Higher Order Properties of GM and Generalized Empirical Likelihood Estimators
Whitney Newey, Massachusetts Institute of Technology
Richard Smith, University of Bristol

Abstract

In an effort to improve the small sample properties of GMM, a number of alternative estimators have been suggested. These include the empirical likelihood (EL), continuous updating and exponential tilting estimators. We show that these estimators share a common structure, being members of a class of Generalized Empirical Likelihood (GEL) estimators. We use this structure to compare their higher-order asymptotic properties. We find that the asymptotic bias of EL often does not grow with the number of moment restrictions, while that of GMM and other GEL estimators grows without bound. We also use the formulae to derive bias corrected GMM and GEL estimators. We find that bias corrected EL inherits the higher-order property of maximum likelihood, that is asymptotically efficient relative to the other bias corrected estimators.