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Posterior distribution of nondifferentiable functions

Author

Listed:
  • Toru Kitagawa

    (Institute for Fiscal Studies and University College London)

  • Jose Luis Montiel Olea

    (Institute for Fiscal Studies and New York University)

  • Jonathan Payne

    (Institute for Fiscal Studies)

Abstract

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

Suggested Citation

  • Toru Kitagawa & Jose Luis Montiel Olea & Jonathan Payne, 2017. "Posterior distribution of nondifferentiable functions," CeMMAP working papers CWP44/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:44/17
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    File URL: https://www.ifs.org.uk/uploads/CWP441717.pdf
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    Cited by:

    1. Gafarov, Bulat & Meier, Matthias & Montiel Olea, José Luis, 2018. "Delta-method inference for a class of set-identified SVARs," Journal of Econometrics, Elsevier, vol. 203(2), pages 316-327.

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