The Empirical Saddlepoint Likelihood Estimator Applied to Two-Step GMM

The empirical saddlepoint likelihood (ESPL) estimator is introduced. The ESPL provides improvement over one-step GMM estimators by including additional terms to automatically reduce higher order bias. The first order sampling properties are shown to be equivalent to efficient two-step GMM. New tests are introduced for hypothesis on the model's parameters. The higher order bias is calculated and situations of practical interest are noted where this bias will be smaller than for currently available estimators. As an application, the ESPL is used to investigate an overidentified moment model. It is shown how the model's parameters can be estimated with both the ESPL and a conditional ESPL (CESPL), conditional on the overidentifying restrictions being satisfied. This application leads to several new tests for over-identifying restrictions. Simulations demonstrate that ESPL and CESPL have smaller bias than currently available one-step GMM estimators. The simulations also show new tests for over-identifying restrictions that have performance comparable to or better than currently available tests. The computations needed to calculate the ESPL estimator are comparable to those needed for a one-step GMM estimator.