 Wild bootstrap

For independent mean-zero errors in a fixed-design regression, form $y_i^{\ast }=x_i^{\top }\hat{\beta }+{\hat{e}}_i{W}_i$, where the independent multipliers have mean zero and variance one, and refit. Rademacher multipliers taking the values $-1$ and $1$ with equal probability are a simple choice. Conditional on the data, the bootstrap errors have mean zero and preserve observation-specific residual variances, supporting heteroscedasticity rather than imposing a common residual distribution. Appropriate leverage corrections and regularity assumptions are required for accurate studentized inference.