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Regularizing Priors for Linear Inverse Problems

  • Jean-Pierre Florens

    (GREMAQ - Groupe de recherche en économie mathématique et quantitative - CNRS - Institut national de la recherche agronomique (INRA) - UT1 - Université Toulouse 1 Capitole - EHESS - École des hautes études en sciences sociales)

  • Anna Simoni

    ()

    (THEMA - Théorie économique, modélisation et applications - Université de Cergy Pontoise - CNRS)

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    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 distribution 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 so 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.

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    Date of creation: 16 Oct 2013
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    Handle: RePEc:hal:wpaper:hal-00873180
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    1. Xiaohong Chen & Demian Pouzo, 2012. "Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals," Econometrica, Econometric Society, vol. 80(1), pages 277-321, 01.
    2. Stefan Hoderlein & Lars Nesheim & Anna Simoni, 2012. "Semiparametric estimation of random coefficients in structural economic models," CeMMAP working papers CWP09/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Erwann Sbai, 2000. "Identification in Empirical Games," Econometric Society World Congress 2000 Contributed Papers 1896, Econometric Society.
    4. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
    5. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2009. "Regularization of Nonparametric Frontier Estimators," IDEI Working Papers 614, Institut d'Économie Industrielle (IDEI), Toulouse.
    6. Carrasco, Marine & Florens, Jean-Pierre, 2000. "Generalization Of Gmm To A Continuum Of Moment Conditions," Econometric Theory, Cambridge University Press, vol. 16(06), pages 797-834, December.
    7. Xiaohong Chen & Markus Reiss, 2007. "On rate optimality for ill-posed inverse problems in econometrics," CeMMAP working papers CWP20/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. DAROLLES, Serge & FLORENS, Jean-Pierre & RENAULT, Éric, 2002. "Nonparametric Instrumental Regression," Cahiers de recherche 2002-05, Universite de Montreal, Departement de sciences economiques.
    9. Eric Gautier & Yuichi Kitamura, 2009. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Cowles Foundation Discussion Papers 1721, Cowles Foundation for Research in Economics, Yale University.
    10. Richard Blundell & Xiaohong Chen & Dennis Kristensen, 2007. "Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves," Econometrica, Econometric Society, vol. 75(6), pages 1613-1669, November.
    11. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(03), pages 522-545, June.
    12. Liao, Yuan & Jiang, Wenxin, 2011. "Posterior consistency of nonparametric conditional moment restricted models," MPRA Paper 38700, University Library of Munich, Germany.
    13. Florens, Jean-Pierre & Simoni, Anna, 2010. "Regularizing priors for linear inverse problems," TSE Working Papers 10-175, Toulouse School of Economics (TSE).
    14. Florens, Jean-Pierre & Simoni, Anna, 2010. "Regularizing priors for linear inverse problems," IDEI Working Papers 621, Institut d'Économie Industrielle (IDEI), Toulouse.
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