Misspecification-Robust Asymptotic and Bootstrap Inference for Nonsmooth GMM

This paper develops an asymptotic distribution theory for Generalized Method of Moments (GMM) estimators, including the one-step and iterated estimators, when the moment conditions are nonsmooth and possibly misspecified. We consider nonsmooth moment functions that are directionally differentiable—such as absolute value functions and functions with kinks—but not indicator functions. While GMM estimators remain √n-consistent and asymptotically normal for directionally differentiable moments, conventional GMM variance estimators are inconsistent under moment misspecification. We propose a consistent estimator for the asymptotic variance for valid inference. Additionally, we show that the nonparametric bootstrap provides asymptotically valid confidence intervals. Our theory is applied to quantile regression with endogeneity under the location-scale model, offering a robust inference procedure for the GMM estimators in Machado and Santos Silva (2019). Simulation results support our theoretical findings.