# Econometrica

## Journal Of The Econometric Society

### An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: Nov, 2020, Volume 88, Issue 6

# Inference Under Random Limit Bootstrap Measures

https://doi.org/10.3982/ECTA16557
p. 2547-2574

Giuseppe Cavaliere, Iliyan Georgiev

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 nonstationary regressors, CUSUM statistics, conditional Kolmogorov–Smirnov specification tests and tests for constancy of parameters in dynamic econometric models.

## Supplemental Material

### Supplement to "Inference Under Random Limit Bootstrap Measures"

This supplement to Cavaliere and Georgiev (2020), CG hereafter, has four main sections.  In Section S.2 we present the proofs of the results on weak convergence in distribution formulated in Appendix A of CG. In Sections S.3 and S.4 some derivations pertaining to the applications in Sections 2 and 4 of CG are given. Finally, Section S.5 provides a description of the Monte Carlo simulation design used in Section 2 of CG. For notation, see CG. Unless di¤erently specied, all references are to sections, equations and results in CG.

### Supplement to "Inference Under Random Limit Bootstrap Measures"

This zip file contains replication files for the manuscript.