Imposing equilibrium restrictions in the estimation of dynamic discrete games

Abstract

Imposing equilibrium restrictions provides substantial gains in the estimation of dynamic discrete games. Estimation algorithms imposing these restrictions have different merits and limitations. Algorithms that guarantee local convergence typically require the approximation of high‐dimensional Jacobians. Alternatively, the Nested Pseudo‐Likelihood (NPL) algorithm is a fixed‐point iterative procedure, which avoids the computation of these matrices, but—in games—may fail to converge to the consistent NPL estimator. In order to better capture the effect of iterating the NPL algorithm in finite samples, we study the asymptotic properties of this algorithm for data generating processes that are in a neighborhood of the NPL fixed‐point stability threshold. We find that there are always samples for which the algorithm fails to converge, and this introduces a selection bias. We also propose a spectral algorithm to compute the NPL estimator. This algorithm satisfies local convergence and avoids the approximation of Jacobian matrices. We present simulation evidence and an empirical application illustrating our theoretical results and the good properties of the spectral algorithm.

Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

“…These had a mean and MSE of −0.9991 and 0.9991, respectively. Aguirregabiria and Marcoux (2019) explain why the estimates converge to "good" values in some samples even though the equilibrium generating the data is unstable.…”

“…Aguirregabiria and Mira (2007) show that k-NPL estimates are in general not efficient for k ≤ ∞, although they show that the ∞-NPL estimator outperforms the 1-NPL estimator in efficiency when both are consistent. Pesendorfer and Schmidt-Dengler (2010), Kasahara and Shimotsu (2012), Egesdal, Lai, and Su (2015), and Aguirregabiria and Marcoux (2019) show that the sequence may fail to converge to the equilibrium that generated the data, even with very good starting values, so that ∞-NPL may not be consistent.…”

“…Recently, Aguirregabiria and Marcoux (2019) studied the convergence properties of NPL and introduced a variation of the NPL algorithm that updates the conditional choice probabilities using spectral methods applied to the NPL mapping. The goal of their algorithm is to improve convergence properties of NPL for unstable fixed points.…”

“…Aguirregabiria and Marcoux (2019) show that this issue can even arise when the data are generated from a stable equilibrium.…”