Econometric Estimation with High-Dimensional Moment Equalities

Abstract

We consider a nonlinear structural model in which the number of moments is not limited by the sample size. The econometric problem here is to estimate and perform inference on a finite-dimensional parameter. To handle the high dimensionality, we must systematically choose a set of informative moments; in other words, delete the uninformative ones. In nonlinear models, a consistent estimator is a prerequisite for moment selection. We develop in this paper a novel two-step procedure. The first step achieves consistency in high-dimensional asymptotics by relaxing the moment constraints of empirical likelihood. Given the consistent estimator, in the second step we propose a computationally efficient algorithm to select the informative moments from a vast number of candidate combinations, and then use these moments to correct the bias of the first-step estimator. We show that the resulting second-step estimator is root-n-asymptotic normal, and achieves the lowest variance under a sparsity condition. To the best of our knowledge, we provide the first asymptotically normally distributed estimator in such an environment. The new estimator is shown to have favorable finite sample properties in simulations, and it is applied to estimate an international trade model with massive China datasets.