GMM Overidentification Test with First Order Underidentification

This paper revisits the asymptotic theory of GMM when the moment conditions identify a unique parameter true value $\theta^0$ but the rank condition of the Jacobian matrix at $\theta^0$ fails. The possibility in case of nonlinear moment restrictions of such simultaneous global identification but first order under-identification has already been pointed out by Sargan (1983). The contribution of this paper is to provide a general asymptotic theory when one can maintain an assumption of second order identification. While this issue has been addressed in a maximum likelihood context by Lee and Chesher (1986) and Rotnitzky, Cox, Bottai and Robbins (2000), we set the focus on the asymptotic behaviour of the GMM overidentification test statistic $J_T$. We show that with $H$ moment conditions, when the Jacobian matrix of the moment conditions evaluated at $\theta^0$ is of rank $p-1$, where $p$ is the number of parameters, the asymptotic distribution of $J_T$ is a half-half mixture of $\chi^2_{H-p}$ and $\chi^2_{H-(p-1)}$ instead of the standard $\chi^2_{H-p}$. In other words, the distribution of $J_T$ for large sample sizes $T$ has fatter tails, leading to over-rejection of the null of valid moments when using standard critical values. The practical significance of this oversize problem is illustrated by Monte Carlo experiments in the context of a test for common GARCH features proposed by Engle and Kozicki (1993).