Pooling data across markets in dynamic Markov games

Otsu, Taisuke; Pesendorfer, Martin; Takahashi, Yuya Article Pooling data across markets in dynamic Markov games Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Otsu, Taisuke; Pesendorfer, Martin; Takahashi, Yuya (2016) : Pooling data across markets in dynamic Markov games, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 7, Iss. 2, pp. 523-559, https://doi.org/10.3982/QE612 This Version is available at: https://hdl.handle.net/10419/150416 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Quantitative Economics 7 (2016), 523–559 1759-7331/20160523 Pooling data across markets in dynamic Markov games Taisuke Otsu Department of Economics, London School of Economics Martin Pesendorfer Department of Economics, London School of Economics Yuya Takahashi Department of Economics, Johns Hopkins University This paper proposes several statistical tests for finite state Markov games to examine whether data from distinct markets can be pooled. We formulate homogeneity tests of (i) the conditional choice and state transition probabilities, (ii) the steady-state distribution, and (iii) the conditional state distribution given an initial state. The null hypotheses of these homogeneity tests are necessary conditions (or maintained assumptions) for poolability of the data. Thus rejections of these null imply that the data cannot be pooled across markets. Acceptances of these null are considered as supporting evidences for the maintained assumptions of estimation using pooled data. In a Monte Carlo study we find that the test based on the steady-state distribution performs well and has high power even with small numbers of markets and time periods. We apply the tests to the empirical study of Ryan (2012) that analyzes dynamics of the U.S. Portland cement industry and assess if the data across markets can be pooled. Keywords. Dynamic Markov game, poolability, multiplicity of equilibria, hypothesis testing. JEL classification. C12, C72, D44. 1. Introduction This paper proposes several statistical tests for finite state Markov games to examine whether data from distinct markets can be pooled. Data pooling is employed in a number of empirical applications of the two-step estimation methods for dynamic games Taisuke Otsu: [email protected] Martin Pesendorfer: [email protected] Yuya Takahashi: [email protected] We thank Stephen Ryan, Pierre Dubois, conference participants of the 14th CEPR–JIE Applied IO Conference, and seminar audiences in Alicante, Barcelona (Pompeu Fabra and Autonoma), Johns Hopkins, Kyoto, NYU, Paris School of Economics, Princeton, Texas Austin, UCL, UBC, and Wisconsin for thoughtful comments. We thank André Stenzel for excellent research assistance. Takahashi’s work was supported by the Deutsche Forschungsgemeinschaft through SFB/TR 15. Copyright ©2016 Taisuke Otsu, Martin Pesendorfer, and Yuya Takahashi. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://www.qeconomics.org. DOI: 10.3982/QE612 524 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) recently developed.1These two-step estimators estimate players’ policies and state transition probabilities in a first stage directly from the data as functions of observable state variables. The second stage conducts a search for the structural model parameters that best rationalize observed behaviors of players and state transitions using the first-stage policy estimates as estimates for the equilibrium beliefs. A typical application may not have long time series data for a single market. Researchers are tempted to pool data from different markets (or games) to perform the first-stage policy function estimation. To do so, researchers assume that the data are generated from a single and identical equilibrium in every market. This assumption has become popular in a number of recent papers.2To be more precise, the assumption commonly imposed requires that the game describing players’ behavior is identical in all markets and that a single and identical equilibrium of that game is played in all markets. It also requires that the econometric model controls for all observable or unobservable market-level elements. A violation of the assumption results in inconsistent policy estimates and inconsistent structural parameter estimates. A violation of the assumption can arise because of equilibrium multiplicity. The single and identical equilibrium assumption may be very restrictive even if the markets are identical as multiplicity of equilibria is a well known feature inherent to games. Incorrectly imposing this assumption leads to erroneous inference. A maintained assumption for estimation based on the pooled data is that the data generating processes are identical across markets. We propose three tests to assess homogeneity of the data generating processes. The first test compares directly the set of conditional choice or state transition probabilities estimated from the pooled (across markets) sample with those estimated from each market separately. The second test is based on the result that there is a unique steady-state distribution associated with a transition matrix of states under the assumption of communicating states. Based on this result, the second test compares the steady-state distribution estimated from the pooled sample with that from each market. Our third test statistic is based on the conditional state distribution given the initial (observed) state. We contrast the observed relative frequencies of states to the theoretical predictions given the initial state. It turns out that the third test does not require several assumptions on Markov chains that are imposed for other tests. Each test has its own advantage. One advantage across all three tests is that we do not need to impose any mixing structure. 1Several papers, including Jofre-Bonet and Pesendorfer (2003), Aguirregabiria and Mira (2007), Bajari, Benkard, and Levin (2007), Pakes, Ostrovsky, and Berry (2007), Pesendorfer and Schmidt-Dengler (2008), Arcidiacono and Miller (2011), Kasahara and Shimotsu (2012), and Srisuma and Linton (2012), proposed two-step estimation methods for dynamic Markov games under varying assumptions. They led to a number of empirical papers that apply these methods to empirically analyze dynamic interactions between multiple players. 2Examples include Beresteanu, Ellickson, and Misra (2010), Collard-Wexler (2013), Dunne, Klimek, Roberts, and Xu (2013), Fan and Xiao (2014), Jeziorski (2014), Lin (2015), Maican and Orth (2014), Minamihashi (2012), Nishiwaki (2015), Ryan (2012), Sanches and Silva Junior (2013), Snider (2009), Suzuki (2013), and Sweeting (2013). They impose the assumption of a single and identical equilibrium in all markets either explicitly or implicitly. The empirical sections of Aguirregabiria and Mira (2007) and Arcidiacono, Bayer, Blevins, and Ellickson (2015) and the Monte Carlo exercise in Arcidiacono and Miller (2011)alsoimpose the same assumption. Quantitative Economics 7 (2016) Pooling data across markets 525 Since the null hypotheses of our homogeneity tests are necessary conditions or maintained assumptions for estimation based on pooled data across markets, a rejection of the null suggests that the data cannot be pooled. A violation can arise because (i) multiple equilibria are played across markets, (ii) the game form describing players’ behavior and interactions differs across markets, and (iii) the specified model is not sufficiently rich as it does not control for all observable or unobservable market-level heterogeneity adequately. It is difficult to distinguish these alternative explanations although we shall illustrate tests accounting for unobservable market-level heterogeneity as in Arcidiacono and Miller (2011) in more detail below. Our test is aimed at checking the validity of the maintained assumption for data pooling commonly imposed in the literature. A rejection of the null points to an inconsistency of the first-stage estimates that arises from pooling different markets. Naturally, since the framework of this paper nests single agent settings as a special case with only one player, our tests can also be thought of as testing whether data can be pooled in the single agent case. To illustrate the finite sample performance of our tests, we first apply the tests to simulated data using an example of multiple equilibria in Pesendorfer and Schmidt-Dengler (2008). Our tests, particularly the one based on the steady-state distribution, perform well and have high power even with small numbers of markets and time periods. We then apply our tests to the empirical study of Ryan (2012) that analyzes dynamics of the U.S. Portland cement industry. Our tests reject the null hypothesis that the data from distinct markets are generated from an identical data generating process. To the best of our knowledge, this is the first paper that proposes tests to assess the validity of data pooling in a general class of dynamic Markov games. Our tests may give a researcher guidance on whether she can pool different markets to estimate policy functions in the first stage. A rejection of the null hypothesis suggests that one or more modeling assumption differs across markets. In the context of static games with incomplete information, de Paula and Tang (2012) propose a test of multiplicity of equilibria that requires conditional independence between players’ actions. Since our tests exploit the panel structure of the data and rely on the way that the game and states evolve, our tests are fundamentally different from theirs. One notable difference is that while de Paula and Tang (2012) maintain the assumption of independent-across-players private shocks, we can allow for within-period correlation in players’ actions and for unobserved state variables. This paper is organized as follows. Section 2lays out a class of general dynamic Markov games we work with and provides some background on Markov chains. Section 3proposes several test statistics. In Section 4we conduct a Monte Carlo study to examine finite sample properties. Section 5applies our tests to data of Ryan (2012). Section 6concludes. The Appendix contains technical details. Replication files are available in a supplementary file on the journal website, http://www.qeconomics.org/supp/612/ code_and_data.zip. 2. Model This section describes elements of a general dynamic Markov game with discrete time t=12. We focus on the description of players’ state variables and actions. These 526 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) states and actions are the observable outcome variables for some underlying dynamic game, which we do not observe. We leave the details of the game unspecified. Instead we shall focus on testable implications of the observed outcomes. Our setting includes the single agent case as a special case when there is one agent per market. We first describe the framework, which applies for all markets j=1M. Players. A typical player is denoted by i=1N. The single agent case arises when N=1. The number of players is fixed and does not change over time. Every period the econometrician observes a profile of states and actions described as follows. States. Each player is endowed with state variables st i∈{1L}in finite support. The state variable st iis publicly observed by all players. We maintain the assumption that the econometrician also observes st i. The vector of all players’ public state variables is denoted by st=(st 1st N)∈S={1L}Nwhose cardinality is ms=LN.InSection 3.5, we discuss the case where some of the public state variables are unobservable by the econometrician. Actions. Each player chooses an action at i∈{01K}in finite support. The decisions are made after the state is observed. The decisions can be made simultaneously or sequentially. The decision may also be taken after an idiosyncratic random utility (or a random profit shock) is observed. We leave the details of the decision process unspecified. Our specification encompasses the random-utility modeling assumptions, and allows for within-period correlation in the random utility component across actions and across players. The vector of joint actions in period tis denoted by at=(at 1at N)∈ A={01K}Nwhose cardinality is ma=(K +1)N. We assume actions are publicly observed by all players and the econometrician. Choice probability matrix.Letσ(a|s)=Pr{at=a|st=s}denote the conditional probability that an action profile awill be chosen conditionally on a state s. Throughout the paper, we assume that σis time invariant and is conditionally independent from other past actions and states. The matrix of conditional choice probabilities is denoted by σ, which has dimension ms×(mams). It consists of conditional probabilities σ(a|s)in row s, column (as),andzerosinrows, column (as)with s=s. State–action transition matrix. Let g(s|as)=Pr{st+1=s|at=ast=s}denote the state–action transition probability that a state sis reached when the current action profileandstatearegivenby(as). We also assume that gis time invariant and is conditionally independent from other past actions and states. We use the symbol Gto denote the (mams)×ms-dimensional state–action transition matrix in which column s∈Sconsists of the vector of probabilities {g(s|as)}a∈As∈S. State transition matrix. Under the above assumptions on σand G, the state variables stobey a (first-order) Markov chain with the (stationary) state transition matrix P=σG whose dimension is ms×ms.Atypicalelementp(s|s)=a∈Aσ(a|s)g(s|as)of Pequals the probability that state sis reached when the current state is given by s. Hereafter we focus on the first-order Markov chain. However, our testing procedures can be extended to higher-order Markov chains since higher-order Markov chains can be reformulated as first-order ones by modifying the state space (see, e.g., Billingsley (1961)). Limiting steady-state distribution. When the limit exists, let Q(ss)=limT→∞T−1× T t=11{st=ss0=s}denote the long run proportion of time that the Markov chain Quantitative Economics 7 (2016) Pooling data across markets 527 Pspends in state swhen starting at the initial state s0=s,where1{·} is the indicator function. Suppose the unconditional long run proportion of time Q(s)= limT→∞T−1T t=11{st=s}that the Markov chain Pspends in state ssatisfies Q(·)= Q(·s)for all initial states s. Then the ms-dimensional row vector of probabilities Q={Q(s)}s∈Sis called the steady-state distribution of the Markov chain. Observe that the state space is finite and Qdescribes a multinomial distribution. The properties of Markov chains are well known. We next describe some property useful for our purpose. To do so, we introduce the concept of communicating states. Communicating states. We say that a state sis reachable from sif there exists an integer Tso that the chain Pwill be at state safter Tperiods with positive probability. If sis reachable from s,andsis reachable from s, then the states sand sare said to communicate. Lemma 1. Suppose all states of the Markov chain Pcommunicate.3Then the steady-state distribution Qexists and is unique.It satisfies Q(s)>0for all s∈Sand Q=QP. This lemma guarantees existence and uniqueness of the steady-state distribution, and states that the long run proportion of time that the Markov chain Pspends in state sis strictly positive for any state s∈Sand the equation Q=QP must hold. A proof of the above properties is given in Levin, Peres, and Wilmer (2009, Proposition 1.14 and Corollary 1.17), for example. Communicating states are typically invoked in applied work; see Ericson and Pakes (1995). Communicating states naturally emerge in dynamic discrete choice models using a random utility specification; see McFadden (1973). The random component having full support in the real numbers implies that all actions arise with strictly positive probability for any state s∈S. Thus, states will communicate if the state–action transition matrix allows that state s, respectively s, can in principle be reached when starting from state s, respectively s, for any pair of states ss∈S. The feature that all states communicate may also emerge when actions are chosen with probability 1 for some (or all) states. Our setup includes these settings as well. What is required for states to communicate in this case is that there exists a sequence of state– action profiles {(a1s1)(atst)}so that the chain starting at state swill be at state s after tperiods for any ss∈S. 3. Homogeneity tests for poolability This section describes hypotheses that we aim to test and proposes statistical tests for those hypotheses. For each market j, a sequence of action–state profiles (at jst j)t=1T is observed, where Tis the length of time periods in the data set. Our null hypothesis is that the observed profiles are generated from an identical data generating process in all markets, and the alternative is that the data generating process is distinct for some markets. This null hypothesis is a maintained assumption for estimation based on pooled data. Based on the setup described in the previous section, the data generating process 3This is also called that the Markov chain Pis ergodic or irreducible. 528 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) of the profiles (at jst j)t=1T is characterized by the conditional choice probability matrix σjand state–action transition matrix Gjthat imply the transition matrix of states Pj=σjGj. In particular, we focus on homogeneity of σjand Pjacross markets, and test the null hypotheses Hσ 0:σ1=···=σM (1) HP 0:P1=···=PM; the alternatives are their negations. The null hypothesis Hσ 0is based on the idea that the equilibrium choice probabilities are identical across markets. The null HP 0has a similar motivation given that the state–action transition is identical across markets. Economic models may have the feature that the state–action transition matrix Gis exogenously given and by construction is identical across markets. In such cases, testing the conditional choice probabilities has the same interpretation as testing the state transition probabilities. However, in general, the tests may not be equivalent. A rejection of the null HP 0could arise either because of nonidentical choice probabilities σjor because of heterogeneous state–action transition matrices Gj. Which test is most suitable depends on the economic application at hand and each test has its own rationale. If all states of the Markov chain Pcommunicate, then by Lemma 1,thereexists a unique steady-state distribution Qand the identical equilibrium hypothesis may be tested by homogeneity of the steady-state distribution, HQ 0:Q1=···=QM(2) As discussed in the next subsection, if the cardinality of the action or state space is large, then the power of the test for Hσ 0or HP 0tends to be low relative to that for HQ 0because a decrease in the degrees of freedom can be expected. Thus, the power of the homogeneity test can be increased by testing the steady-state distribution. Lemma 1says that the null HP 0of equal transition matrices implies the null HQ 0of equal steady-state distributions. Thus, a rejection of HQ 0provides strong evidence for a rejection of HP 0. By testing HQ 0first, we may exploit the property that the power of testing the null HQ 0is typically higher than the power of testing the null HP 0. However, it should be noted that the converse is not true: the equivalence of the steady-state distribution across markets does not necessarily imply that of the transition matrix. To test the above hypotheses, we consider the situation where for each market j,we observe the action–state profiles (at jst j)t=1T with sufficiently large T. The test procedures discussed in the next subsection are theoretically justified when the time length T increases to infinity. However, the researcher may face the situation where the length of time periods Tis relatively short compared to the number of markets M.Insuchascenario, it would be natural to treat the action–state profiles with fixed Tacross markets as an independent and identically distributed (i.i.d.) sample (over j=1M)from the distribution parametrized by a common choice probability σor a common transition matrix P. For example, testing may be based on the conditional state distribution st|s1=sgiven the initial state sfor t=2T. By conditioning on the initial state we do Quantitative Economics 7 (2016) Pooling data across markets 529 not require that states communicate so that the industry at hand can reach the steadystate distribution. This situation arises naturally in new or growing industries. Using the transition matrix P, the conditional distribution st|s1=sis described by ι sPt,whereιs takes 1at the element corresponding to sand 0otherwise. There are many ways to compare the vector of conditional probabilities {Pr{st=s|s1=s}}s∈Swith the theoretical prediction ι sPt. For example, at a given initial state s, we can consider the null hypothesis in the form of Hs 0:1 T−1 T  t=2 Prst=s|s1=ss∈S=1 T−1 T  t=2 ι sPt(3) The left hand side is a vector of model-free conditional probabilities. The right hand side is the model-based prediction for those probabilities. Note that the hypothesis Hs 0 is implied from two assumptions: (i) the data (st j)t=1T for j=1M are i.i.d. over j, which allows us to express the hypothesis Hs 0without using a market index j,and (ii) the Markov chain is first-order and time-homogeneous. Thus, a rejection of Hs 0may be interpreted as violation of the i.i.d. assumption (perhaps associated with multiplicity of equilibrium) or misspecification of the Markov chain (such as time inhomogeneity or higher order). The left hand side denotes the empirical frequency (across markets) of visiting state sin periods t=2T conditional on the initial state s1=s. The right hand side is the theoretical predicted counterpart under the null of homogeneity across markets. A violation of (3) would indicate that the empirical frequency distribution (across markets) differs from that predicted by the theoretical model. Hypothesis (3) focuses on the average probabilities of visiting each state given the initial state s. We may do so for selected initial states. Alternatively, one may consider all possible initial states jointly by testing the null H0:Pr{st=s|s1=s}=ι sPtfor all s∈Sand tor its linear combinations. We note that the null Hs 0tests the validity of the i.i.d. parametric model for (sj)j=1M with sj=(s1 jsT j)for fixed T. As mentioned above, a rejection of the null can arise from multiple equilibria, the game form differing across markets, and/or unobservable market-level heterogeneity. Our framework nests single agent settings as a special case. In case of rejection, the first possibility (multiple equilibria) is naturally excluded so the interpretation of the rejection would be simpler. Therefore, our tests can be thought of as testing whether richer heterogeneity among agents should be considered in the single agent case. 3.1 Testing choice and transition probabilities Let us first consider testing for Hσ 0and HP 0in (1) based on the conditional choice and transition probabilities, respectively. We form a generally applicable chi-squared test statistic based on the conditional choice or transition probability, that is TP= M  j=1 d∈D Wj(d) Pj(d)− P(d)2(4) 530 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) where  Pj(d)is a nonparametric estimator of the probability of interest for a market j without imposing the null hypothesis of interest,  P(d)is another nonparametric estimator under the null of homogeneity of  Pj(d)across markets, and Wj(d)is a weight or standardization to obtain a standard limiting distribution. For example, to test homogeneity of the conditional choice probabilities Hσ 0,weset d=(as)and D=A×S.Letfj(as)=T t=11{at j=ast j=s}be the frequency of action– state profile (as)in market jand let fj(s)=T t=11{st j=s}be the frequency of state sin market j. Then we estimate the conditional choice probabilities for the action profile a given the current state sin market j,σj(a|s), by the relative frequencies  P(d)= M  j=1 fj(as) M  j=1 fj(s)  Pj(d)=fj(as) fj(s)(5) with and without imposing Hσ 0, respectively. To obtain the chi-squared limiting distribution, we set the weight as Wj(d)=fj(s)/ P(d). Also, to test the equivalence of the transition matrices HP 0,wesetd=(ss)and D=S×S.Letf1 j(ss)=T−1 t=11{st+1 j=sst j=s}and f1 j(s)=T−1 t=11{st j=s}.Thenwe estimate the transition probability pj(s|s)by  P(d)= M  j=1 f1 jss M  j=1 f1 j(s)  Pj(d)=f1 jss f1 j(s)(6) with and without imposing HP 0, respectively. The weight is set as Wj(d)=f1 j(s)/ P(d). The limiting null distribution of the statistic TPis obtained in the following proposition (see Appendix A.1 for the proof). Proposition 1. Consider the setup of Section 2.Suppose that all states of the Markov chain Pjcommunicate for each j=1M and that the observations (at jst j)t=1T are mutually independent over j=1M.Then under Hσ 0(or,respectively,H P 0), the statistic TPconverges in distribution to the chi-squared distribution with degrees of freedom (M −1)ms(ma−1)(or,respectively,(M −1)ms(ms−1))asthelengthoftimeperiodsT increases to infinity. Bootstrap critical value. The chi-squared limiting distributions of the statistic TP gives us critical values to control the asymptotic null rejection probabilities. Alternatively one may compute critical values by some bootstrap method. For example, to test the null HP 0, we can randomly pick an initial state s0∈Sand then draw the bootstrap counterpart f1b j(ss)of f1 j(ss)from the estimated conditional probability  P(d)in (6)forss∈S,j=1M,andb=1B. Note that we Quantitative Economics 7 (2016) Pooling data across markets 537 across markets by applying the methods in Arcidiacono and Miller (2011)andKasahara and Shimotsu (2009). Based on these estimates, we obtain an estimator of P,say ˜ P.Then we can apply the test statistic Tsin (13) by replacing  Pwith ˜ P,thatis, ˜ Ts=M˜ C s˜ V− s˜ Cs(14) where ˜ C s=(T −1)−1(T t=2 Qt s−ι sT t=2˜ Pt)and ˜ V− sis a generalized inverse of an estimator of the asymptotic variance of √M˜ Csunder Hs 0. Similar to Ts, this statistic converges to a χ2distribution under Hs 0as M→∞while Tis fixed. Third, we illustrate how to extend the test for HP 0in (1) to accommodate unobservable time-invariant state variables. Again, for simplicity of exposition suppose s2is binary. We can modify the null hypothesis as ˜ HP 0:sjis a Markov chain from P(a) or P(b) for all j As M→∞, we can consistently estimate P(a) and P(b) using the pooled sample across markets by applying Arcidiacono and Miller (2011)orKasahara and Shimotsu (2009). Let ˜ P(a) and ˜ P(b) be such estimators. On the other hand, as T→∞, the estimator  Pjdefined in (6) consistently estimates the transition for each market jand thus converges to P(a) or P(b) under ˜ HP 0. Based on these observations, a test statistic for ˜ HP 0may be constructed as ˜ TP=M j=1˜ TPj,where ˜ TPj =min ss∈S f1 j(s) ˜ P(a)ss Pjss−˜ P(a)ss2 (15)  ss∈S f1 j(s) ˜ P(b)ss Pjss−˜ P(b)ss2 This construction of the test statistic (i.e., aggregate the statistic ˜ TPj over cross-section units j=1M) appears often in the literature of large-Tpanel data analysis (see, e.g., Baltagi (2008, Chapter 12)). In this literature, it is common to take the sequential limits (i.e., take T→∞first to derive the limiting distribution of ˜ TPj for each j, and then take M→∞to establish the limiting distribution of ˜ TP) to analyze the asymptotic properties of test statistics, such as panel unit root tests. Phillips and Moon (1999) provided additional requirements to strengthen the sequential limit theory to the joint one, where T and Mcangrowinanarbitraryway.However,inoursetup,thestatistic ˜ TPj for market jdepends on both M(for ˜ P(a) and ˜ P(b))andT(for  Pj(s)). Therefore, the existing techniques of large-Tpanel data analysis are not directly applicable. Although the complete analysis of the asymptotic theory for ˜ TPis beyond the scope of this paper, we can adjust the construction of the test statistic to fit into the sequential asymptotic framework. To this end, we choose the sample size to estimate ˜ P(a) and ˜ P(b) as a function of T,say CT. Also we assume CT/T →∞as T→∞, which guarantees that the estimation errors ˜ P(a) −P(a) and ˜ P(b) −P(b) are negligible. Since ˜ P(a) and ˜ P(b) are typically computed by a pooled sample across markets, the requirement CT/T →∞is mild. Under these 538 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) additional requirements, the statistic ˜ TPj depends only on Tand satisfies ˜ TPj =min ss∈S f1 j(s) ˜ P(a)ss Pjss−P(a)ss2  ss∈S f1 j(s) ˜ P(b)ss Pjss−P(b)ss2+op(1) d →χ2 ms(ms−1)as T→∞under ˜ HP 0 for every j. Therefore, we can obtain the limiting distribution of ˜ TP=M j=1˜ TPj under the sequential limit, that is, ˜ TP−Mms(ms−1) 2Mms(ms−1) d →N(01) as T→∞followed by M→∞sequentially. This sequential limiting result may be strengthened to the joint result by verifying additional conditions in Phillips and Moon (1999, Lemma 6). In practice, the test for ˜ HP 0based on ˜ TPis used as follows. If we reject ˜ HP 0, the maintained assumption for pooling the whole data is violated and it is recommended to look for a subset that preserves homogeneity. On the other hand, acceptance of the null ˜ HP 0is considered as supporting evidence for the researcher to pool the data across markets to implement two-step estimation for parameters, where the first-step estimates are constructed by using ˜ P(a) and ˜ P(b). 4. Monte Carlo This section examines the practical aspects of the proposed tests in a Monte Carlo study. We consider a simple and transparent dynamic oligopoly game with multiple equilibria. The game was illustrated and analyzed in more detail in Pesendorfer and SchmidtDengler (2008). It has the following features. There are two players: binary actions at i∈{01}and binary states st i∈{01}. The distribution of the profitability shocks is the standard normal. The discount factor is fixed at 09. The state transition law is given by st+1 i=at i. Period payoffs are symmetric and parametrized as π(aiajsi)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0if ai=0;si=0, 01if ai=0;si=1, π1−02if ai=1;aj=0;si=0, π2−02if ai=1;aj=1;si=0, π1if ai=1;aj=0;si=1, π2if ai=1;aj=1;si=1, where π1=12and π2=−12. The period payoffs can be interpreted as stemming from a game with switching costs and/or as entry/exit game. A player who selects action 1 Quantitative Economics 7 (2016) Pooling data across markets 539 receives monopoly profits π1if she is the only active player, and she receives duopoly profits π2otherwise. Additionally, a player who switches states from 0 to 1 incurs the entry cost 02, while a player who switches from 1 to 0 receives the exit value 01. Multiplicity. The game illustrates the possibility of multiple equilibria, which is a feature inherent to games. The following analysis focuses on two asymmetric equilibria of the three equilibria described in Pesendorfer and Schmidt-Dengler (2008). In equilibrium (i), player two is more likely to choose action 0 than is player one in all states. The ex ante probability vectors for both players are given by σ(a1=0| s1s2)=(027039020025)and σ(a2=0|s2s1)=(072078058071), where the order of the elements in the probability vectors corresponds to the state vector (s1s2)∈ {(00)(01)(10) (11)}. In equilibrium (ii), player two is more likely to choose action 0 than is player one in all states with the exception of state (10). The probability vectors are given by σ(a1=0| s1s2)=(038069017039)and σ(a2=0|s2s1)=(047070016042). Design. The simulated data are generated by randomly drawing a time series of actions from the calculated equilibrium choice probabilities described above for each of the equilibria (i) and (ii), respectively. The initial state is taken as (00)and we start the sampling process after 100 periods. The number of markets and the length of the time series is varied in the experiment with the aim of staying close to typical industry applications. We choose M=2040640 and T=510640. The parameter λdenotes the fraction of markets that adopt equilibrium (i), while 1−λdenotes the fraction of markets that adopt equilibrium (ii). Implementation. The Monte Carlo study considers the conditional choice probability multiplicity test by TP, its optimal version by T∗ P, the steady-state distribution test by TQ, and the conditional state distribution test by Tsas described in Section 3.Inthis example, at=st+1and the state transition probabilities Pequal the conditional choice probabilities σ. Therefore, the null hypotheses Hσ 0and HP 0and their tests are identical. To implement TPin (4)andT∗ Pin (7), we employ the formula in (6).9The steady-state probabilities Qare estimated by the relative frequencies. For the steady-state distribution test by TQ, we use the identity matrix for the variance matrix in (12). For the conditional state distribution test by Ts, we consider the sum TS=s∈STsinstead of focusing on a particular initial state. To compute Ts,wereplacethevariancematrix Vsin (13)with the identity matrix. The critical values of these test statistics are calculated using a bootstrap procedure. For every bootstrap iteration b, we simulate choice/state profiles {sb j}from the transition matrix based on  P(d)defined in (6) for every market j. For the first three tests (i.e., the tests by TP,T∗ P,andTQ), as in the data generating process, the initial state is taken as (00)and we start the sampling process after 100 periods. For the test by Ts,foreach market, we use the same initial state as is observed in the simulated sample and start the game from that state. The bootstrap counterparts of the test statistics are calculated for b=1B. The critical values are obtained by the 95th percentile of the bootstrapped statistics. 9When T t=11{st j=s}=0(or M j=1T t=11{st j=s}=0), we remove such states from the summand of the test statistics. 540 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) Results. The experiment is based on B=999 repetitions for the bootstrap sample and 1,000 Monte Carlo repetitions. Tables 1–4report the results of the experiments. These tables report the percentages of rejections of our tests for selected values of M,T,andλ. We first study the size properties of our tests. Tables 1and 2consider the cases of λ=1and λ=0, respectively. For these cases, there is a unique equilibrium and the null hypotheses are satisfied. All tests perform reassuringly well leading to a 5% rejection frequency as Tand/or Mincrease. We next assess the power properties of our tests. Table 3considers the case of λ=05, where the first and second equilibria arise with equal probability. It shows that as the number of time periods Tand/or markets Mincreases, all the tests typically reject the null more frequently. The two conditional choice probability tests (TPand T∗ P)andthe steady-state distribution test (TQ) perform better than the conditional state distribution test (Ts) for moderate values of M(e.g., M=20 or 40). When Mbecomes large (M= 320 or 640), Tsdominates TPand T∗ Pespecially when Tis relatively small. Comparing the conditional choice probability tests and the steady-state distribution test, we find that TQperforms better than TPand T∗ P. A possible reason is that TQuses fewer cells than TPand T∗ P.TestTQis based on mscells while TPand T∗ Pare based on (msma)cells. Table 3also illustrates that for a typical industry application with about 40 markets and 20 time periods the performance of TQis satisfying. Also the test by TPand the optimal test by T∗ Phave similar performance. For a better comparison based on the result in Proposition 2, we compute the size-adjusted power for TPand T∗ P. We find that the sizeadjusted power for T∗ Pis higher than that for TPin most cases.10 To further investigate the power properties of these tests, Table 4considers the case of λ=09.Thatis,thefirst equilibrium is played in 90% of Mmarkets. While all the tests have lower power than in Table 3, the relative performances of these tests appear the same. Test TQstill has the best performance among all tests. Overall, our Monte Carlo study illustrates that the steady-state distribution test by TQperforms well for moderate sample sizes of Tand M. It seems well suited for typical industry applications.11,12 10For example, when M=40,T=20, and λ=05,Table3suggests that the power for TPis higher than the power for T∗ P. On the other hand, the size-adjusted power for TPis 161, while the size-adjusted power for T∗ Pis 218. 11The number of markets Mand time periods Tin several important papers in the literature are (M= 1,600,T=24)inCollard-Wexler (2013), (M=639,T=5)inDunne et al. (2013), (M=23,T=19)inRyan (2012), and (M=102,T=4)inSweeting (2013). 12We also check the performance of ˜ Tsand ˜ TPusing the following simple simulation design. Suppose there are only two states. Consider the three state transition matrices P(a) =0307 0307P(b) =0703 0703P(c) =0802 0802 Under the null, each market follows P(a) or P(b) with equal probability. Under the alternative, each market follows P(a) or P(b) with probability 025, and follows P(c) with probability 05. We compute the size and power of the two test statistics with varying numbers of markets and time periods. Overall, the size approaches 5% quickly for both statistics. For the power, ˜ TPperforms better than ˜ Ts. For example, the power of ˜ TPwhen (M=500,T=10), (M=500,T=30), and (M=500,T=50)is93,801, and 997%, respectively. On the other hand, the corresponding figures for ˜ Tsare 50,70, and 91%, respectively. The details of this exercise are available upon request. Quantitative Economics 7 (2016) Pooling data across markets 541 Table 1. Monte Carlo results: λ=1. MTTPT∗ PTQTs 20 5 132591332 20 10 70452539 20 20 44503549 20 40 51624340 20 80 57665029 20 320 44444834 20 640 61534935 40 5 65231337 40 10 38272950 40 20 43343541 40 40 45533448 40 80 53535730 40 320 52544553 40 640 53544944 80 5 53151248 80 10 32122548 80 20 52352543 80 40 39393557 80 80 47465047 80 320 49555451 80 640 42415357 160 5 49062051 160 10 34092142 160 20 33244145 160 40 48483933 160 80 45465447 160 320 54576238 160 640 47425344 320 5 50051445 320 10 36083245 320 20 43193853 320 40 45463954 320 80 48414353 320 320 48565051 320 640 60585657 640 5 43040742 640 10 32091943 640 20 47293653 640 40 48434434 640 80 54484050 640 320 51494445 640 640 56555743 542 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) Table 2. Monte Carlo results: λ=0. MTTPT∗ PTQTs 20 5 1351270548 20 10 79842543 20 20 58713453 20 40 48543638 20 80 51524455 20 320 50495954 20 640 35364744 40 5 80641247 40 10 52492352 40 20 62693540 40 40 39533637 40 80 56453839 40 320 50475257 40 640 51524240 80 5 46371652 80 10 56511947 80 20 49573855 80 40 47513144 80 80 53504848 80 320 33374547 80 640 40414338 160 5 40141351 160 10 47392660 160 20 45452937 160 40 63553351 160 80 55523449 160 320 46483351 160 640 53514142 320 5 41170849 320 10 48281456 320 20 46363844 320 40 50443860 320 80 58614561 320 320 64654260 320 640 52555347 640 5 42221258 640 10 49201750 640 20 40393248 640 40 53463760 640 80 44484960 640 320 47453563 640 640 52565153 Quantitative Economics 7 (2016) Pooling data across markets 543 Table 3. Monte Carlo results: λ=05. MT TPT∗ PTQTs 20 5 103832959 20 10 6574202137 20 20 278274639235 20 40 797761979477 20 80 999998 1000724 20 320 1000 1000 1000971 20 640 1000 1000 1000982 40 5 47416981 40 10 7455378158 40 20 446362890365 40 40 974943999644 40 80 1000 1000 1000838 40 320 1000 1000 1000981 40 640 1000 1000 1000998 80 5 3323124103 80 10 10858643279 80 20 685555991532 80 40 1000999 1000843 80 80 1000 1000 1000958 80 320 1000 1000 1000999 80 640 1000 1000 1000999 160 5 2909228187 160 10 12458895489 160 20 923786 1000826 160 40 1000 1000 1000959 160 80 1000 1000 1000995 160 320 1000 1000 1000 1000 160 640 1000 1000 1000 1000 320 5 2211449352 320 10 20868995771 320 20 997963 1000980 320 40 1000 1000 1000 1000 320 80 1000 1000 1000 1000 320 320 1000 1000 1000 1000 320 640 1000 1000 1000 1000 640 5 1506780693 640 10 332105 1000980 640 20 1000 1000 1000 1000 640 40 1000 1000 1000 1000 640 80 1000 1000 1000 1000 640 320 1000 1000 1000 1000 640 640 1000 1000 1000 1000 544 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) Table 4. Monte Carlo results: λ=09. MT TPT∗ PTQTs 20 5 107602464 20 10 6548114147 20 20 117128301200 20 40 327353646292 20 80 758765942446 20 320 1000 1000 1000715 20 640 1000 1000 1000829 40 5 452531100 40 10 5442190170 40 20 160148455260 40 40 490501871415 40 80 935925999588 40 320 1000 1000 1000894 40 640 1000 1000 1000942 80 5 341743107 80 10 5932289211 80 20 233197713339 80 40 728732980535 80 80 997996 1000732 80 320 1000 1000 1000952 80 640 1000 1000 1000984 160 5 400988168 160 10 6021466289 160 20 382306925470 160 40 934924 1000681 160 80 1000 1000 1000857 160 320 1000 1000 1000989 160 640 1000 1000 1000995 320 5 2910147222 320 10 9137733450 320 20 602497999686 320 40 998997 1000884 320 80 1000 1000 1000967 320 320 1000 1000 1000997 320 640 1000 1000 1000 1000 640 5 2705321409 640 10 12348933699 640 20 866772 1000910 640 40 1000 1000 1000982 640 80 1000 1000 1000 1000 640 320 1000 1000 1000 1000 640 640 1000 1000 1000 1000 Quantitative Economics 7 (2016) Pooling data across markets 545 5. Empirical application Recently, a number of empirical papers have applied a dynamic game to data and estimate parameters of the game using two-step methods. These papers include Ryan (2012), Collard-Wexler (2013), Sweeting (2013), Beresteanu, Ellickson, and Misra (2010), and the empirical section of Aguirregabiria and Mira (2007), among others. Panel data frequently contain a number of markets over a relatively short time period. Researchers tend to pool different markets together to estimate policy functions in the first stage. To do this pooling, an important assumption is that a single equilibrium is played in every market. This section tests the homogeneity hypotheses for poolability using the data of Ryan (2012). We chose Ryan (2012) because it is one of a few papers already published and because the number of state variables is relatively small so that it fits our illustrative purpose well. To evaluate the welfare costs of the 1990 Amendments to the Clean Air Act on the Portland cement industry in the United States, Ryan (2012) develops a dynamic oligopoly model based on Ericson and Pakes (1995) and estimates the model using a two-step method developed by Bajari, Benkard, and Levin (2007). In his application, there are 23 geographically separated markets. To estimate firms’ policy functions in the first stage, Ryan (2012) assumes that the data are generated by a single Markov perfect equilibrium. We apply our test to check this assumption. One caveat is that we use a discrete state space framework, while Ryan (2012) uses a continuous state space. Thus, we have to discretize the state variables in Ryan’s (2012) application to perform the test. For a fine grid, however, little differences between the two frameworks are expected in practice. We first summarize Ryan’s (2012) model. Then we explain the procedure of our test in this context. 5.1 Ryan’s (2012)model Ryan (2012)assumesthatNfirms play a dynamic oligopoly game in each regional cement market. Firms make decisions to maximize the discounted sum of expected profits. The timing of the decisions is as follows. At the beginning of each period, incumbent firms draw a private scrap value and decide whether to exit the market or not. Then potential entrants receive a private draw of entry costs and investment costs. At the same time, incumbent firms that have not decided to exit the market draw private costs of investment and divestment. Then all entry and investment decisions are made simultaneously. Firms compete in the product market and profits are realized. Finally, firms enter and exit, and their capacity levels change according to the investment/divestment decisions in this period. Let s=(s1sN)∈Sbe the capacity levels of Nfirms and let εibe a vector of all private shocks to firm i. Assuming that εiis i.i.d. over time and focusing on pure Markovian strategies, firm i’s strategy is a mapping from states and private shocks to actions. The game payoff for firm iis defined as the discounted sum of expected period payoffs given the beliefs now and in the future. The collection of strategies and beliefs is a Markov perfect equilibrium if (i) for all i,firmi’s strategy is a best response to its rivals’ strategies 546 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) given the beliefs at all states s∈Sand (ii) for all i, the beliefs of firm iare consistent with the strategies. The existence of pure strategy equilibria in a class of dynamic games is provided in Doraszelski and Satterthwaite (2010). The model of Ryan (2012) also falls in this class. Furthermore, multiplicity of equilibria is prevalent. Ryan (2012) follows the two-step method developed by Bajari, Benkard, and Levin (2007). In the first stage, Ryan (2012) estimates the entry, exit, and investment policies as a function of states. Because of the issue of multiplicity, different equilibria may be played in different markets. However, since Ryan (2012)hasonly19 years of time series compared to a large state space, estimating policy functions market by market is not practical. Thus, he imposes the following assumption: Assumption 1. The same equilibrium is played in all markets. Based on this assumption Ryan pools all markets when estimating policy functions. Our aim is to test the validity of this assumption. In addition to Assumption 1,Ryan (2012) assumes flexible functional forms for the policy functions. First, the probability of entry is modeled as a probit regression, Pr{firm ienters in period t|si=0s}(16) =ψ1+ψ2 j=i st j+ψ31{t>1990} where (·)is the cumulative distribution function (c.d.f.) of the standard normal. The dummy 1{t>1990}is introduced to account for the change in firms’ behavior after the introduction of the 1990 Amendments. Second, the exit probability is also modeled as a probit, Pr{firm iexits in period t|si>0s}(17) =ψ4+ψ5st i+ψ6 j=i st j+ψ71{t>1990} Finally, the investment policy is modeled using the empirical model of the (Ss) rule by Attanasio (2000). Specifically, firms adjust the current capacity level to a target level of capacity when current capacity exceeds one of the bands around the target level. The target level s∗t iis given by lns∗t i=λ 1b1st i+λ 2b2 j=i st j+u∗t i(18) where u∗t iis i.i.d. normal with zero mean and a homoscedastic variance, and the functions b1(·)and b2(·)denoteacubicb-spline, which is to capture flexible functional forms in the variables st iand j=ist j, respectively. The lower and upper bands are given by st i=s∗t i−expλ 3b1st i+λ 4b2 j=i st j+ubt i(19) Quantitative Economics 7 (2016) Pooling data across markets 553 Appendix A.1 Proofs Proof of Proposition 1. We first consider the statistic TPfor Hσ 0defined by (5), that is, TP= M  j=1 (as)∈A×Sfj(as)−fj(s)σ(a|s)2 fj(s)σ(a|s) where σ(a|s)=M j=1fj(as) M j=1fj(s).Letξj(as)={fj(as)−fj(s)σj(a|s)}/fj(s)1/2and define the (mams)-dimensional vector ξj={ξj(as)a∈A}∈s∈S. Since at j|st jis conditionally independent from past values, the Markov chain Pis stationary, and all states of Pcommunicate, the same argument in the proof of Billingsley (1961, Theorem 3.1) implies ξj d →N0diagVj(s)s∈S for each j=1M,where[Vj(s)](kl) =1{k=l}σj(ak|s)−σj(ak|s)σj(al|s)for kl = 1ma. Thus, we obtain  (as)∈A×Sfj(as)−fj(s)σj(a|s)2 fj(s)σj(a|s) d →χ2ms(ma−1)(24) for each j=1M. Note that under the setup of Section 2,σ(a|s)is the maximum likelihood estimator of σ(a|s)under Hσ 0:σ1=···=σM=σusing the full sample (at jst j)t=1T for j=1M. Therefore, based on (24), the asymptotic theory of the chi-squared statistic (e.g., Lemma 17.3 of van der Vaart (1998)) implies the conclusion. We now consider the statistic TPfor HP 0defined by (6), that is, TP= M  j=1 (ss)∈S×Sf1 jss−f1 j(s) ps|s2 f1 j(s) ps|s where  p(s|s)=M j=1f1 j(ss) M j=1f1 j(s).Inthiscase,Billingsley (1961, Theorem 3.1) directly implies the asymptotic normality of {f1 j(ss)−f1 j(s)pj(s|s)}/f 1 j(s)1/2. Thus, a similar argument yields the conclusion.  Proof of Proposition 2. WeprovetheoptimalityforT∗ Pto test HP 0. The case for testing Hσ 0is shown in the same manner although the notation becomes more complicated. Let ωj=(s1 jsT j)∈Ωjand Ω=Ω1×···×ΩMbe the sample space of the observables ω=(ω1ωM). The sample space Ωis partitioned into different types {Λl}l=1L,where{Λl}l=1L is a collection of disjoint subsets of Ωsatisfying Ω= 554 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) L l=1Λland any element in Λlyields the same joint counts {f1 j(··)}j=1M .Atestisdefined as a partition (ΩAΩR)of Ω,whereΩAand ΩRmean the acceptance and rejection regions, respectively. First, we show that for any test (ΩAΩR), there exists a test (˜ ΩA˜ ΩR)based only on the joint counts {f1 j(··)}j=1M such that lim T→∞ 1 T−1logPr˜ ΩR:HP 0≤lim T→∞ 1 T−1logPrΩR:HP 0 (25) lim T→∞ 1 T−1logPr˜ ΩA:HP 1≤lim T→∞ 1 T−1logPrΩA:HP 1 Note that the subset ΩAor ΩRcontains at least half of the elements in Λlfor each l=1L.Thus,forany(ΩAΩR),wecandefine(˜ ΩA˜ ΩR)as follows. For each l= 1L,ifΩA(or respectively ΩR) contains at least half of the elements in Λl, then let ˜ ΩA(or respectively ˜ ΩR) include all elements in Λl.Observethat(˜ ΩA˜ ΩR)depends only on {f1 j(··)}j=1M by construction. Now pick any type Λlsuch that Λl⊂˜ ΩR. It holds PrΩR:HP 0≥PrΩR∩Λl:HP 0≥1 2PrΛl:HP 0(26) =1 2 M  j=1 PrΛlj :HP 0 where the first inequality follows from the set inclusion relationship, the second inequality follows from the facts that at least half of elements of Λlis contained in ΩR(due to Λl⊂˜ ΩR) and that all elements in Λloccur with same probability, and the equality follows from independence of (ω1ωM)and Λl=Λl1×···×ΛM.ByGutman (1989, Lemma 1), if the initial values (s0 1s0 M)are fixed, for any probability measure Pon Ω given by a Markov chain, there exists a positive sequence δT=O(T−1log T)such that exp−(T −1)K(qjlp)+δT≤Pr{Λlj :P}(27) ≤exp−(T −1)K(qjlp)−δT where qjl(··)is the two-period joint empirical measure given by the type Λlj,p(··)is the two-period joint measure given by P,and K(qjlp)= s∈S qjl(s) s∈S qjls|slog qjls|s ps|s is the Kullback–Leibler divergence for qjl and p. Combining (26)and(27), PrΩR:HP 0≥1 2exp−(T −1)M  j=1 K(qjlp)+δ1T (28) Quantitative Economics 7 (2016) Pooling data across markets 555 for some δ1T=O(T−1log T).Herepis the common joint measure under HP 0.Thus,we have Pr˜ ΩR:HP 0=PrΩR:HP 0+ l:Λl⊂˜ ΩR PrΩA∩Λl:HP 0 ≤PrΩR:HP 0+ l:Λl⊂˜ ΩR exp−(T −1)M  j=1 K(qjlp)−δ2T ≤PrΩR:HP 0+ l:Λl⊂˜ ΩR PrΩR:HP 0exp(T −1)δ3T =PrΩR:HP 01+LRexp(T −1)δ3T for some δ2Tδ3T=O(T−1log T), where the first equality follows from the construction of ˜ ΩR, the first inequality follows from Pr{ΩA∩Λl:HP 0}≤Pr{Λl:HP 0}and (27), the second inequality follows from (28), and the last equality follows from the definition of LR= L l=11{Λl⊂˜ ΩR}. Therefore, the first inequality in (25) follows by (T −1)−1logLR→0. The second inequality in (25) is obtained in the same manner (by replacing ˜ ΩR,ΩR,and HP 0with ˜ ΩA,ΩA,andH P 1, respectively). By (25), we can focus on the test defined by the joint counts {f1 j(··)}j=1M . Next we show (10). Pick any test (˜ ΩA˜ ΩR)based only on {f1 j(··)}j=1M that satisfies (9). Then there exists δ4T=O(T−1logT)such that e−α(T−1)≥Pr˜ ΩR:HP 0= l:Λl⊂˜ ΩR M  j=1 PrΛlj :HP 0(29) ≥exp−(T −1)M  j=1 K(qjlp)+δ4T for any lsatisfying Λl⊂˜ ΩRand all Tlarge enough, where the first inequality follows from (9), the equality follows from independence of (ω1ωM)and Λl=Λl1×···× ΛMand the fact that ˜ ΩRdepends only on the types, and the second inequality follows from (27). Thus, if the rejection by ˜ ΩRoccurs, then the observed joint empirical measure {qj}j=1M satisfies (29), and setting pas the joint empirical measure qtotal(··)= 1 M(T−1)M j=1f1 j(··)in (29)implies α−δ4T≤ M  j=1 K(qjqtotal)=T∗ P 2(T −1) for all Tlarge enough, and (10) follows. Finally, we show (8). Define the entropy of a two-period joint measure q(··)as H(q) =− s∈S q(s) s∈S qs|slogqs|s 556 Otsu, Pesendorfer, and Takahashi Quantitative Economics 7 (2016) Then by the definition of K(··), the test statistic is written as T∗ P 2(T −1)=MH(qtotal)− M  j=1 H(qj) (30) Let Ω∗ Rbe the rejection region of the test 1{T∗ P≥2(T −1)(α −δ4T)}.Alsoletqωj j(··)be the two-period joint empirical measure based on ωjand qω total(··)=M−1M j=1qωj j(··). We have PrΩ∗ R:HP 0= ω∈Ω∗ R M  j=1 Prωj:HP 0 ≤ ω∈Ω∗ R exp−(T −1)MHqω total ≤exp−(T −1)(α −δ4T) ω∈Ω∗ R exp−(T −1) M  j=1 Hqωj j ≤exp−(T −1)(α −δ4T)M  j=1 ωj∈Ωj exp−(T −1)Hqωj j ≤exp−(T −1)(α −δ4T)+(T −1)OT−1log T where the equality follows from independence of (ω1ωM), the first inequality follows from the fact that under HP 0the log likelihood M j=1logPr{ωj:HP 0}of observed ωis maximized by qω total with maximum −M(T −1)H(qω total), the second inequality follows from ω∈Ω∗ Rand (30)(i.e.,MH(qω total)−M j=1H(qωj j)≥2(α −δ4T)), the third inequality follows from the Jensen inequality and Ω∗ R⊂Ω, and the last inequality follows from the upper bounds of the entropy and number of types of Markov chains in Davisson, Longo, and Sgarro (1981, Theorem 1 combined with Eq. (4)). Therefore, (8) follows.  A.2 Detail for the test statistic TQ The asymptotic variance Vjin (11) has the (kl)th element vj kl =1{k=l}qj k−qj kqj l+qj k ∞  m=1pj(m) kl −ql+qj l ∞  m=1pj(m) lk −qk qj kis the kth element of Qj,andpj(m) kl is the (k l)th element of (Pj)m. It should be noted that rank(Vj)=ms−1due to the linear constraint (11)Fj=T−1.UnderH Q 0,it holds that V=V1=···=VMand the common asymptotic variance Vcan be estimated Quantitative Economics 7 (2016) Pooling data across markets 557 by, for example, Newey and West’s (1987) estimator  Vwhose (kl)th element is defined as  vkl =1{k=l} qk− qk ql+ qk bT  m=1 p(m) kl − ql+ ql bT  m=1 p(m) lk − qk where qkis the kth element of 1 M(T−1)M j=1Fj, p(m) kl is the (k l)th element of Pm,and P= {1 M(T−1)M j=1f1 j(ss)}ss∈S. Also the bandwidth bTsatisfies bT→∞and T−1/2bT→0. References Aguirregabiria, V. and P. 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