Structural Models Involving Highly Dimensional Fixed Point Problems: An Asymptotically Efficient Two-Stage Estimator

In this paper we consider a class of structural econometric models in which the distribution of the endogenous variables can be defined as the solution of a fixed-point problem. We propose a two-stage estimator which avoids repeated solution of the fixed point problem. The idea behind our estimator builds on previous work by Hotz and Miller (1993), Manski (1993) and Aguirregabiria and Mira (1999) in the context of dynamic discrete choice models. The first stage involves non-parametric estimation of the distribution of endogenous variables. In the second stage a pseudo-likelihood function is maximized. Given parameter values, the pseudo-likelihood function is computed by a single iteration on the fixed point operator, using the first stage non-parametric estimates as starting values. Our two-stage estimator is the value of the structural parameters that maximizes this pseudo-likelihood function. We state regularity conditions for consistency of this estimator and provide additional sufficient conditions under which the two-stage estimator can be used instead of a maximum likelihood estimator with no loss of asymptotic efficiency. We also show that these sufficient conditions will be satisfied if a Newton operator is used to solve the fixed point problem. Finally, we study the performance of this estimator in finite samples for several specific examples: a dynamic programming discrete choice model, a model of oligopolistic competition in a differentiated product market and a model of irreversible investment.