Convergence Rates of GMM Estimators with Nonsmooth Moments under Misspecification

TL;DR

This paper investigates the convergence rates of GMM estimators with nonsmooth moments under misspecification, revealing slower rates for two-step estimators and unexpected robustness of one-step estimators.

Contribution

It provides new theoretical insights and simulation evidence on the convergence behavior of GMM estimators with nonsmooth moments when models are misspecified.

Findings

Two-step GMM estimators have at most $n^{1/3}$ convergence rate under misspecification.

One-step GMM estimators with identity weights maintain $ oot n$ convergence rate despite misspecification.

Simulations confirm theoretical predictions about convergence rates.

Abstract

The asymptotic behavior of GMM estimators depends critically on whether the underlying moment condition model is correctly specified. Hong and Li (2023, Econometric Theory) showed that GMM estimators with nonsmooth (non-directionally differentiable) moment functions are at best $n^{1/3}$-consistent under misspecification. Through simulations, we verify the slower convergence rate of GMM estimators in such cases. For the two-step GMM estimator with an estimated weight matrix, our results align with theory. However, for the one-step GMM estimator with the identity weight matrix, the convergence rate remains $\sqrt{n}$, even under severe misspecification.