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Posterior distribution of nondifferentiable functions

Kitagawa, T; Montiel Olea, J; Payne, J; Velez, A; (2020) Posterior distribution of nondifferentiable functions. Journal of Econometrics , 217 (1) pp. 161-175. 10.1016/j.jeconom.2019.10.009. Green open access

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Abstract

This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function g(θ), where θ is a finite-dimensional parameter of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable estimator θ^n, its bootstrap approximation, and the Bayesian posterior for θ all agree asymptotically. It is shown that whenever g is locally Lipschitz, though not necessarily differentiable, the posterior distribution of g(θ) and the bootstrap distribution of g(θ^n) coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for g(θ) as approximately valid posterior inference in a large sample. Another implication—built on known results about bootstrap inconsistency—is that credible intervals for a nondifferentiable parameter g(θ) cannot be presumed to be approximately valid confidence intervals (even when this relation holds true for θ).

Type: Article
Title: Posterior distribution of nondifferentiable functions
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.jeconom.2019.10.009
Publisher version: https://doi.org/10.1016/j.jeconom.2019.10.009
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Bootstrap, Bernstein–von Mises Theorem, Directional Differentiability, Posterior Inference
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL SLASH
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of S&HS
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of S&HS > Dept of Economics
URI: https://discovery.ucl.ac.uk/id/eprint/10085670
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