  We consider bootstrap inference for estimators which are (asymptotically)
biased. We show that, even when the bias term cannot be consistently estimated,
valid inference can be obtained by proper implementations of the bootstrap.
Specifically, we show that the prepivoting approach of Beran (1987, 1988),
originally proposed to deliver higher-order refinements, restores bootstrap
validity by transforming the original bootstrap p-value into an asymptotically
uniform random variable. We propose two different implementations of
prepivoting (plug-in and double bootstrap), and provide general high-level
conditions that imply validity of bootstrap inference. To illustrate the
practical relevance and implementation of our results, we discuss five
examples: (i) inference on a target parameter based on model averaging; (ii)
ridge-type regularized estimators; (iii) nonparametric regression; (iv) a
location model for infinite variance data; and (v) dynamic panel data models.
