. In conthat ’s trast, with a countably infinite signal space, the corresponding Markov chain interpretation lacks the relevant contraction mapping structure and common learning may fail. 2. A MODEL OF MULTIAGENT LEARNING 2.1. Learning Time is discrete and periods are denoted by t = 0 1 2 Before period zero, nature selects a parameter θ from the finite set Θ according to the prior distribution p. Remark 4 explains the extent to which we can relax the assumption that agents share a common prior. For notational simplicity, we restrict attention to two agents, denoted by = 1 (he) and 2 (she). Our positive results (Propositions 2 and 3) hold for an arbitrary finite number of agents (see Remarks 3 and 5). θ θ ∞ Conditional on θ, a stochastic process ζ θ ≡ {ζtθ }∞ t=0 ≡ {ζ1t ζ2t }t=0 generates a signal profile zt ≡ (z1t z2t ) ∈ Z1 × Z2 ≡ Z for each period t, where Z is the set of possible period-t signals for agent = 1 2.3 For each θ ∈ Θ, the signal process {ζtθ }∞ t=0 is independent and identically distributed across t. When 2
This perspective is inspired by Samet (1998). Monderer and Samet (1995) established a common learning result for agents who observe one realization of a private signal and then observe only public signals. 3
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convenient, we let {θ} denote the event {θ} × Z ∞ , and often write θ rather than {θ} when the latter appears as an argument of a function. A state consists of a parameter and a sequence of signal profiles, with the set of states given by Ω ≡ Θ × Z ∞ . We use P to denote the measure on Ω induced by the prior p and the signal processes (ζ θ )θ∈Θ , and use E[ · ] to denote expectations with respect to this measure. Let P θ denote the measure conditional on a given parameter and let E θ [·] denote expectations with respect to this measure. A period-t history for agent is denoted by ht ≡ (z0 z1 zt−1 ). We let Ht ≡ (Z )t denote the space of period-t histories for agent and let {Ht }∞ t=0 denote the filtration induced on Ω by agent ’s histories. The random variables {P(θ | Ht )}∞ t=0 , giving agent ’s beliefs about the parameter θ at the start of each period, are a bounded martingale with respect to the measure P, for each θ, and so the agents’ beliefs converge almost surely (Billingsley (1979, Theorem 35.4)). For any state ω, ht (ω) ∈ Ht is the agent- period-t history induced by ω. As usual, P(θ | Ht )(ω) is often written P(θ | ht (ω)) or P(θ | ht ) when ω is understood. For any event F ⊂ Ω, the Ht -measurable random variable E[1F | Ht ] is the probability agent attaches to F given her information at time t. We define q Bt (F) ≡ ω ∈ Ω : E[1F | Ht ](ω) ≥ q q
Thus, Bt (F) is the set of states where at time t agent attaches at least probability q to event F . DEFINITION 1—Individual Learning: Agent learns parameter θ if conditional on parameter θ, agent ’s posterior on θ converges in probability to 1, that is, if for each q ∈ (0 1) there is T such that for all t > T , (1)
q
P θ (Bt (θ)) > q
Agent learns Θ if learns each θ ∈ Θ. REMARK 1: Individual learning is equivalent to (2)
q
lim P θ (Bt (θ)) = 1
t→∞
∀q ∈ (0 1)
We have formulated individual learning using convergence in probability rather than almost sure convergence to facilitate the comparison with common learning. While convergence in probability is in general a weaker notion than almost sure convergence, since P(θ | Ht ) converges almost surely to some random variable, (2) is equivalent to P(θ | Ht ) → 1 P θ -a.s. We assume throughout that each agent individually learns the parameter— there is no point considering common learning when individual learning fails.
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Our aim is to identify the additional conditions that must be imposed to ensure that the agents commonly learn the parameter. q The event that F ⊂ Ω is q-believed at time t, denoted by Bt (F), occurs if each agent attaches at least probability q to F , that is, q
q
q
Bt (F) ≡ B1t (F) ∩ B2t (F) The event that F is common q-belief at date t is q q Ct (F) ≡ [Bt ]n (F) n≥1 q
q
Hence, on Ct (F), the event F is q-believed, and this event Bt (F) is itself q-believed and so on. We are interested in common q-belief as a measure of approximate common knowledge because this notion ensures the continuity of equilibrium behavior in incomplete information games (Monderer and Samet (1989)). We say that the agents commonly learn the parameter θ if, for any probability q, there is a time such that, with high probability when the parameter is θ, it is common q-belief at all subsequent times that the parameter is θ: DEFINITION 2—Common Learning: The agents commonly learn parameter θ ∈ Θ if for each q ∈ (0 1) there exists a T such that for all t > T , q
P θ (Ct (θ)) > q The agents commonly learn Θ if they commonly learn each θ ∈ Θ. REMARK 2: Common learning is equivalent to q
lim P θ (Ct (θ)) = 1
t→∞ q
∀q ∈ (0 1)
q
Because Ct (θ) ⊂ Bt (θ), common learning implies individual learning (cf. (2)). q
Rather than working with the countable collection of events {[Bt ]n (θ)}n≥1 directly, it is easier to work with q-evident events. An event F is q-evident at time t if it is q-believed when it is true, that is, q
F ⊂ Bt (F) Monderer and Samet (1989, Definition 1 and Proposition 3) showed the following proposition: PROPOSITION 1—Monderer and Samet: F is common q-belief at ω ∈ Ω and time t if and only if there exists an event F ⊂ Ω such that F is q-evident at time t q and ω ∈ F ⊂ Bt (F ).
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We use the following immediate implication: COROLLARY 1: The agents commonly learn Θ if and only if for all θ ∈ Θ and q ∈ (0 1), there exists a sequence of events Ft and a period T such that for all t > T, (i) θ is q-believed on Ft at time t, (ii) P θ (Ft ) > q, and (iii) Ft is q-evident at time t. 2.2. Special Cases: Perfect Correlation and Independence Suppose first the signals are public, as commonly assumed in the literature. ˆ beliefs, and we have Then agent knows everything there is to know about ’s P(θ | H1t ) = P(θ | H2t ) for all θ and t—and hence beliefs are always common. Individual learning then immediately implies common learning. At the other extreme, we have independent signals. Here, the fact that ˆ signals ensures common learning. agent learns nothing about agent ’s PROPOSITION 2: Suppose each agent learns Θ and that for each θ ∈ Θ, the stoθ ∞ chastic processes {ζ1tθ }∞ t=0 and {ζ2t }t=0 are independent. Then the agents commonly learn Θ. √ q
PROOF: Fix θ. Abbreviating {θ} × Z ∞ to {θ}, we set Ft ≡ {θ} ∩ Bt (θ) and apply Corollary 1. √ q q (i) Because Ft ⊂ Bt (θ) ⊂ Bt (θ), parameter θ is q-believed on Ft at time t. √ √ q q θ θ P (B (θ))P (B (θ)). By (1), we can (ii) Independence implies P θ (Ft ) = 1t 2t √ √ q θ choose T sufficiently large that P (Bt (θ)) > q for all and all t > T , and hence P θ (Ft ) > q. q (iii) To show that Ft is q-evident, we must show that Ft ⊂ Bt (Ft ) for = 1 2. We have q Bt (Ft ) = ω : E 1B√q (θ) 1B√q (θ)∩{θ} Ht ≥ q t ˆ t √ √ = ω : 1B q (θ) E 1B q (θ)∩{θ} Ht ≥ q ˆ t
t
√
q
q t
√ q ˆ t
= Bt (θ) ∩ B (B (θ) ∩ {θ}) √
q
√ q
Ft ⊂ Bt (θ), where the second equality uses Bt (θ) ∈ Ht . By construction, √ q q and so it suffices for Ft ⊂ Bt (Ft ) that on Ft , we have P(Btˆ (θ) ∩ {θ} | Ht ) > q for ˆ√ = . As above, (1) allows us to choose T sufficiently large that √ q P θ (Bt (θ)) > q for all and all t > T . The conditional independence of
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ˆ agents’ signals implies that, given θ, agent ’s history is uninformative about ’s signals, and hence (3)
√ q
P θ (Btˆ (θ) | Ht ) >
√
q
for all agent- histories.4 But on Ft , we have P(θ | Ht ) > again on Ft , (4)
√ q
√ q. Consequently,
√ q
P(Btˆ (θ) ∩ {θ} | Ht ) = P θ (Btˆ (θ) | Ht )P(θ | Ht ) > q
and we have the desired result. Q.E.D. 2 REMARK 3—Arbitrary Finite Number of Agents: The proof of Proposition √ nq covers an arbitrary finite number of agents once we redefine Ft as {θ} ∩ Bt (θ), where n is the number of agents. One would expect common learning to be more likely the more informaˆ beliefs. Perhaps surprisingly, however, correlation that is tion has about ’s less than perfect can generate information that disrupts common learning. The danger is that agent 1 may have observed signal frequencies that are typical of parameter θ but which lead 1 to believe 2 has seen signals less typical of θ. This destroys the uniform (across 1’s histories) bound on 1’s beliefs about 2’s beliefs (cf. (3)) that played a central role in establishing q-evidence and hence common learning with independent signals. We show by example in Section 4 that common learning can fail as a result. The following section shows that common learning still obtains when signal sets are finite, though the event Ft used in the proof of Proposition 2 no longer suffices to show that the conditions of Corollary 1 are satisfied. 3. SUFFICIENT CONDITIONS FOR COMMON LEARNING 3.1. Finite Signals Imply Common Learning For our positive result, we assume that the signal sets are finite. ASSUMPTION 1—Finite Signal Sets: Agents 1 and 2 have finite signal sets, I and J, respectively. 4 Since conditional probabilities are only unique for P-almost all states, the set Ft depends on the choice of version of √the relevant conditional probabilities. In the proof, we have selected the √ q q constant function P θ (Btˆ (θ)) as the version of P θ (Btˆ (θ) | Ht ). For other versions of conditional probabilities, the definition of Ft must be adjusted to exclude appropriate zero probability subsets.
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We use I and J to also denote the cardinality of sets I and J, trusting the context will prevent confusion. We denote the probability distribution of the agents’ signals conditional on θ by (π θ (ij))i∈Ij∈J ∈ (I × J). Hence, π θ (ij) is the probability that (z1t z2t ) = (i j) under parameter θ in period t. For each θ ∈ Θ, let
θ θ π (ij) > 0 I ≡ i∈I: j
and
π θ (ij) > 0 Jθ ≡ j ∈ J : i
be the sets of signals that appear with positive probability under parameter θ. θ θ j∈J θ by Π . Denote (π θ (ij))i∈I θ θ We let φ (i) ≡ j π (ij) denote the marginal probability of agent 1’s signal i and let ψθ (j) = i π θ (ij) denote the marginal probability of agent 2’s signal j. We let φθ = (φθ (i))i∈I θ and ψθ = (ψθ (j))j∈J θ be the row vectors of expected frequencies of the agents’ signals under parameter θ. Notice that we restrict attention to those signals that appear with positive probability under parameter θ in defining the vectors φθ and ψθ . Given Assumption 1, the following statement is equivalent to (1). ASSUMPTION 2—Individual Learning: For every θ = θ , the marginal distrib
utions are distinct, that is, φθ = φθ and ψθ = ψθ . Our main result is the following: PROPOSITION 3: Under Assumptions 1 and 2, the agents commonly learn Θ. REMARK 4—The Role of the Common Prior and Agreement on π θ : Though we have presented Proposition 3 in terms of a common prior, the analysis applies with little change to a setting where the two agents have different but commonly known priors. Indeed, the priors need not be commonly known—it is enough that there be a commonly known bound on the minimum probability any parameter receives in each agent’s prior. We simply modify Lemma 1 to still find a neighborhood of signal frequencies in which every “type” of agent i will assign high probability to the true parameter. The rest of the proof is unchanged. Our model also captures settings in which the agents have different beliefs about the conditional signal-generating distributions (π θ (ij))i∈Ij∈J . In particular, such differences of opinion can be represented as different beliefs about a
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parameter ρθ that determines the signal-generating process given θ. Our analysis then applies to a reformulated model in which agents are uncertain about the joint parameter (θ ρθ ) (but know the signal-generating process conditional on this parameter). Our work is complementary to Acemoglu, Chernozhukov, and Yildiz (2006), who considered environments in which even arbitrarily large samples of common data may not reconcile disagreements in agents’ beliefs. Acemoglu, Chernozhukov, and Yildiz (2006) stressed the possibility that the agents in their model may not know the signal-generating process (π θ (ij))i∈Ij∈J , but we have just argued that this is not an essential distinction in our context. The key difference is that the signal-generating processes considered by Acemoglu, Chernozhukov, and Yildiz (2006) need not suffice for individual learning. In our context, it is unsurprising that common learning need not hold when individual learning fails. 3.2. Preliminaries The main idea of the proof is to classify histories in terms of the realized frequencies of the signals the agents have observed and to work with events exhibiting appropriate signal frequencies. Let ft (ij) denote the number of periods in which agent 1 has received the signal i and agent 2 has received the signal j before period t. Defining f1t (i) ≡ f (ij) and f (j) ≡ 2t j t i ft (ij), the realized frequencies of the signals are given by the row vectors φˆ t ≡ (f1t (i)/t)i∈I and ψˆ t ≡ (f2t (j)/t)j∈J . Finally, let φˆ θt = (f1t (i)/t)i∈I θ denote the realized frequencies of the signals that appear with positive probability under parameter θ, with a similar convention for ψˆ θ . Denote by M1θ the I θ × J θ matrix whose ijth element is π θ (ij)/φθ (i), that is, the conditional probability under parameter θ of signal j given signal i. At any date t, when agent 1 has realized frequency distribution φˆ t , his estimate (expectation) of the frequencies observed by agent 2 conditional on parameter θ is given by the matrix product φˆ θt M1θ The corresponding matrix for agent two, M2θ , is the J θ × I θ matrix with jith element π θ (ij)/ψθ (j). We now make a key observation relating φθ , ψθ , M1θ , and M2θ . Let Dθ1 be the θ I × I θ diagonal matrix with ith diagonal element (φθ (i))−1 and let e be a row vector of 1’s. It is then immediate that (5)
φθ M1θ = φθ Dθ1 Π θ = eΠ θ = ψθ
A similar argument yields (6)
ψθ M2θ = φθ
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Note that the product φˆ θt M1θ M2θ gives agent 1’s expectation of agent 2’s expectation of the frequencies observed by agent 1 (conditional on θ). Hence, θ ≡ M1θ M2θ is a Markov transition matrix on the set I θ of signals for agent 1 M12 with stationary distribution φθ . θ may be zero, requiring further considerSome elements of the matrix M12
ation. Two signals i and i belong to the same recurrence class under the tranθ if and only if the probability of a transition from i to i (in sition matrix M12 some finite number of steps) is positive. We let (Rθ1 (k))Kk=1 denote the collection of recurrence classes, and we order the elements of I θ so that the recurrence classes are grouped together and in the order of their indices. The matrix θ Dθ1 = Dθ1 Π θ Dθ2 [Π θ ]T Dθ1 , where Dθ2 is the diagonal matrix whose jth diagoM12 nal element is (ψθ (j))−1 for j ∈ J θ , is obviously symmetric. This implies that the θ with initial distribution φθ is reversible.5 Consequently, Markov process M12 θ the process has φ as a stationary distribution when run backward as well as forward, and hence (since φθ (i) > 0 for all i ∈ I θ ) has no transient states. The relation defined by belonging to a recurrence class is thus an equivalence and the recurrence classes partition I θ .6 θ Similarly, the matrix M21 ≡ M2θ M1θ is a Markov transition matrix with stationθ ary distribution ψ on the set J θ that we can partition into recurrence classes (Rθ2 (k))Kk=1 . Define a mapping ξ from (Rθ1 (k))Kk=1 to (Rθ2 (k))Kk=1 by letting ξ(Rθ1 (k)) = θ R2 (k ) if there exist signals i ∈ Rθ1 (k) and j ∈ Rθ2 (k ) with π θ (ij) > 0. Then ξ is a bijection (as already reflected in our notation). It is convenient therefore to group the elements of J θ by their recurrence classes in the same order as was done with I θ . We use the notation Rθ (k) to refer to the kth recurrence class in either I θ or J θ when the context is clear. This choice of notation also reflects the equalities of the probabilities of Rθ1 (k) and Rθ2 (k) under θ, that is, φθ (Rθ1 (k)) ≡ (7) φθ (i) = ψθ (j) ≡ ψθ (Rθ2 (k)) i∈Rθ1 (k)
j∈Rθ2 (k)
Since agent 1 observes a signal in Rθ1 (k) under parameter θ if and only if agent 2 observes a signal in Rθ2 (k), conditional on θ the realized frequencies of the recurrence classes also agree. Let φˆ θk denote the distribution over I θ obtained by conditioning φˆ on the kth recurrence class Rθ (k) (for those cases in which φˆ θ (Rθ (k)) > 0), and define φθk , ψθk , and ψˆ θk t analogously. 5
θ As M12 Dθ1 is symmetric, the detailed balance equations at φθ hold, that is, θ θ (ii ) = φθ (i )M12 (i i) φθ (i)M12
(Brémaud (1999, p. 81)). 6 Since the Markov process has no transient states, if the probability of a (finite-step) transition from i to i is positive, then the probability of a (finite-step) transition from i to i is also positive.
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3.3. Outline of the Proof Fix a parameter θ. As in the proof of Proposition 2, we show the agents commonly learn θ by identifying a set Ft that satisfies the sufficient conditions of Corollary 1. Once again, Ft is built around requirements that each agent attaches high probability to θ and attaches high probability to ˆ attaching high probability to θ. Our argument begins, in Lemma 1, by showing that there is a δ > 0 so that whenever 1’s observed frequency distribution φˆ t is within a distance δ of φθ , his marginal signal distribution under θ, the posterior probability he assigns to θ approaches 1 over time. Let F1t (0) denote this δ-neighborhood of φθ :7 F1t (0) ≡ {ω : φˆ θt − φθ < δ} Next, from (5), the continuity of the linear map M1θ implies that if 1’s frequencies are in a neighborhood of φθ , then 1 expects that 2’s frequencies are in the neighborhood of ψθ , and hence that 2 assigns high probability to θ. To pass from “expecting” that 2’s signals are typical of θ to assigning high probability to this event, we must bound the error in agent 1’s estimate of 2’s frequencies. Lemma 3 provides such a bound, showing that conditional on θ, given any ε1 > 0 and ε2 > 0, there is a time T after which the probability that 2’s realized frequencies are more than ε1 away from 1’s estimate (φˆ θt M1θ ) is less than ε2 . A crucial detail here is that this bound applies uniformly across all histories for 1. There is thus a neighborhood of φθ such that if 1’s frequency φˆ θt falls in this neighborhood for sufficiently large t, then agent 1 assigns high probability to the event that 2 assigns high probability to θ. Let F1t (1) denote this neighborhood, which we can equivalently think of as a neighborhood of ψθ into which φˆ θt M1θ must fall, that is, F1t (1) ≡ {ω : φˆ θt M1θ − ψθ < δ − ε} where ε is small and determined below. Let F1t (0) ∩ F1t (1) ≡ F1t . A natural starting point for the set Ft would be F1t ∩ F2t (where F2t is defined similarly to F1t for agent 2). It simplifies the argument for q-evidence to intersect these sets with the event {θ}, so that Ft ≡ F1t ∩ F2t ∩ {θ}. It is intuitive (and indeed true) that θ is q-believed on Ft at time t and that P θ (Ft ) > q for sufficiently large t. It remains to verify that the q q q set Ft is q-evident at time t, that is, Ft ⊂ Bt (Ft ) = B1t (Ft ) ∩ B2t (Ft ). It suffices for q-evidence to argue that (with a symmetric argument for agent 2) q
F1t ∩ {θ} ⊂ B1t (F1t ∩ F2t ∩ {θ}) 7
For any x ∈ RN , x ≡
N k=1
|xk | is the variation norm of x.
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Define Fˆ1t (1) ≡ {ω : φˆ θt M1θ − ψˆ θt < ε}, the event that 2’s realized frequencies are close to 1’s estimate. Then an application of the triangle inequality yields F1t (1) ∩ Fˆ1t (1) ⊂ F2t (0) θ θ is strictly positive. In that case, M12 is Suppose that every element of M12 θ θ a contraction when viewed as a mapping on I with fixed point φ . Hence, θ θ θ for some r ∈ (0 1), ω ∈ F1t (0) implies φˆ θt M12 − φθ M12
= φˆ θt M12 − φθ < θ θ θ θ ˆ ˆ rδ. Since M2 is a stochastic matrix, ω ∈ F1t (1) implies φt M1 M2 − ψˆ θt M2θ = θ
φˆ θt M12 − ψˆ θt M2θ < ε. Setting ε small enough that rδ < δ − 2ε, an application of the triangle inequality gives ψˆ θt M2θ − φθ < δ − , implying
F1t (0) ∩ Fˆ1t (1) ⊂ F2t (1) Hence, F1t ∩ Fˆ1t (1) ⊂ F2t and so F1t ∩ Fˆ1t (1) ∩ {θ} ⊂ F2t ∩ {θ}. But the discusq sion preceding the definition of F1t (1) implies F1t ∩ {θ} ⊂ B1t (Fˆ1t (1) ∩ {θ}) for large t. Consequently, F1t ∩ {θ} ⊂ B1t (F1t ∩ Fˆ1t (1) ∩ {θ}) ⊂ B1t (F1t ∩ F2t ∩ {θ}) q
q
and we are done. The proof of Proposition 3 must account for the possibility that some eleθ may not be strictly positive. However, as we show in Lemma 4, ments of M12 θ since M12 is irreducible when restricted to a recurrence class, some power of this restricted matrix is a contraction. The proof then proceeds as outlined above, with the definition of Ft now taking into account the need to take powθ ers of M12 . 3.4. Frequencies Suffice for Beliefs Our first result shows that when agent 1’s signal frequencies are sufficiently close to those expected under any parameter θ, the posterior probability he attaches to parameter θ approaches 1. LEMMA 1: There exist δ ∈ (0 1), β ∈ (0 1), and a sequence ξ : N → [0 1] with ξ(t) → 1 such that P(θ | h1t ) ≥ ξ(t) θk for all θ ∈ Θ and h1t satisfying P(θ | h1t ) > 0, φˆ θk t − φ < δ, and β < θ θ θ θ −1 φˆ t (R (k))/φ (R (k)) < β for all k. An analogous result holds for agent 2.
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PROOF: Fix a parameter θ and δ˜ < miniθ {φθ (i) : φθ (i) > 0}. Then φˆ θk t − φ < δ˜ for all k only if φˆ t puts strictly positive probability on every signal i ∈ I θ . For θ and h1t with P(θ | h1t ) > 0, define the ratio θk
λθθ 1t ≡ log
φθ (it−1 )P(θ | h1t−1 ) P(θ | h1t ) = log P(θ | h1t ) φθ (it−1 )P(θ | h1t−1 )
We now show that β and δ ≤ δ˜ can be chosen so that there exists η > 0 with the property that
θθ ∀θ = θ λθθ 1t ≥ λ10 + tη θθ
θk ˜ for all histories h1t for which φˆ θk t − φ < δ for all k and for which λ1t is
defined. Notice that λθθ 10 = p(θ)/p(θ ) is the log-likelihood ratio at time zero, that is, the ratio of prior probabilities. ˜ implying that every signal i ∈ I θ has appeared in the history Our choice of δ,
θ θ
h1t , ensures that P(θ |h1t ) > 0 (and hence λθθ 1t is well defined) only if I ⊂ I . This in turn ensures that the following expressions are well defined (in particular, having nonzero denominators). Because signals are distributed indepen
dently and identically across periods, λθθ 1t can be written as θθ
1t
λ
θθ
10
=λ
+
t−1 s=0
φθ (is ) log φθ (is )
We find a lower bound for the last term. Let
φθ (i) θθ
θ H ≡ E log θ
>0 φ (i)
denote the relative entropy of φθ with respect to φθ . Then t−1 φθ (i )
s θθ log − tH φθ (is ) s=0 θ
φθ (i) φ (i) θ = f1t (i) log φ (i) log −t
θ φ (i) φθ (i) i∈I θ i∈I θ θ θ φ (i) θ ˆ = t (φt (i) − φ (i)) log φθ (i) i∈I θ θ (φˆ θ (i) − φθ (i)) log φ (i) ≤t t θ φ (i) θ i∈I
≤ t φˆ θt − φθ log b
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for b = maxiθθ ∈Θ {φθ (i)/φθ (i) : φθ (i) > 0}. (By Assumption 2, b > 1.) Thus, θθ
θθ
λθθ − φˆ θt − φθ log b 1t ≥ λ10 + t H
We now argue that δ ≤ δ˜ and β can be chosen so that H θθ −log b φˆ θt −φθ > η for all θ θ and some η > 0. For this, it is enough to observe that the mapping θ θ φˆ t (R (k)) k {φˆ θk t }k φˆ θ (Rθ (k))φˆ θk (i) − φθ (i) = φˆ θ − φθ → t t t
k
i∈k
is continuous, and equals zero if and only if φˆ θt (Rθ (k)) = φθ (Rθ (k)) and θk for all k. φˆ θk t =φ We thus have δ and β such that for θ and h1t satisfying the hypotheses of the
θθ
lemma and for θ with P(θ | h1t ) > 0, it must be that λθθ 1t ≥ λ10 + tη and hence p(θ ) P(θ |h1t ) tη ≥ e p(θ) P(θ|h1t ) Noting that this inequality obviously holds for θ with P(θ | h1t ) = 0, we can sum over θ = θ and rearrange to obtain p(θ) tη P(θ | h1t ) ≥ e 1 − P(θ | h1t ) 1 − p(θ) giving the required result.
Q.E.D.
We next note that with high probability, observed frequencies match their expected values. Together with Lemma 1, this implies that each agent learns Θ. LEMMA 2: For all ε > 0 and θ, P θ ( φˆ θt − φθ < ε) → 1 and P θ ( ψˆ θt − ψθ < ε) → 1 as t → ∞. PROOF: This follows from the weak law of large numbers (Billingsley (1979, p. 86)). Q.E.D. 3.5. Beliefs About Others’ Frequencies We now show that each agent believes that, conditional on any parameter θ, his or her expectation of the frequencies of the signals observed by his or her opponent is likely to be nearly correct. Recall that φˆ θt M1θ is agent 1’s expectation of 2’s frequencies ψˆ θt and that ψˆ θt M2θ is agent 2’s expectation of 1’s frequencies φˆ θt .
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LEMMA 3: For any ε1 > 0, ε2 > 0, there exists T such that for all t > T and for every ht with P θ (ht ) > 0, (8) P θ φˆ θt M1θ − ψˆ θt < ε1 | h1t > 1 − ε2 and (9)
P θ ψˆ θt M2θ − φˆ θt < ε1 | h2t > 1 − ε2
PROOF: We focus on (8); the argument for (9) is identical. Defining ψ¯ θt ≡ φˆ θt M1θ , the left side of (8) is bounded below:
θ θ ε1 θ θ θ θ θ ˆ ¯ ˆ ˆ P φt M1 − ψt < ε1 | h1t ≥ 1 − (10) P |ψt (j) − ψt (j)| ≥ θ h1t J θ j∈J
Conditional on θ and h1t , agent 2’s signals are independently, but not identically, distributed across time. In period s, given signal is , agent 2’s signals are distributed according to the conditional distribution (π θ (is j)/φθ (is ))j . However, we can bound the expression on the right side of (10) using a related process obtained by averaging the conditional distributions. The average probability that agent 2 observes signal j over the t periods {0 1 t − 1}, conditional on h1t , is 1 π θ (is j) ˆ π θ (ij) = = ψ¯ θt (j) (i) φ t θ (i) t s=0 φθ (is ) φ i t−1
agent 1’s expectation of the frequency that 2 observed j. Consider now t independent and identically distributed draws of a random variable distributed on J θ according to the “average” distribution ψ¯ θt ∈ (J θ ); we refer to this process as the average process. Denote the frequencies of signals generated by the average process by ηt ∈ (J θ ). The process that generates the frequencies ψˆ t attaches the same average probability to each signal j over periods 0 t − 1 as does the average process, but does not have identical distributions (as we noted earlier). We use the average process to bound the terms in the sum in (10). By Hoeffding (1956, Theorem 4, p. 718), the original process is more concentrated about its mean than is the average process, that is,8
ε1 ε1 θ θ θ θ ˜ ¯ ˆ ¯ P |ψt (j) − ηt (j)| ≥ θ ≥ P |ψt (j) − ψt (j)| ≥ θ h1t j ∈ J θ J J 8 For example, 100 flips of a (p 1 − p) coin generate a more dispersed distribution than 100p flips of a (1 0) coin and 100(1 − p) flips of a (0 1) coin.
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where P˜ is the measure associated with the average process. Applying this upper bound to (10), we have
θ θ ε1 θ θ θ ˜ ˆ ˆ ¯ (11) P φt M1 − ψt < ε1 | h1t ≥ 1 − P |ψt (j) − ηt (j)| ≥ θ J θ j∈J
The event {|ψ¯ θt (j) − ηt (j)| > ε1 /J θ } is the event that the realized frequency of a Bernoulli process is far from its mean. By a large deviation inequality ((42) in Shiryaev (1996, p. 69)),
ε1 2 θ 2 θ ˜ ¯ P |ψt (j) − ηt (j)| > θ ≤ 2e−2tε1 /(J ) J Using this bound in (11), we have 2 θ 2 P θ φˆ θt M1θ − ψˆ θt < ε1 | h1t ≥ 1 − 2J θ e−2tε1 /(J ) This inequality holds for any history h1t . We can thus choose t large enough so that the right side is less than ε2 and the statement of the lemma follows. Q.E.D. 3.6. Proof of Proposition 3 We fix an arbitrary parameter θ and define a sequence of events Ft (suppressing notation for the dependence of Ft on θ), and show that Ft has the three requisite properties from Corollary 1 for sufficiently large t. Let γ θk denote a probability distribution over I θ that takes positive values only on the kth recurrence class Rθ (k), and denote the set of such distributions by Rθ (k). LEMMA 4: There exist r < 1 and a natural number n such that for all k ∈ {1 K} and for all γ θk γ˜ θk in Rθ (k), (12)
θ n θ n ) − γ˜ θk (M12 ) ≤ r γ θk − γ˜ θk
γ θk (M12
θ n and similarly for (M21 ) . θ Dθ1 = Dθ1 Π θ Dθ2 [Π θ ]T Dθ1 has a nonzero diagonal, imPROOF: The matrix M12 θ plying that M12 has a nonzero diagonal and hence is aperiodic. The restriction θ to any given recurrence class is irreducible and hence ergodic. Thus, of M12 because signals are grouped by their recurrence classes, there exists a natθ n ) is block-diagonal, with each block containing ural number n such that (M12 only strictly positive entries. The blocks consist of the nonzero n-step transition probabilities between signals within a recurrence class. The product of γ θk
COMMON LEARNING
925
θ n ) is just the product of γ θk restricted to Rθ (k) with the kth block with (M12 θ n of (M12 ) . Because it has all nonzero entries, the kth block is a contraction mapping (Stokey and Lucas (1989, Lemma 11.3)). In particular, there exists an r < 1 such that (12) holds. Q.E.D.
The Event Ft : Let δ ∈ (0 1) and β ∈ (0 1) be the constants identified in Lemma 1. Pick ε > 0 such that rδ < δ − 2nε, where r < 1 and n are identified in Lemma 4. For each date t, we define the event Ft as follows. First, we ask that agent 1’s realized frequency of signals from I θ and 2’s realized frequency of signals from J θ be close to the frequencies expected under θ. For each k, define the events (13)
θk F1tk (0) ≡ {ω : φˆ θk t − φ < δ}
and (14)
θk F2tk (0) ≡ {ω : ψˆ θk t − ψ < δ}
θ n θ n k θk Lemma 4 ensures that φˆ θk t (M12 ) − φ (M12 ) is smaller than δ on F1t (0). θ We define our event so that the same is true for all powers of M12 between 0 and n. For any l ∈ {1 n} and for each k, let θ l−1 (15) M1θ − ψθk < δ − (2l − 1)ε F1tk (2l − 1) ≡ ω : φˆ θk t (M12 )
and (16)
θ l θk F1tk (2l) ≡ ω : φˆ θk t (M12 ) − φ < δ − 2lε
Similarly, for agent 2, (17)
θ l−1 F2tk (2l − 1) ≡ ω : ψˆ θk M2θ − φθk < δ − (2l − 1)ε t (M21 )
and (18)
θ l θk F2tk (2l) ≡ ω : ψˆ θk t (M21 ) − ψ < δ − 2lε
Next, define the events F1t ≡
K 2n−1
F1tk (κ) ≡
k=1 κ=0
F2t ≡
K 2n−1 k=1 κ=0
K
K k=1
F1t (κ)
κ=0
k=1
F2tk (κ) ≡
2n−1
F1tk ≡
2n−1
F2tk ≡
F2t (κ)
κ=0
[θ] ≡ {ω ∈ {θ} × (I × J)∞ : P θ (ht ) > 0 ∈ {1 2} t = 0 1 }
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CRIPPS, ELY, MAILATH, AND SAMUELSON
and (19) (20)
φˆ t (Rθ (k)) Gθt ≡ [θ] ∩ β < θ θ < β−1 ∀k ≡ [θ] ∩ G1t φ (R (k))
ψˆ t (Rθ (k)) −1 < β ∀k ≡ [θ] ∩ G2t = [θ] ∩ β < θ θ ψ (R (k))
The equality of the two descriptions of Gθt follows from (7) and the following discussion. Finally, we define the event Ft ≡ F1t ∩ F2t ∩ Gθt In the analysis that follows, we simplify notation by using { · < ε} to denote the event {ω: · < ε}. θ is q-believed on Ft : By definition Ft ⊂ F1t (0) ∩ F2t (0) ∩ Gθt . Lemma 1 then q implies that for any q < 1, we have Ft ⊂ Bt (θ) for all t sufficiently large. Ft is likely under θ: If φˆ t = φθ and ψˆ t = ψθ , then the inequalities (13)–(20) appearing in the definitions of the sets F1t , F2t , and Gθt are strictly satisfied (because φθk M1θ = ψθk and ψθk M2θ = φθk for each k). The (finite collection of) inequalities (13)–(20) are continuous in φˆ t and ψˆ t , and independent of t. Hence, (13)–(20) are satisfied for any φˆ t and ψˆ t sufficiently close to φθ and ψθ . We can therefore choose ε† > 0 sufficiently small such that { φˆ t − φθ < ε† ψˆ t − ψθ < ε† } ∩ [θ] ⊂ Ft
∀t
By Lemma 2, the P θ -probability of the set on the left side approaches 1 as t gets large, ensuring that for all q ∈ (0 1), P θ (Ft ) > q for all large enough t. Ft is q-evident: We now argue that for any q, Ft is q-evident when t is sufficiently large. Recalling that ε and β were fixed in defining Ft , choose ε1 ≡ εβ minj∈J θ ψθ (j). Note that ε1 /ψˆ θ (Rθ (k)) < ε on the events F1t ∩ Gθt and F2t ∩ Gθt . STEP 1: The first step is to show that if the realized frequencies of agent 1’s signals are close to their expected frequencies under θ and his expectations of agent 2’s frequencies are not too far away from agent 2’s realized frequencies, then (conditional on θ) the realized frequencies of agent 2’s signals are also close to their expected frequencies under θ. In particular, we show (21)
F1t ∩ Gθt ∩ { φˆ θt M1θ − ψˆ θt < ε1 } ⊂ F2t
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First, fix k and note that for each l = 1 n, (22)
θ ˆ θk F1tk (2l) ∩ { φˆ θk t M1 − ψt < ε} θ l ˆ θk θ l−1 M2θ < ε ⊂ F1tk (2l) ∩ φˆ θk t (M12 ) − ψt (M21 ) θ l θk = φˆ θk t (M12 ) − φ < δ − 2lε θ l ˆ θk θ l−1 M2θ < ε ∩ φˆ θk t (M12 ) − ψt (M21 ) θ l−1 ⊂ ψˆ θk M2θ − φθk < δ − (2l − 1)ε t (M21 )
= F2tk (2l − 1) θ l−1 The first inclusion uses the fact that (M21 ) M2θ is a stochastic matrix. The k equalities use the definitions of F1t (2l) and F2tk (2l − 1). The last inclusion is a consequence of the triangle inequality. Similarly, for l = 1 n, we have θ k ˆ θk F1tk (2l − 1) ∩ { φˆ θk t M1 − ψt < ε} ⊂ F2t (2(l − 1))
This suffices to conclude that
2n−1
(23)
θ ˆ θk F1tk (κ) ∩ { φˆ θk t M1 − ψt < ε} ⊂
κ=1
2n−2
F2tk (κ)
κ=0
We next note that (24)
θ k ˆ θk θ ˆ θk ˆ θk F1tk (0) ∩ { φˆ θk t M1 − ψt < ε} ⊂ F1t (2n) ∩ { φt M1 − ψt < ε}
⊂ F2tk (2n − 1) θ n ) = φθk , Lemma 4, and where F1tk (0) ⊂ F1tk (2n) is an implication of φθk (M12 our choice of ε and n, while the second inclusion follows from (22) (for l = n). Combining (23) and (24) for k = 1 K, we have θ ˆ θk (25) F1t ∩ { φˆ θk t M1 − ψt < ε} ⊂ F2t k
As the matrix M1θ maps recurrence classes to recurrence classes, on Gθt we have, for any k, φˆ θ (Rθ (k ))φˆ θk M θ − ψˆ θ (Rθ (k ))ψˆ θk
φˆ θt M1θ − ψˆ θt = t t t 1 t k
θ ˆθ θ ˆ θk ≥ φˆ θt (Rθ (k))φˆ θk t M1 − ψt (R (k))ψt θ ˆ θk = ψˆ θt (Rθ (k)) φˆ θk t M1 − ψt
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CRIPPS, ELY, MAILATH, AND SAMUELSON
since φˆ θt (Rθ (k)) = ψˆ θt (Rθ (k)) on [θ] (recall the discussion following (7)). Our choice of ε1 then yields, on F1t ∩ Gθt ,
φˆ θt M1θ − ψˆ θt < ε1
⇒
ε>
ε1 ψˆ θt (Rθ (k))
θ ˆ θk > φˆ θk ∀k t M1 − ψt
Therefore, F1t ∩ Gθt ∩ { φˆ θt M1θ − ψˆ θt < ε1 } ⊂ F1t ∩
θ ˆ θk { φˆ θk t M1 − ψt < ε}
k
and by (25) we have proved (21). √ STEP 2: We now conclude the proof of q-evidence. Pick p ∈ ( q 1) and set ε2 = 1 − p in Lemma 3. Consider the event F1t ∩ Gθt . For t sufficiently large, given any history consistent with a state in F1t ∩ Gθt , agent 1 attaches at least probability p to θ p (F1t ∩ Gθt ⊂ B1t (θ)) (Lemma 1). Conditional on θ we have, by Lemma 3, that for large t, agent 1 attaches probability at least p to φˆ θt M1θ − ψˆ θt < ε1 . Hence p2 F1t ∩ Gθt ⊂ B1t { φˆ θt M1θ − ψˆ θt < ε1 } ∩ [θ] Since F1t ∩ G1t is measurable with respect to H1t and Gθt = [θ] ∩ G1t , we have p2 F1t ∩ Gθt ⊂ B1t F1t ∩ Gθt ∩ { φˆ θt M1θ − ψˆ θt < ε1 } and, hence, from (21), (26)
p2
p2
F1t ∩ Gθt ⊂ B1t (F1t ∩ F2t ∩ Gθt ) = B1t (Ft ) p2
A similar argument for agent 2 gives F2t ∩ Gθt ⊂ B2t (Ft ) and thus Ft ⊂ p2 q Bt (Ft ) ⊂ Bt (Ft ) for sufficiently large t. REMARK 5 —Arbitrary Finite Number of Agents: The restriction to two agents simplifies the notation, but the result holds for any finite number of agents. We illustrate the argument for three agents (and keep the notation as similar to the two agent case as possible). Denote agent 3’s finite set of signals by K. The joint probability of the signal profile ijk ∈ I × J × K under θ is π θ (ijk). In addition to the marginal distributions φθ and ψθ for 1 and 2, the marginal distribution for 3 is ϕθ . As before, M1θ is the I θ × J θ matrix with ijth element k π θ (ijk)/φθ (i) (and similarly for M 2 ). For the pair 1-3, we denote by N1θ the I θ × K θ matrix with ikth element j π θ (ijk)/φθ (i) (and similarly for N3θ ). Finally, for the pair 2-3, we have analogous definitions for the matrices Q2θ and Q3θ . As before, φθ is a stationary distribution of M1θ M2θ , but now also of
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N1θ N3θ ; similar statements hold for ψθ and the transitions M2θ M1θ and Q2θ Q3θ , as well as for ϕθ and the transitions N3θ N1θ and Q3θ Q2θ . Suppose (for exposition only) that every element of the various Markov transition matrices is nonzero, and let r < 1 now be the upper bound on the modulus of contraction of the various contractions. The argument of the outline still applies, once we redefine F1t (1) ≡ {ω : φˆ θt M1θ − ψθ < δ − ε} ∩ {ω : φˆ θt N1θ − ϕθ < δ − ε} and Fˆ1t (1) ≡ {ω : φˆ θt M1θ − ψˆ θ < ε} ∩ {ω : φˆ θt N1θ − ϕˆ θ < ε} (with similar definitions for the other two agents). The proof of Proposition 3 similarly applies to the n agent case after analogous modifications. 4. A COUNTEREXAMPLE TO COMMON LEARNING This section presents an example in which Assumption 1 fails and common learning does not occur, although the agents do privately learn. There are two values of the parameter, θ and θ
, satisfying 0 < θ < θ
< 1. Signals are nonnegative integers. The distribution of signals is displayed in Figure 2.9 If we set θ = 0 and θ
= 1, then we can view one period of this process as an instance of the signals in Rubinstein’s (1989) electronic mail game, where the signal corresponds to the number of “messages” received.10 It is immediate that the agents Probability θ ε(1 − θ) (1 − ε)ε(1 − θ) (1 − ε)2 ε(1 − θ) (1 − ε)3 ε(1 − θ) (1 − ε)4 ε(1 − θ) (1 − ε)5 ε(1 − θ)
Agent-1 signal 0 1 1 2 2 3 3
Agent-2 signal 0 0 1 1 2 2 3
FIGURE 2.—The distribution of signals for the counterexample given parameter θ ∈ {θ θ
}, where ε ∈ (0 1). 9 It would cost only additional notation to replace the single value ε in Figure 2 with heterogeneous values, as long as the resulting analogue of (27) is a collection whose values are bounded away from 0 and 1. 10 Rubinstein (1989) was concerned with whether a single signal drawn from this distribution allows agents to condition their action on the parameter, while we are concerned with whether an arbitrarily large number of signals suffices to commonly learn the parameter. Interestingly, repeated observation of the original Rubinstein process (i.e., θ = 0 and θ
= 1) leads to common learning. In particular, consider the event Ft at date t that the parameter is θ
and no messages have ever been received. This event is q(t)-evident, where q(t) approaches 1 as t approaches infinity, since agent 1 assigns probability 1 and agent 2 assigns a probability approaching 1 to Ft whenever it is true. Similarly, the event that the parameter is θ and each agent has received at least one signal is q(t) evident for q(t) approaching 1.
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CRIPPS, ELY, MAILATH, AND SAMUELSON
faced with a sequence of independent draws from this distribution learn Θ. We now show that common learning does not occur. What goes wrong when trying to establish common learning in this context, and how does this depend on the infinite set of signals? Establishing common q-belief in parameter θ requires showing that if agent 1 has observed signals just on the boundary of inducing probability q that the parameter is θ, then agent 1 nonetheless believes 2 has seen signals inducing a similar belief (and believes that 2 believes 1 has seen such signals, and so on). In the case of finite signals, a key step in this argument is the demonstration that (an appropriate θ is a contraction. In the current power of) the Markov transition matrix M12 case, the corresponding Markov process is not a contraction (though the marginal distribution is still stationary). As a result, agent can observe signals on the boundary of inducing probability q of parameter θ while believing that agent ˆ has observed signals on the “wrong side” of this boundary. We require the notion of iterated q-belief. The event that F is iterated qbelief is defined to be q
q
q
q
q
q
q
It (F) ≡ B1t (F) ∩ B2t (F) ∩ B1t B2t (F) ∩ B2t B1t (F) ∩ · · · Morris (1999, Lemma 14) showed that iterated belief is (possibly strictly) weaker than common belief: q
q
LEMMA 5—Morris: Ct (F) ⊂ It (F). See Morris (1999, p. 388) for an example with strict inclusion. Now let
(1 − ε) ε(1 − θ
) (27) q ≡ min
θ + ε(1 − θ
) (2 − ε) Note that regardless of the signal observed by agent 1, he always believes with probability at least q that 2 has seen the same signal, and regardless of the signal observed by 2, she always believes with probability at least q that 1 has seen a higher signal. We show that for all t sufficiently large there is (independently of the observed history) a finite iterated q-belief that θ is the true parameter. This implies that θ
can never be iterated p-believed for any p > 1 − q, with Lemma 5 then implying that θ
can never be common p-belief. That is, we will show that q p p for t large enough, It (θ ) = Ω and so It (θ
) = ∅ and hence Ct (θ
) = ∅ for all p > 1 − q. Define for each k, the event that agent observes a signal of at least k before time t: Dt (k) ≡ {ω : zs ≥ k for some s ≤ t}
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Note that Dt (0) is equal to Ω (the event that any t-length history occurs). For every k ≥ 0 the definition of q implies q
D1t (k) ⊂ B1t (D2t (k)) and q
D2t (k − 1) ⊂ B2t (D1t (k)) which together imply q
q
D2t (k − 1) ⊂ B2t B1t (D2t (k)) By induction, for all 0 ≤ m ≤ k, (28)
q
q
D2t (m) ⊂ (B2t B1t )k−m (D2t (k))
Now, for any K and any list (k1 k2 kK ), where ks ≥ ks−1 , define the event that agent observes distinct signals of at least ks before time t: Dt (k1 k2 kK ) ≡ {ω : ∃ distinct τs ≤ t s = 1 K s.t. zτs ≥ ks } Note that for K ≤ t, Dt (0 k2 kK ) = Dt (k2 kK ). Whenever agent 1 observes a signal k, he knows that agent 2 has seen a signal at least k − 1. Hence, for k2 ≥ 1, q
D1t (k1 k2 kK ) ⊂ B1t (D2t (k1 k2 − 1 k3 − 1 kK − 1)) and, by similar reasoning, q
D2t (k1 k2 kK ) ⊂ B2t (D1t (k1 + 1 k2 k3 kK )) so that for all n, if 0 ≤ k1 ≤ k2 − 2n, then (29)
q
q
D2t (k1 k2 kK ) ⊂ (B2t B1t )n D2t (k1 +n k2 −n k3 −n kK −n)
From (28), (30)
q
q
t−1
Ω = D2t (0) ⊂ (B2t B1t )2 D2t (2t−1 )
and, for t ≥ 2, from (29), (31)
q
q
t−2
D2t (2t−1 ) = D2t (0 2t−1 ) ⊂ (B2t B1t )2 D2t (2t−2 2t−2 )
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CRIPPS, ELY, MAILATH, AND SAMUELSON q
q
t−1
t−2
Inserting (31) into (30) gives Ω ⊂ (B2t B1t )2 +2 D2t (2t−2 2t−2 ). Continuing in this fashion and noting that 2t−1 + 2t−2 + · · · + 2t−t = 2t − 1, we obtain (32)
t
t
Ω ⊂ (B2t B1t )2 −1 D2t (2t−t 2t−t 2t−t ) = (B2t B1t )2 −1 D2t (1 1 1) q
q
q
q
t times
t times
Now choose t large enough so that after a t-length history in which signal 0 was never observed, agent 2 assigns probability at least q to θ , that is,11 D2t (1 1 1) ⊂ B2t (θ ) q
t times q
q
t
q
Using (32), we then have Ω ⊂ (B2t B1t )2 −1 B2t (θ ) and hence have shown that for t large enough, regardless of the history, there cannot be iterated p-belief p p in θ
for any p > 1 − q, that is, It (θ
) = ∅. Now by Lemma 5, Ct (θ
) = ∅. REMARK 6—Why Did Common Learning Fail?: The counterexample is not driven by the cardinality of the signals per se. For example, if Θ is finite, but the private signals are normally distributed conditional on θ, it can be shown that common learning still holds (recall also from Proposition 2 that common learning holds with finite Θ and independent signals, irrespective of the size of Z ). On the other hand, when both the parameter and the signals are normally distributed, arguments mimicking those from global games (Carlsson and van Damme (1993)) show that common learning fails. While there is common learning in every discrete parameter space approximation to the normal– normal example, we expect that the time by which the parameter is common q-belief goes to infinity as the approximation improves, and so there would be no discontinuity. Dept. of Economics, University College London, Gower Street, London WC1E 6BT, United Kingdom;
[email protected], Dept. of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60208-2600, U.S.A.;
[email protected], Dept. of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19103-6297, U.S.A.;
[email protected], and Economics Department, Yale University, 30 Hillhouse Avenue, New Haven, CT 06520-8268, U.S.A.;
[email protected]. Manuscript received August, 2006; final revision received March, 2008. 11
This is possible because after such a history
P(θ |h2t ) p(θ ) 1 − θ t = 1 − P(θ |h2t ) p(θ
) 1 − θ
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REFERENCES ACEMOGLU, D., V. CHERNOZHUKOV, AND M. YILDIZ (2006): “Learning and Disagreement in an Uncertain World,” Unpublished Manuscript, MIT. [917] BILLINGSLEY, P. (1979): Probability and Measure (First Ed.). New York: Wiley. [912,922] BRÉMAUD, P. (1999): Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. New York: Springer-Verlag. [918] CARLSSON, H., AND E. VAN DAMME (1993): “Global Games and Equilibrium Selection,” Econometrica, 61, 989–1018. [932] CRIPPS, M. W., G. J. MAILATH, AND L. SAMUELSON (2007): “Disappearing Private Reputations in Long-Run Relationships,” Journal of Economic Theory, 134, 287–316. [910] HOEFFDING, W. (1956): “On the Distribution of the Number of Successes in Independent Trials,” Annals of Mathematical Statistics, 27, 713–721. [923] MONDERER, D., AND D. SAMET (1989): “Approximating Common Knowledge With Common Beliefs,” Games and Economic Behavior, 1, 170–190. [910,913] (1995): “Stochastic Common Learning,” Games and Economic Behavior, 9, 161–171. [911] MORRIS, S. (1999): “Approximate Common Knowledge Revisited,” International Journal of Game Theory, 28, 385–408. [930] RUBINSTEIN, A. (1989): “The Electronic Mail Game: Strategic Behavior Under Almost Common Knowledge,” American Economic Review, 79, 385–391. [929] SAMET, D. (1998): “Iterated Expectations and Common Priors,” Games and Economic Behavior, 24, 131–141. [911] SHIRYAEV, A. N. (1996): Probability (Second Ed.). New York: Springer-Verlag. [924] STOKEY, N., AND R. E. LUCAS, JR. (1989): Recursive Methods in Economic Dynamics. Cambridge, MA: Harvard University Press. [925] WISEMAN, T. (2005): “A Partial Folk Theorem for Games With Unknown Payoff Distributions,” Econometrica, 73, 629–645. [910]
Econometrica, Vol. 76, No. 4 (July, 2008), 935–939
ANNOUNCEMENTS 2008 AUSTRALASIAN MEETING: MARKETS AND MODELS—POLICY FRONTIERS IN THE PHILLIPS TRADITION
THE 2008 ECONOMETRIC SOCIETY AUSTRALASIAN MEETING will be held in on 9–11 July, 2008 in Wellington, New Zealand and serves as a symposium to mark the life and work of AWH Phillips. The New Zealand Association of Economists (NZAE) and the Australasian Standing Committee of the Econometric Society (ESASC) felt that 2008, the 50th anniversary of the Phillips curve paper, would be an appropriate time to celebrate the life and work of the late Bill Phillips. Of course, Phillips’ career embraced much more than just the curve that came to carry his name. His work embraced economic modelling, including stabilisation policy, economic dynamics, continuous time modelling, economic development, with a particular interest in China, and of course, the famous MONIAC machine. The program will consist of a number of keynote speakers as well as contributed papers and special sessions broadly themed around Phillips fields of interest. Further information is available on the conference website: http://www.esam08.org.nz/. The Program Committee is co-chaired by Bob Buckle (New Zealand Treasury) and John Gibson (University of Waikato). The local Steering Committee welcomes you on behalf of the International Advisory Board for this event. We hope you will plan to participate in an outstanding event, to hear world class speakers and visit New Zealand’s exciting capital city. At least one co-author must be a member of the Econometric Society or must join prior to submission. This can be done electronically at http://www.econometricsociety. org. 2008 FAR EASTERN AND SOUTH ASIAN MEETING
THE 2008 FAR EASTERN AND SOUTH ASIAN MEETING of the Econometric Society (FEMES-SAMES 2008) will be hosted by the School of Economics of Singapore Management University, on July 16–18, 2008. The venue of the meeting is the campus of Singapore Management University in the central business district of Singapore. The program will consist of plenary, invited, and contributed sessions in all fields of economics. Partial subsidy is available for travel and local expenses of young economists presenting a paper in the conference. Please visit the conference website at http://www. economics.smu.edu.sg/femes/2008/index.asp for additional meeting and registration details. Confirmed Keynote and Invited Speakers: Clive Granger, Torsten Persson, Peter Phillips, Yacine Ait-Sahalia, Torben Andersen, T. W. Anderson, Donald Andrews, Yongsung Chang, Vincent Crawford, Francis X. Diebold, David Hendry, Cheng Hsiao, Atsushi Kajii, Yuichi Kitamura, Oliver Lin935
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ton, Whitney Newey, Werner Ploberger, Peter Robinson, Neil Shephard, and Eytan Sheshinski Advisory Committee: Fumio Hayashi, University of Tokyo Cheng Hsiao, University of Southern California Chung-Ming Kuan, Institute of Economics, Academia Sinica Joon Y. Park, Texas A&M University Lawrence J. Lau, The Chinese University of Hong Kong Program Committee: Roberto S. Mariano (Chair), Singapore Management University and University of Pennsylvania Jushan Bai, New York University Parkash Chander, National University of Singapore Yongsung Chang, University of Rochester Shurojit Chatterji, University of Warwick Yin-Wong Cheung, University of California, Santa Cruz In Choi, Hong Kong University of Science and Technology Bhaskar Dutta, University of Warwick Taiji Furusawa, Hitotsubashi University Yong-miao Hong, Cornell University Hoon Hian Teck, Singapore Management University Hidehiko Ichimura, University of Tokyo Atsushi Kajii, Kyoto University Winston Koh, Singapore Management University Naoto Kunitomo, University of Tokyo Edwin Lai, Federal Reserve Bank, Dallas Myoung-Jae Lee, Korea University Euston Quah, Nanyang Technological University Guillaume Rocheteau, Federal Reserve Bank of Cleveland Arunava Sen, Indian Statistical Institute Liangjun Su, Peking University Wing Suen, University of Hong Kong Sun Yeneng, National University of Singapore Tse Yiu Kuen, Singapore Management University Whang Yoon-Jae, Seoul National University Zhijie Xiao, Boston College Jun Yu, Singapore Management University Asad Zaman, International Islamic University Junsen Zhang, The Chinese University of Hong Kong Local Organizing Committee: Yang Zhenlin (Chair), Singapore Management University Tilak Abeysinghe, National University of Singapore Choy Keen Meng, Nanyang Technological University
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Tomoki Fujii, Singapore Management University Aurobindo Ghosh, Singapore Management University Nicolas Jacquet, Singapore Management University Lee Gea Myoung, Singapore Management University Shahidur Rahman, Nanyang Technological University Sam Ouliaris, National University of Singapore Anthony Tay, Singapore Management University 2008 EUROPEAN MEETING
THE 2008 EUROPEAN MEETING of the Econometric Society (ESEM) will take place in Milan, Italy, from 27 to 31 August, 2008. The Meeting is organized by Bocconi University and it will run in parallel with the Congress of the European Economic Association (EEA). Participants will be able to attend all sessions of both events. The Program Committee Chairs are Martin Cripps (University College London) for Theoretical and Applied Economics, and Thierry Magnac (University of Toulouse) for Econometrics and Empirical Economics. The Local Arrangements Chair is Donato Masciandaro (Bocconi University) and the Co-Chairs are Michele Polo (Bocconi University) and Pier Paolo Battigalli (Bocconi University). For further information please visit the conference website at http://www.eeaesem2008.org. 2008 EUROPEAN WINTER MEETINGS
THE 2008 EUROPEAN WINTER MEETINGS of the Econometric Society will take place at Churchill College, University of Cambridge, Cambridge, United Kingdom, from October 31 to November 1. The Programme Committee consists of the Regional Consultants with Professor Richard J. Smith as Programme Committee Chair. Senior members of the Econometric Society may propose candidates to the consultant of their region. Candidates are PhD students or recent PhD graduates who are intending to go on the job market during academic year 2008–2009. Eighteen participants are selected by the Regional Consultants to present their research and to discuss the research of the other participants at the meetings. The selection of participants proceeds in two stages. In the first stage each Regional Consultant selects one participant from his/her region and proposes a shortlist of candidates. In the second stage the remaining ten participants are elected by the Regional Consultants from these shortlists. Regional Consultants: Chairperson: Prof. Jean-Marc Robin; Paris School of Economics, Université de Paris 1, and University College London; e-mail:
[email protected]; Areas of expertise: Labor economics, Microeconometrics; Region: France, Portugal and Spain Prof. Giuseppe Bertola; Università di Torino; e-mail:
[email protected]; Areas of expertise: Macroeconomics, Labor economics, Financial theory and policy; Region: Greece, Italy and Switzerland
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Prof. Georg Kirchsteiger; Université Libre de Bruxelles; e-mail:
[email protected]; Areas of expertise: Microeconomic theory, Experimental economics, Public economics; Region: Benelux Prof. Tom Krebs; University of Mannheim; e-mail:
[email protected]; Areas of expertise: Macroeconomics, Financial economics, Economic theory; Region: Austria and Germany Prof. Zvika Neeman; Boston University and Tel Aviv University; e-mail: zvika@bu. edu; Areas of expertise: Microeconomic theory, Game theory, Mechanism design; Region: Israel and Other Areas Prof. Richard J. Smith; University of Cambridge; e-mail:
[email protected]; Areas of expertise: Econometric theory, Estimation and inference in econometrics, Hypothesis testing, Model selection; Region: Great Britain and Ireland Prof. Kjetil Storesletten; University of Oslo; e-mail:
[email protected]; Areas of expertise: Macroeconomics, Political economy; Region: Scandinavia Prof. Ákos Valentinyi; University of Southampton; e-mail:
[email protected]; Areas of expertise: Macroeconomic Theory, Economic growth, Transition economics; Region: Eastern European countries Proposals of candidates should be sent by e-mail to the Consultant of the relevant region for the candidate before September 5, 2008. The e-mail should include the candidate’s CV, the complete paper she/he would like to present together with an introduction detailing the qualities of the candidate and, if the paper is co-authored, the contribution of the candidate. Successful candidates will be informed by October 3, 2008. Information concerning these meetings will be made available in due course on the conference website http://www.econ.cam.ac.uk/esewm08/. 2008 LATIN AMERICAN MEETING
THE 2008 LATIN AMERICAN MEETING of the Econometric Society (LAMES) will take place in Rio de Janeiro, Brasil, November 20–22. The meetings are jointly hosted by Fundação Getulio Vargas (FGV/EPGE) and Instituto Nacional de Matemática Pura e Aplicada (IMPA) and will take place at IMPA. The 2008 LAMES will run in parallel to the Latin American and Caribbean Economic Association (LACEA). Registered participants will be welcome to attend all sessions for both meetings. Economists and professionals working in related fields, including those who currently are not members of the Econometric Society, are invited to submit theoretical and applied papers in all areas of economics for presentation at the Congress. Each author may submit only one paper to the LAMES and only one paper to the LACEA. The same paper may not be submitted to both the LAMES and LACEA Congress. At least one co-author must be a member of the corresponding Society or Association. Authors may join the Econometric Society at http://www.econometricsociety.org. The Program Committee Chair is Aloisio Araujo.
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On behalf of the Program Committee, Co-Chairs are—Theory and Applied Economics: Humberto Moreira (EPGE/FGV), Econometrics: Sergio Firpo (PUC-Rio). For further information please visit the conference website at lacealames2008.fgv.br or contact us via e-mail at
[email protected]. 2009 NORTH AMERICAN WINTER MEETING
THE 2009 NORTH AMERICAN WINTER MEETING of the Econometric Society will be held in San Francisco, CA, from January 3 to 5, 2009, as part of the annual meeting of the Allied Social Science Associations. The program will consist of contributed and invited papers. It is hoped that the research presented will represent a broad spectrum of applied and theoretical economics and econometrics. The program committee will be chaired by Steven Durlauf of University of Wisconsin–Madison. Program Committee: Steven Durlauf, University of Wisconsin–Madison Program Chair David Austen-Smith, Northwestern University (Political Economy) Dirk Bergemann, Yale University (Information Economics) Lawrence Blume, Cornell University (Game Theory) Moshe Buchinsky, University of California, Los Angeles (Applied Econometrics) Dennis Epple, Carnegie Mellon University (Public Economics) Oded Galor, Brown University (Economic Growth) Jinyong Hahn, University of California, Los Angeles (Econometric Theory) Caroline Hoxby, Stanford University (Social Economics) Guido Kuersteiner, University of California, Davis (Time Series) Jonathan Levin, Stanford University (Industrial Organization) Shelly Lundberg, University of Washington (Labor Economics) James Rauch, University of California, San Diego (International Trade) Hélène Rey, Princeton University (International Finance) Manuel Santos, University of Miami (Computational Economics) Christina Shannon, University of California, Berkeley (Mathematical Economics) Steven Tadelis, University of California, Berkeley (Market Design) Petra Todd, University of Pennsylvania (Microeconometrics/Empirical Microeconomics) Toni Whited, University of Wisconsin (Finance) Noah Williams, Princeton University (Macroeconomics) Justin Wolfers, Wharton (Behavioral Economics/Experimental Economics) Tao Zha, Federal Reserve Bank of Atlanta (Macroeconomics) Lin Zhou, Arizona State University (Social Choice Theory/Microeconomic Theory)
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FORTHCOMING PAPERS THE FOLLOWING MANUSCRIPTS, in addition to those listed in previous issues, have been accepted for publication in forthcoming issues of Econometrica. ABADIE, ALBERTO, AND GUIDO W. IMBENS: “On the Failure of the Bootstrap for Matching Estimators.” BARNDORFF-NIELSEN, OLE E., PETER REINHARD HANSEN, ASGER LUNDE, AND NEIL SHEPHARD: “Designing Realised Kernels to Measure the ex-post Variation of Equity Prices in the Presence of Noise.” BLUNDELL, RICHARD, MARTIN BROWNING, AND IAN CRAWFORD: “Best Nonparametric Bounds on Demand Responses.” CHAMBERLAIN, GARY: “Decision Theory Applied to an Instrumental Variables Model: Comment.” DE CLIPPEL, GEOFFROY, AND ROBERTO SERRANO: “Marginal Contributions and Externalities in the Value.” FLORENS, J. P., J. J. HECKMAN, C. MEGHIR, AND E. VYTLACIL: “Identification of Treatment Effects Using Control Functions in Models With Continuous, Endogenous Treatment and Heterogeneous Effects.” HÖRNER, JOHANNES, AND NICOLAS VIEILLE: “Public vs. Private Offers in the Market for Lemons.” JIA, PANLE: “What Happens When Wal-Mart Comes to Town: An Empirical Analysis of the Discount Retailing Industry.” LENTZ, RASMUS, AND DALE MORTENSEN: “An Empirical Model of Growth Through Product Inovation.” MARQUEZ, ROBERT, AND BILGE YILMAZ: “Information and Efficiency in Tender Offers.” MATZKIN, ROSA L.: “Identification in Nonparametric Simultaneous Equations Models.” NEWEY, WHITNEY K., AND FRANK WINDMEIJER: “GMM With Many Weak Moment Conditions.” OLSZEWSKI, WOJCIECH, AND ALVARO SANDRONI: “Manipulability of FutureIndepedent Tests.” PITARAKIS, JEAN-YVES: “Threshold Autoregressions With a Unit Root Revisited.” RIGOTTI, LUCA, CHRIS SHANNON, AND TOMASZ STRZALECKI: “Subjective Beliefs and ex-ante Trade.” TEAGUE, VANESSA: “Problems With Coordination in Two-Player Games.” VANBERG, CHRISTOPH: “Why Do People Keep Their Promises? An Experimental Test of Two Explanations.” WEINTRAUB, GABRIEL Y., C. LANIER BENKARD, AND BENJAMIN VAN ROY: “Markov Perfect Industry Dynamics With Many Firms.”
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SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES FOR MONOGRAPHS IN ECONOMIC THEORY, a PDF of the manuscript should be emailed to Professor George J. Mailath at
[email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor George J. Mailath Department of Economics 3718 Locust Walk University of Pennsylvania Philadelphia, PA 19104-6297, U.S.A. For monographs in theoretical and applied econometrics, a PDF of the manuscript should be emailed to Professor Andrew Chesher at
[email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor Andrew Chesher Department of Economics University College London Gower Street London WC1E 6BT, U.K. They must be accompanied by a letter of submission and be written in English. Authors submitting a manuscript are expected not to have submitted it elsewhere. It is the authors’ responsibility to inform the Editors about these matters. There is no submission charge. The Editors will also consider proposals consisting of a detailed table of contents and one or more sample chapters, and can offer a preliminary contingent decision subject to the receipt of a satisfactory complete manuscript. All submitted manuscripts should be double spaced on paper of standard size, 8.5 by 11 inches or European A, and should have margins of at least one inch on all sides. The figures should be publication quality. The manuscript should be prepared in the same manner as papers submitted to Econometrica. Manuscripts may be rejected, returned for specified revisions, or accepted. Once a monograph has been accepted, the author will sign a contract with the Econometric Society on the one hand, and with the Publisher, Cambridge University Press, on the other. Currently, monographs usually appear no more than twelve months from the date of final acceptance.
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