Nearly Singular Design in GMM and Generalized Empirical Likelihood Estimators

Abstract:     
This article analyzes Generalized Empirical Likelihood (GEL) estimators and GMM under nearly singular design. This design relaxes the nonsingularity assumption of the limit weight matrix in GMM, and the nonsingularity of the limit variance matrix for the first order conditions in GEL. The sample versions of these matrices are nonsingular, but one or more of the eigenvalues are near zero. In large samples we assume these sample matrices converge to a singular matrix. Usage of the generalized inverses for the sample variance estimate does not solve this problem since the limit variance is singular not the sample version. This kind of problem can result in large size distortions for the overidentifying restrictions test and poor small sample performance of the estimators. This nearly singular design may occur because of the usage of similar instruments in these matrices. However, since the sample versions of these matrices are nonsingular practitioners may ignore deleting the problematic instruments. In this paper, we derive the large sample theory for GMM and GEL estimators under nearly singular design. We show that the rate of convergence of the estimators and the Lagrange Multiplier in GEL is slower than root n. This rate depends on the nature of the problem. However, the limits are the same as in the standard case. The test of overidentifying restrictions has the same limit. We also derive higher order expansions and show that bias converges to zero slowly compared with standard GEL and GMM. Empirical likelihood estimator has still the best bias property among other GEL estimators, this is very clear in the nearly singular design.

Number of Pages in PDF File: 33

Keywords: Singular Matrix, Rate of Convergence, Small Sample Properties

JEL Classification: C13, C30

working papers series