We study the first-order asymptotic properties of a class of estimators of the structural parameters in dynamic discrete choice games. We consider K-stage policy iteration (PI) estimators, where K denotes the number of policy iterations employed in the estimation. This class nests several estimators proposed in the literature. By considering a “pseudo likelihood” criterion function, our estimator becomes the K-PML estimator in Aguirregabiria and Mira (2002, 2007). By considering a “minimum distance” criterion function, it defines a new K-MD estimator, which is an iterative version of the estimators in Pesendorfer and Schmidt-Dengler (2008) and Pakes et al. (2007). First, we establish that the K-PML estimator is consistent and asymptotically normal for any K ∈ ℕ. This complements findings in Aguirregabiria and Mira (2007), who focus on K = 1 and K large enough to induce convergence of the estimator. Furthermore, we show under certain conditions that the asymptotic variance of the K-PML estimator can exhibit arbitrary patterns as a function of K. Second, we establish that the K-MD estimator is consistent and asymptotically normal for any K ∈ ℕ. For a specific weight matrix, the K-MD estimator has the same asymptotic distribution as the K-PML estimator. Our main result provides an optimal sequence of weight matrices for the K-MD estimator and shows that the optimally weighted K-MD estimator has an asymptotic distribution that is invariant to K. The invariance result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for K-PML estimators. Our main result implies two new corollaries about the optimal 1-MD estimator (derived by Pesendorfer and Schmidt-Dengler (2008)). First, the optimal 1-MD estimator is efficient in the class of K-MD estimators for all K ∈ ℕ. In other words, additional policy iterations do not provide first-order efficiency gains relative to the optimal 1-MD estimator. Second, the optimal 1-MD estimator is more or equally efficient than any K-PML estimator for all K ∈ ℕ. Finally, the appendix provides appropriate conditions under which the optimal 1-MD estimator is efficient among regular estimators.
The literature on dynamic discrete games often assumes that the conditional choice probabilities and the state transition probabilities are homogeneous across markets and over time. We refer to this as the "homogeneity assumption" in dynamic discrete games. This homogeneity assumption enables empirical studies to estimate the game's structural parameters by pooling data from multiple markets and from many time periods. In this paper, we propose a hypothesis test to evaluate whether the homogeneity assumption holds in the data. Our hypothesis is the result of an approximate randomization test, implemented via a Markov chain Monte Carlo (MCMC) algorithm. We show that our hypothesis test becomes valid as the (user-defined) number of MCMC draws diverges, for any fixed number of markets, time-periods, and players.We apply our test to the empirical study of the U.S. Portland cement industry in Ryan (2012).
We study the asymptotic properties of a class of estimators of the structural parameters in dynamic discrete choice games. We consider K-stage policy iteration (PI) estimators, where K denotes the number of policy iterations employed in the estimation. This class nests several estimators proposed in the literature. By considering a "pseudo likelihood" criterion function, our estimator becomes the K-PML estimator in Aguirregabiria and Mira (2002, 2007). By considering a "minimum distance" criterion function, it defines a new K-MD estimator, which is an iterative version of the estimators in Pesendorfer and Schmidt-Dengler (2008) and Pakes et al. (2007).First, we establish that the K-PML estimator is consistent and asymptotically normal for any K. This complements findings in Aguirregabiria and Mira (2007), who focus on K = 1 and K large enough to induce convergence of the estimator. Furthermore, we show under certain conditions that the asymptotic variance of the K-PML estimator can exhibit arbitrary patterns as a function of K.Second, we establish that the K-MD estimator is consistent and asymptotically normal for any K. For a specific weight matrix, the K-MD estimator has the same asymptotic distribution as the K-PML estimator. Our main result provides an optimal sequence of weight matrices for the K-MD estimator and shows that the optimally weighted K-MD estimator has an asymptotic distribution that is invariant to K. The invariance result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for K-PML estimators. Our main result implies two new corollaries about the optimal 1-MD estimator (derived by Pesendorfer and Schmidt-Dengler (2008)). First, the optimal 1-MD estimator is optimal in the class of K-MD estimators for all K. In other words, additional policy iterations do not provide asymptotic efficiency gains relative to the optimal 1-MD estimator. Second, the optimal 1-MD estimator is more or equally asymptotically efficient than any K-PML estimator for all K. Finally, the appendix provides appropriate conditions under which the optimal 1-MD estimator is asymptotically efficient. We thank the editor and three anonymous referees for comments and suggestions that have greatly improved the manuscript. We also thank Peter Arcidiacono, Patrick Bayer, Joe Hotz, Matt Masten, Arnaud Maurel, Takuya Ura, and seminar participants at various institutions for useful comments and suggestions. Of course, all errors are our own.
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