Residual bootstrap tests in linear models with many regressors

Abstract

This paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors. I 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. One of the conditions for these results is that the regressors come from an asymptotically balanced design. Depending on the number of restrictions tested and on the errors’ distribution, violation of that condition might render the bootstrap tests asymptotically invalid. In that case, I propose bootstrapping Calhoun’s (2011) G statistic or modified versions of the LR and LM statistics, and show that these procedures remain asymptotically valid.Monte Carlo simulations indicate that the bootstrap tests often outperform asymptotic ones. However, analyzing the approximate third cumulant of the F statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements. Nevertheless, it is found that the bootstrap third cumulant partially matches the population third cumulant, which might explain the bootstrap’s good finite sample performances, especially when the errors come from a symmetric distribution.