  High-dimensional linear models with endogenous variables play an increasingly
important role in recent econometric literature. In this work we allow for
models with many endogenous variables and many instrument variables to achieve
identification. Because of the high-dimensionality in the second stage,
constructing honest confidence regions with asymptotically correct coverage is
non-trivial. Our main contribution is to propose estimators and confidence
regions that would achieve that. The approach relies on moment conditions that
have an additional orthogonal property with respect to nuisance parameters.
Moreover, estimation of high-dimension nuisance parameters is carried out via
new pivotal procedures. In order to achieve simultaneously valid confidence
regions we use a multiplier bootstrap procedure to compute critical values and
establish its validity.
