  This paper concerns statistical inference for the components of a
high-dimensional regression parameter despite possible endogeneity of each
regressor. Given a first-stage linear model for the endogenous regressors and a
second-stage linear model for the dependent variable, we develop a novel
adaptation of the parametric one-step update to a generic second-stage
estimator. We provide conditions under which the scaled update is
asymptotically normal. We then introduce a two-stage Lasso procedure and show
that the second-stage Lasso estimator satisfies the aforementioned conditions.
Using these results, we construct asymptotically valid confidence intervals for
the components of the second-stage regression coefficients. We complement our
asymptotic theory with simulation studies, which demonstrate the performance of
our method in finite samples.
