On GMM Inference: Partial Identification, Identification Strength, and Non-Standard

This paper analyses aspects of GMM inference in moment equality models when the moment Jacobian is allowed to be rank deficient. In this setting first order identification may fail, and the singular values of the Jacobian are not constrained, thereby allowing for varying levels of identification strength. No specific structure is imposed on the functional form of the moment conditions, the long-run variance of the moment conditions can be singular, and the GMM criterion function weighting matrix may also be chosen sub-optimally. Explicit analytic formulations for the asymptotic distributions of estimable functions of the resulting GMM estimator and the asymptotic distributions of GMM criterion test statistics are derived under relatively mild assumptions. The distributions can be computed using standard software without recourse to bootstrap or simulation methods. The practical operation of the theoretical results, and the relationship between lack of identification and identification strength, is illustrated via numerical examples involving instrumental variables estimation of a structural equation with endogenous regressors. The results suggest that although the presence and origin of identification problems can in practice be obscure, the applied researcher can take comfort from the fact that probabilities and quantile values calculated using the new asymptotic sampling distributions of statistics constructed from the standard GMM criterion function will give accurate approximations in the presence of identification issues, irrespective of the latter's source.