  The linear regression model is widely used in empirical work in Economics,
Statistics, and many other disciplines. Researchers often include many
covariates in their linear model specification in an attempt to control for
confounders. We give inference methods that allow for many covariates and
heteroskedasticity. Our results are obtained using high-dimensional
approximations, where the number of included covariates are allowed to grow as
fast as the sample size. We find that all of the usual versions of Eicker-White
heteroskedasticity consistent standard error estimators for linear models are
inconsistent under this asymptotics. We then propose a new heteroskedasticity
consistent standard error formula that is fully automatic and robust to both
(conditional)\ heteroskedasticity of unknown form and the inclusion of possibly
many covariates. We apply our findings to three settings: parametric linear
models with many covariates, linear panel models with many fixed effects, and
semiparametric semi-linear models with many technical regressors. Simulation
evidence consistent with our theoretical results is also provided. The proposed
methods are also illustrated with an empirical application.
