Bootstrap tests in linear models with many regressors

This paper is concerned with bootstrap hypothesis testing in high dimensional linear regression models. Using a theoretical framework recently introduced by Anatolyev (2012), we show that bootstrap F, LR and LM tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size. These results are derived for models with iid error terms, but Monte Carlo evidence suggests that they extend to the wild bootstrap in the presence of heteroskedasticity and to bootstrap methods for heavy tailed data.