  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.
