Bruce E. Hansen
"Edgeworth expansions for the Wald and GMM statistics for nonlinear restrictions"
Econometric Theory and Practice, (2006), edited by Dean Corbae, Steven N. Durlauf, and Bruce E. Hansen.
An Edgeworth expansion is derived for the GMM distance statistic for a real-valued nonlinear restriction on a normal linear regression. We also provide a refinement of the Edgeworth expansion for the Wald statistic derived by Park and Phillips (1988). Our calculations show that the leading coefficient is the same in these two expansions, and thus the GMM distance statistic has a superior approximation to the chi-square distribution than does the Wald statistic.
We also update the Monte Carlo simulation of Gregory and Veall (1985) to include both heteroskedasticity-robust covariance matrix estimation and the GMM distance statistic. We find that if the robust covariance matrix is calculated under the null, the GMM statistic has near perfect finite sample Type I error in our experiments, even in sample sizes as small as n=20.
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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111.
Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.