Convergence Rates of GMM Estimators with Nonsmooth Moments under Misspecification

The asymptotic behavior of generalized method of moments (GMM) estimators depends critically on whether the underlying moment condition model is correctly specified. Hong and Li (2024) showed that GMM estimators with nonsmooth (nondirectionally differentiable) moment functions are at best n1/3 consistent under misspecification. Through simulations, we verify the decelerated convergence rate of GMM estimators in such cases. For the two-step GMM estimator with an estimated weight matrix, our results align with the theory. However, for the one-step GMM estimator with the identity weight matrix, the convergence rate remains n even under severe misspecification.