GMM with many weak moment conditions

Using many moment conditions can improve efficiency but makes the usual GMM inferences inaccurate. Two step GMM is biased. Generalized empirical likelihood (GEL) has smaller bias but the usual standard errors are too small in instrumental variable settings. In this paper we give a new variance estimator for GEL that addresses this problem. It is consistent under the usual asymptotics and under many weak moment asymptotics is larger than the usual one, and is consistent. We also show that the Kleibergen (2005) Lagrange multiplier and conditional likelihood ratio statistics are valid under many weak moments. In addition we introduce a jackknife GMM estimator, but find that GEL is asymptotically more efficient under many weak moments. In Monte Carlo examples we find that t-statistics based on the new variance estimator have nearly correct size in a wide range of cases. JEL Classification: C12, C13, C23