This article develops the adaptive elastic net generalized method of moments (GMM) estimator in large-dimensional models with potentially (locally) invalid moment conditions, where both the number of structural parameters and the number of moment conditions may increase with the sample size. The basic idea is to conduct the standard GMM estimation combined with two penalty terms: the adaptively weighted lasso shrinkage and the quadratic regularization. It is a one-step procedure of valid moment condition selection, nonzero structural parameter selection (i.e., model selection), and consistent estimation of the nonzero parameters.

The procedure achieves the standard GMM efficiency bound as if the authors know the valid moment conditions ex ante, for which the quadratic regularization is important. They also study the tuning parameter choice, with which they show that selection consistency still holds without assuming Gaussianity. The authors apply the new estimation procedure to dynamic panel data models, where both the time and cross-section dimensions are large. The new estimator is robust to possible serial correlations in the regression error terms.