The large sample behaviour of the generalized method of moments estimator in misspecified models

This paper presents the limiting distribution theory for the GMM estimator when the estimation is based on a population moment condition which is subject to non-local (or fixed) misspecification. It is shown that if the parameter vector is overidentified then the weighting matrix plays a far more fundamental role than it does in the corresponding analysis for correctly specified models. Specifically, the rate of convergence of the estimator depends on the rate of convergence of the weighting matrix to its probability limit. The analysis is presented for four particular choices of weighting matrix which are commonly used in practice. In each case the limiting distribution theory is different, and also different from the limiting distribution in a correctly specified model. Statistics are proposed which allow the researcher to test hypotheses about the parameters in misspecified models.