Nonparametric Likelihood Methods for Estimation and Inference in Moment Condition Models with Weak Instruments

In this paper, I examine the properties of the class of generalized empirical likelihood estimators of moment-condition models. These nonparametric likelihood estimators satisfy exactly the moment conditions and automatically remove any bias due to a lack of centering. Moreover, the bias of the empirical likelihood estimator has been found (Newey and Smith (2000)) to be the same as that for the infeasible optimal GMM, where the coefficients of the optimal linear combinations do not have to be estimated. I examine the finite sample properties of these alternative estimators in the presence of weakly identified parameters and show their robustness. Most importantly, the proposed inference procedure does not involve explicit estimation of the variance-covariance matrix, which can be problematic, especially in small samples with dependent data. The confidence sets for the parameters of interest are constructed by inverting the criterion test, whose limiting chi-square distribution at the true values of the parameters is preserved in the presence of weak instruments.