Inference under random limit bootstrap measures

TL;DR

This paper redefines bootstrap validity to include cases where the limit bootstrap measure is random, showing it can still provide reliable inference in large samples under certain conditions.

Contribution

It introduces new conditions for bootstrap validity when the limit distribution is random, expanding the understanding of bootstrap inference in econometrics.

Findings

Bootstrap validity can hold with random limit measures if certain averaging conditions are met.

The framework applies to non-stationary regressors and functional test statistics.

Bootstrap can be valid for conditional inference even with random limit distributions.

Abstract

Asymptotic bootstrap validity is usually understood as consistency of the distribution of a bootstrap statistic, conditional on the data, for the unconditional limit distribution of a statistic of interest. From this perspective, randomness of the limit bootstrap measure is regarded as a failure of the bootstrap. We show that such limiting randomness does not necessarily invalidate bootstrap inference if validity is understood as control over the frequency of correct inferences in large samples. We first establish sufficient conditions for asymptotic bootstrap validity in cases where the unconditional limit distribution of a statistic can be obtained by averaging a (random) limiting bootstrap distribution. Further, we provide results ensuring the asymptotic validity of the bootstrap as a tool for conditional inference, the leading case being that where a bootstrap distribution estimates consistently a conditional (and thus, random) limit distribution of a statistic. We apply our framework to several inference problems in econometrics, including linear models with possibly non-stationary regressors, functional CUSUM statistics, conditional Kolmogorov-Smirnov specification tests, the `parameter on the boundary' problem and tests for constancy of parameters in dynamic econometric models.