An Improved Approximation to the Distributions in GMM Estimation

The empirical saddlepoint distribution provides an approximation to the sampling distributions for the GMM parameter estimates and the statistics that test the overidentifying restrictions. The empirical saddlepoint distribution permits asymmetry, non-normal tails, and multiple modes. If identification assumptions are satisfied as the sample size grows the empirical saddlepoint distributions converges to the familiar asymptotic normal distributions. Formulas are given to transform from the moment conditions used in GMM estimation to the estimation equations needed for the saddlepoint approximation. Unlike the absolute errors associated with the asymptotic normal distributions and the bootstrap, the empirical saddlepoint has a relative error. This provides a more accurate approximation to the sampling distribution, particularly in the tails. The calculation of the empirical saddlepoint approximation is computer intensive. The calculations are comparable to the bootstrap and requires repeatedly solving nonlinear equations. The structure of the saddlepoint equation permit analytically first and second derivatives.