Wild Bootstrap for Regression Inference

The ordinary (pairs or residual) bootstrap resamples whole observations or residuals as if they were interchangeable, which assumes a constant error variance. The wild bootstrap instead keeps each residual fixed to its own observation and only flips its sign with a random weight, preserving the per-observation variance and making the inference valid under heteroscedasticity.

It is superior precisely when the error variance is non-constant (heteroscedastic) or the data have a cluster structure. In those cases the ordinary bootstrap distorts the standard errors, while the wild bootstrap remains accurate.

It is the variant where the random sign is drawn once per cluster rather than per observation, so all residuals within a group are flipped together. This preserves within-cluster correlation and is the recommended approach when observations are grouped and the number of clusters is small.

Below about 20 observations the residual rescaling becomes unreliable and a standard bootstrap is preferable. Below 10 observations the procedure is essentially meaningless and a permutation test should be used instead.