This paper studies the Generalized Method of Moments (GMM) estimation and inference problem that occurs when the Jacobian of the moment conditions is degenerate. Dovonon and Renault (2013, Econometrica) recently raised a local identification issue stemming from this degenerate Jacobian. The local identification issue leads to a slow rate of convergence of the GMM estimator and a non-standard asymptotic distribution of the over-identification tests. We show that the degenerate Jacobian matrix may contain non-trivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identification tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identification tests are investigated under the same simulation designs used in Dovonon and Renault (2013).

This is joint work with Zhipeng Liao (UCLA).