Second Order Approximations to GMM Statistics

Abstract

This thesis uses second order approximations to study the finite sample behavior of statistics that are employed under the GMM setting.

We present a Nagar (1959) approximation to the MSE of the IV estimates, when the disturbances are elliptically distributed. The accuracy of the approximation is illustrated through a comparison of the Nagar-type expansion with the exact finite sample MSE, that was derived in Knight (1985).

The comparison suggests that second order approximations can be quite accurate, even when the sample size is 60 observations. This, alongside with the fact that exact results are more difficult to derive and harder to interpret, suggests that second-order approximations are powerful alternatives to the standard, first-order, asymptotic approximations.

We proceed by analyzing the finite sample behavior of the LMstatistic, as it is employed under the GMM setting. This is achieved through a second order expansion, known as Edgeworth Expansion, of the distribution of the LM statistic. Our analysis suggests that the passage from the finite to the limiting distribution of the LM test is based on several measures, such as the variance-covariance matrix of the moments and its first derivative, the fourth product moment of the population moment condition, the covariance between the moments and their variance, the number of parameters, and the number of moments. We conclude with a simulation study that illustrates how these measures drive the passage from the finite sample to the asymptotic distribution.