Sequential Estimation of Dynamic Discrete Games: A Comment∗

Aguirregabiria and Mira (2007), henceforth AM (2007), study pseudo maximum likelihood estimators of dynamic games and propose an iterative nested pseudo maximum likelihood method. This comment revisits the asymptotic properties of the sequential method. We illustrate that the method may not be consistent. We provide an example in which the sequential method converges to a fixed number distinct from the true parameter value with probability approaching one. Example. Consider a repeated game with t = 1, 2, ...,∞. Every period t two firms, indexed by i = 1, 2, simultaneously decide whether to be active or not. Firm i’s period payoff is equal to: εi if firm i is active and firm 3 − i is not active; θ + εi if both firms are active; and ε 2 i if firm i is not active. The true parameter θ0 is contained in the interior of a compact interval Θ with Θ = [−1,−10]. The tuple of random variables ( εi , ε 2 i ) are such that the difference εi = εi−εi is drawn independently every period from the distribution function Fα and observed privately by firm i prior to making the choice with