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.

The Empirical Saddlepoint Estimator

We define a moment-based estimator that maximizes the empirical saddlepoint (ESP) approximation of the distribution of solutions to empirical moment conditions. We call it the ESP estimator. We prove its existence, consistency and asymptotic normality, and we propose novel test statistics. We also show that the ESP estimator corresponds to the MM (method of moments) estimator shrunk toward parameter values with lower estimated variance, so it reduces the documented instability of existing moment-based estimators. In the case of just-identified moment conditions, which is the case we focus on, the ESP estimator is different from the MM estimator, unlike the recently proposed alternatives, such as the empirical-likelihood-type estimators

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.Generalized method of moments estimator; test of overidentifying restrictions; sampling distribution; empirical saddlepoint approximation; asymptotic distribution

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

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

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 overidentifying restrictions. Simulations demonstrate that ESPL and CESPL have smaller bias than currently available one-step GMM estimators. The simulations also show new tests for overidentifying 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

The empirical saddlepoint estimator

peer reviewedWe define a moment-based estimator that maximizes the empirical saddlepoint (ESP) approximation of the distribution of solutions to empirical moment conditions. We call it the ESP estimator. We prove its existence, consistency and asymptotic normality, and we propose novel test statistics. We also show that the ESP estimator corresponds to the MM (method of moments) estimator shrunk toward parameter values with lower implied estimated variance, so it reduces the documented instability of existing moment-based estimators. In the case of just-identified moment conditions, which is the case we focus on, the ESP estimator is different from the MM estimator, unlike the more recent alternatives, such as the empirical-likelihood-type estimators