Bootstrap Inference in the Presence of Bias

AbstractWe 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.Keywords: Asymptotic biasbootstrapincidental parameter biasmodel averagingnonparametric regressionprepivotingDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgementsWe thank Federico Bandi, Matias Cattaneo, Christian Gourieroux, Philip Heiler, Michael Jansson, Anders Kock, Damian Kozbur, Marcelo Moreira, David Preinerstorfer, Mikkel Sølvsten, Luke Taylor, Michael Wolf, and participants at the AiE Conference in Honor of Joon Y. Park, 2022 Conference on Econometrics and Business Analytics (CEBA), 2023 Conference on Robust Econometric Methods in Financial Econometrics, 2022 EC2 conference, 2nd ‘High Voltage Econometrics’ workshop, 2023 IAAE Conference, 3rd Italian Congress of Econometrics and Empirical Economics, 3rd Italian Meeting on Probability and Mathematical Statistics, 19th School of Time Series and Econometrics, Brazilian Statistical Association, 2023 Société Canadienne de Sciences Économiques, 2022 Virtual Time Series Seminars, as well as seminar participants at Aarhus University, CREST, FGV - Rio, FGV - São Paulo, Ludwig Maximilian University of Munich, Queen Mary University, Singapore Management University, UFRGS, University of the Balearic Islands, University of Oxford, University of Pittsburgh, University of Victoria, York University, for useful comments and feedback. Cavaliere thanks the the Italian Ministry of University and Research (PRIN 2017 Grant 2017TA7TYC) for financial support. Gonçalves thanks the Natural Sciences and Engineering Research Council of Canada for financial support (NSERC grant number RGPIN-2021-02663). Nielsen thanks the Danish National Research Foundation for financial support (DNRF Chair grant number DNRF154).Conflict of interest statementThe authors report there are no competing interests to declare.Notes1 Note that we write Tn∗−B̂n→d∗pξ1 to mean that Tn∗−B̂n has (conditionally on Dn ) the same asymptotic distribution function as the random variable ξ1. We could alternatively write that Tn∗−B̂n→d∗pξ1∗ and Tn−Bn→dξ1 where ξ1∗ and ξ1 are two independent copies of the same distribution, i.e. P(ξ1≤u)= P(ξ1∗≤u). We do not make this distinction because we care only about distributional results, but it should be kept in mind.2 The same result follows in terms of weak convergence in distribution of Tn∗|Dn. Specifically, because Tn∗=(Tn∗−B̂n)+(B̂n−Bn)+Bn, where Tn∗−B̂n→d∗pξ1∗ and (jointly) B̂n−Bn→dξ2 with ξ1∗∼ξ1 independent of ξ2, we have that Tn∗|Dn→w(B+ξ1∗+ξ2)|ξ2.