This paper develops efficient GMM estimation under misspecified moment conditions. The influence function of the standard GMM estimator under misspecification depends on both the moment conditions and their Jacobian, motivating a new class of estimators based on augmented, recentered moment conditions. Standard GMM is a special case 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 estimand. In linear models, its asymptotic variance takes the textbook form $(G'W^{*}G)^{-1}$, where $W^{*}$ is the inverse variance of the residualized moments after projection on the Jacobian $G$. We propose a feasible bootstrap estimator, which is a misspecification-robust and efficient version of the Hall and Horowitz (1996) bootstrap GMM estimator. Finally, we establish uniform local asymptotic minimax bounds over a class of weighting matrices. We illustrate the proposed methods in simulation and two empirical examples.

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