Linear Regression with Many Included Covariates

We consider asymptotic inference for linear regression coefficients when the number of included covariates grows as fast as the sample size. We find a limiting normal distribution with asymptotic variance that is larger than the usual one. We also find that all of the usual versions of heteroskedasticity consistent standard error estimators are inconsistent under this asymptotics. The problem with these standard errors is that they do not make a correct “degrees of freedom” adjustment. We 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 many covariates. We illustrate our findings in three distinct settings: (i) parametric linear models with many covariates, (ii) semiparametric semilinear models with many technical regressors, and (iii) linear panel models with many fixed effects.