The Empirical Saddlepoint Approximation for GMM Estimators

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, the empirical saddlepoint distribution converges to the familiar asymptotic normal distribution. In small sample Monte Carlo simulations, the empirical saddlepoint performs as well as, and often better than, the bootstrap. The formulas necessary to transform the GMM moment conditions to the estimation equations needed for the saddlepoint approximation are provided. Unlike the absolute errors associated with the asymptotic normal distributions and the bootstrap, the empirical saddlepoint has a relative error. The relative error leads to a more accurate approximation, particularly in the tails.