Efficient GMM and Weighting Matrix under Misspecification

TL;DR

This paper introduces a new class of efficient GMM estimators that are robust to misspecification, optimizing asymptotic variance and including bootstrap and split-sample methods, with theoretical and empirical validation.

Contribution

It develops a misspecification-efficient GMM estimator with optimal weighting, recentering, and robust bootstrap methods, improving over standard GMM under misspecification.

Findings

The ME estimator achieves the smallest asymptotic variance under misspecification.

The asymptotic variance reduces to the efficient-GMM formula in linear models.

Simulation and empirical examples demonstrate the effectiveness of the proposed methods.

Abstract

This paper develops efficient GMM estimation when the moment conditions are misspecified. We observe that the influence function of the standard GMM estimator under misspecification depends on both the original moment conditions and their Jacobian, motivating a new class of estimators based on augmented moment conditions with recentering. The standard GMM estimator is a special case within this class, and generally suboptimal. By optimally weighting the augmented system, we obtain a misspecification-efficient (ME) estimator with the smallest asymptotic variance for the same GMM pseudo-true value. In linear models, the asymptotic variance of ME estimator reduces to the textbook efficient-GMM variance formula $(G'W^{*}G)^{-1}$, where $W^{*}$ is the inverse of the variance of residualized moments after projection on the Jacobian $G$. We consider a feasible double-recentered bootstrap estimator, which can be considered as a misspecification-robust and efficient version of Hall and Horowitz (1996) recentered bootstrap GMM estimator, and also consider a split-sample ME estimator. Finally, we establish uniform local asymptotic minimax bounds over a class of weighting matrices. We illustrate the proposed methods in simulation and empirical examples.