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 "maximum likelihood" criterion function, our estimator becomes the K-ML estimator in Aguirregabiria and Mira (  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) . First, we establish that the K-ML estimator is consistent and asymptotically normal for any K. This complements findings in Aguirregabiria and Mira (2007)  . Furthermore, we show  that the asymptotic variance of the K-ML estimator can exhibit arbitrary patterns as a function  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-ML 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. This new result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for K-ML estimators. Our main result implies two new and important 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-ML estimator for all K  .
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 such as those in Aguirregabiria and Mira ( 2002, 2007) , Pesendorfer and Schmidt-Dengler (2008) , and Pakes et al. (2007). First, we establish that the K-PML estimator is consistent and asymptotically normal for all 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 all 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 . 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 .
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