Saddlepoint Approximations and Test Statistics for Accurate Inference in Overidentified Moment Conditions Models

We propose a new class of test statistics inducing accurate dual likelihood ratio tests of parametric constraints in overidentified moment conditions models. These statistics are derived from the dual likelihood implied by the exponent in the saddlepoint approximation of a general GMM estimator and are shown to be asymptotically chi-squared distributed to higher order, with a relative error of order O(1/n). Since these statistics require the knowledge of the moment generating function of the given orthogonality function we introduce an empirical likelihood version of these tests which can be applied in the fully nonparametric setting and which only requires a preliminary GMM parameter estimation to be computed. Therefore, it can be also easily incorporated into available GMM estimation packages. Finally, we provide some numerical Monte Carlo evidence on the accuracy of the new statistics. In these experiments we find that empirical dual likelihood ratio tests provide a higher accuracy than standard GMM test statistics and some recent information theoretic alternatives for a broad class of GMM models.