On the Iterated Estimation of Dynamic Discrete Choice Games[Pseudo maximum likelihood estimation of structural models involving fixed-point problems]

We study the first-order asymptotic properties of a class of estimators of the structural parameters in dynamic discrete choice games. We consider -stage policy iteration (PI) estimators, wheredenotes the number of PIs employed in the estimation. This class nests several estimators proposed in the literature. By considering a “pseudo likelihood” criterion function, our estimator becomes the -pseudo maximum likelihood (PML) estimator in Aguirregabiria and Mira (2002, 2007). By considering a “minimum distance” criterion function, it defines a new -minimum distance (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 -PML estimator is consistent and asymptotically normal for any . This complements findings in Aguirregabiria and Mira (2007), who focus onandlarge enough to induce convergence of the estimator. Furthermore, we show under certain conditions that the asymptotic variance of the -PML estimator can exhibit arbitrary patterns as a function of . Second, we establish that the -MD estimator is consistent and asymptotically normal for any . For a specific weight matrix, the -MD estimator has the same asymptotic distribution as the -PML estimator. Our main result provides an optimal sequence of weight matrices for the -MD estimator and shows that the optimally weighted -MD estimator has an asymptotic distribution that is invariant to . The invariance result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for -PML estimators. Our main result implies two new corollaries about the optimal -MD estimator (derived by Pesendorfer and Schmidt-Dengler (2008)). First, the optimal -MD estimator is efficient in the class of -MD estimators for all . In other words, additional PIs do not provide first-order efficiency gains relative to the optimal -MD estimator. Second, the optimal -MD estimator is more or equally efficient than any -PML estimator for all . Finally, the Appendix provides appropriate conditions under which the optimal -MD estimator is efficient among regular estimators.