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Regularizing Priors For Linear Inverse Problems

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  • Florens, Jean-Pierre
  • Simoni, Anna

Abstract

This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions such that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

Suggested Citation

  • Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
  • Handle: RePEc:cup:etheor:v:32:y:2016:i:01:p:71-121_00
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    1. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
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    Cited by:

    1. Jan Johannes & Anna Simoni & Rudolf Schenk, 2020. "Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models," Annals of Economics and Statistics, GENES, issue 137, pages 83-116.
    2. Siddhartha Chib & Minchul Shin & Anna Simoni, 2022. "Bayesian estimation and comparison of conditional moment models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 740-764, July.
    3. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
    4. Florens, Jean-Pierre & Simoni, Anna, 2012. "Nonparametric estimation of an instrumental regression: A quasi-Bayesian approach based on regularized posterior," Journal of Econometrics, Elsevier, vol. 170(2), pages 458-475.
    5. Laurent Ferrara & Anna Simoni, 2023. "When are Google Data Useful to Nowcast GDP? An Approach via Preselection and Shrinkage," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(4), pages 1188-1202, October.
    6. Liao, Yuan & Jiang, Wenxin, 2011. "Posterior consistency of nonparametric conditional moment restricted models," MPRA Paper 38700, University Library of Munich, Germany.
    7. Jean-Pierre Florens & Anna Simoni, 2021. "Gaussian Processes and Bayesian Moment Estimation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(2), pages 482-492, March.

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    More about this item

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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