GEL-based inference with unconditional moment inequality restrictions

Abstract

This paper studies the properties of generalised empirical likelihood (GEL) methods for the estimation of and inference on partially identi ed parameters in models speci ed by unconditional moment inequality constraints. The central result is, as in moment equality condition models, a large sample equivalence between the scaled optimised GEL objective function and that for generalised method of moments (GMM) with weight matrix equal to the inverse of the e cient GMM metric for moment equality restrictions. Consequently, the paper provides a generalisation of results in the extant literature for GMM for the non-diagonal GMM weight matrix setting. The paper demonstrates that GMM in such circumstances delivers a consistent estimator of the identi ed set, i.e., those parameter values that satisfy the moment inequalities, and derives the corresponding rate of convergence. Based on these results the consistency of and rate of convergence for the GEL estimator of the identi ed set are obtained. A number of alternative equivalent GEL criteria are also considered and discussed. The paper proposes simple conservative consistent con dence regions for the identi ed set and the true parameter vector based on both GMM with a non-diagonal weight matrix and GEL. A simulation study examines the e cacy of the non-diagonal GMM and GEL procedures proposed in the paper and compares them with the standard diagonal GMM method.