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Nonparametric Estimation of An Instrumental Regression: A Quasi-Bayesian Approach Based on Regularized Posterior

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  • Florens, Jean-Pierre
  • Simoni, Anna

Abstract

We propose a Quasi-Bayesian nonparametric approach to estimating the structural relationship ' among endogenous variables when instruments are available. We show that the posterior distribution of ' is inconsistent in the frequentist sense. We interpret this fact as the ill-posedness of the Bayesian inverse problem defined by the relation that characterizes the structural function '. To solve this problem, we construct a regularized posterior distribution, based on a Tikhonov regularization of the inverse of the marginal variance of the sample, which is justified by a penalized projection argument. This regularized posterior distribution is consistent in the frequentist sense and its mean can be interpreted as the mean of the exact posterior distribution resulting from a gaussian prior distribution with a shrinking covariance operator.

Suggested Citation

  • Florens, Jean-Pierre & Simoni, Anna, 2010. "Nonparametric Estimation of An Instrumental Regression: A Quasi-Bayesian Approach Based on Regularized Posterior," IDEI Working Papers 622, Institut d'Économie Industrielle (IDEI), Toulouse.
  • Handle: RePEc:ide:wpaper:22800
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    References listed on IDEAS

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    1. Frédérique Fève & Jean-Pierre Florens, 2010. "The practice of non-parametric estimation by solving inverse problems: the example of transformation models," Econometrics Journal, Royal Economic Society, vol. 13(3), pages 1-27, October.
    2. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(01), pages 71-121, February.
    3. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(03), pages 497-521, June.
    4. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521818728.
    5. S. Darolles & Y. Fan & J. P. Florens & E. Renault, 2011. "Nonparametric Instrumental Regression," Econometrica, Econometric Society, vol. 79(5), pages 1541-1565, September.
    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. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521524131.
    8. Florens, Jean-Pierre & Johannes, Jan & Van Bellegem, Sébastien, 2011. "Identification And Estimation By Penalization In Nonparametric Instrumental Regression," Econometric Theory, Cambridge University Press, vol. 27(03), pages 472-496, June.
    9. 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.
    10. Carrasco, Marine & Florens, Jean-Pierre & Renault, Eric, 2007. "Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization," Handbook of Econometrics,in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 77 Elsevier.
    11. Chunrong Ai & Xiaohong Chen, 2003. "Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions," Econometrica, Econometric Society, vol. 71(6), pages 1795-1843, November.
    12. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521818742.
    13. Jean‐Pierre Florens & Jan Johannes & Sébastien Van Bellegem, 2012. "Instrumental regression in partially linear models," Econometrics Journal, Royal Economic Society, vol. 15(2), pages 304-324, June.
    14. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521524117.
    15. Jean-Pierre Florens & Anna Simoni, 2012. "Regularized Posteriors in Linear Ill-Posed Inverse Problems," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(2), pages 214-235, June.
    16. Richard Blundell & James L. Powell, 2001. "Endogeneity in nonparametric and semiparametric regression models," CeMMAP working papers CWP09/01, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    17. Gagliardini, Patrick & Scaillet, Olivier, 2012. "Tikhonov regularization for nonparametric instrumental variable estimators," Journal of Econometrics, Elsevier, vol. 167(1), pages 61-75.
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    1. repec:bpj:jecome:v:6:y:2017:i:1:p:25:n:5 is not listed on IDEAS
    2. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(01), pages 71-121, February.
    3. Xiaohong Chen & Timothy M. Christensen, 2013. "Optimal uniform convergence rates for sieve nonparametric instrumental variables regression," CeMMAP working papers CWP56/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Xiaohong Chen & Yin Jia Jeff Qiu, 2016. "Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 259-290, October.
    5. Hedibert F. Lopes & Nicholas G. Polson, 2014. "Bayesian Instrumental Variables: Priors and Likelihoods," Econometric Reviews, Taylor & Francis Journals, vol. 33(1-4), pages 100-121, June.
    6. Liao, Yuan & Jiang, Wenxin, 2011. "Posterior consistency of nonparametric conditional moment restricted models," MPRA Paper 38700, University Library of Munich, Germany.
    7. Liao, Yuan & Simoni, Anna, 2012. "Semi-parametric Bayesian Partially Identified Models based on Support Function," MPRA Paper 43262, University Library of Munich, Germany.
    8. Ryo Kato & Takahiro Hoshino, 2018. "Semiparametric Bayes Instrumental Variable Estimation with Many Weak Instruments," Discussion Paper Series DP2018-14, Research Institute for Economics & Business Administration, Kobe University.
    9. Centorrino Samuele & Feve Frederique & Florens Jean-Pierre, 2017. "Additive Nonparametric Instrumental Regressions: A Guide to Implementation," Journal of Econometric Methods, De Gruyter, vol. 6(1), pages 1-25, January.
    10. Horowitz, Joel L., 2014. "Adaptive nonparametric instrumental variables estimation: Empirical choice of the regularization parameter," Journal of Econometrics, Elsevier, vol. 180(2), pages 158-173.
    11. Nalan Basturk & Cem Cakmakli & S. Pinar Ceyhan & Herman K. van Dijk, 2014. "On the Rise of Bayesian Econometrics after Cowles Foundation Monographs 10, 14," Tinbergen Institute Discussion Papers 14-085/III, Tinbergen Institute, revised 04 Sep 2014.
    12. Joel L. Horowitz, 2013. "Adaptive nonparametric instrumental variables estimation: empirical choice of the regularization parameter," CeMMAP working papers CWP30/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Nalan Basturk & Cem Cakmakli & S. Pinar Ceyhan & Herman K. van Dijk, 2013. "Historical Developments in Bayesian Econometrics after Cowles Foundation Monographs 10, 14," Tinbergen Institute Discussion Papers 13-191/III, Tinbergen Institute.
    14. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Sup-norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation," Cowles Foundation Discussion Papers 1923R, Cowles Foundation for Research in Economics, Yale University, revised Apr 2015.

    More about this item

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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