Bootstrap inference in the presence of bias

TL;DR

This paper demonstrates that bootstrap methods, when properly implemented with prepivoting, can provide valid inference even for biased estimators, across various complex models.

Contribution

It introduces a bootstrap approach using prepivoting to achieve valid inference with biased estimators, including practical implementations and general conditions.

Findings

Prepivoting restores bootstrap validity for biased estimators.

Two implementations: plug-in and double bootstrap.

Applicable to diverse models like nonparametric regression and panel data.

Abstract

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.