Block Bootstraps for Time Series With Fixed Regressors

This article examines block bootstrap methods in linear regression models with weakly dependent error variables and nonstochastic regressors. Contrary to intuition, the tapered block bootstrap (TBB) with a smooth taper not only loses its superior bias properties but may also fail to be consistent in the regression problem. A similar problem, albeit at a smaller scale, is shown to exist for the moving and the circular block bootstrap (MBB and CBB, respectively). As a remedy, an additional block randomization step is introduced that balances out the effects of nonuniform regression weights, and restores the superiority of the (modified) TBB. The randomization step also improves the MBB or CBB. Interestingly, the stationary bootstrap (SB) automatically balances out regression weights through its probabilistic blocking mechanism, without requiring any modification, and enjoys a kind of robustness. Optimal block sizes are explicitly determined for block bootstrap variance estimators under regression. Finite sample performance and practical uses of the methods are illustrated through a simulation study and two data examples, respectively. Supplementary materials are available online.