High Dimensional Linear GMM

This paper proposes a desparsified GMM estimator for estimating high-dimensional regression models allowing for, but not requiring, many more endogenous regressors than observations. We provide finite sample upper bounds on the estimation error of our estimator and show how asymptotically uniformly valid inference can be conducted in the presence of conditionally heteroskedastic error terms. We do not require the projection of the endogenous variables onto the linear span of the instruments to be sparse; that is we do not impose the instruments to be sparse for our inferential procedure to be asymptotically valid. Furthermore, the variables of the model are not required to be sub-gaussian and we also explain how our results carry over to the classic linear dynamic panel data model. Simulations show that our estimator has a low mean square error and does well in terms of size and power of the tests constructed based on the estimator.