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Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression

Author

Listed:
  • Dunker, Fabian
  • Florens, Jean-Pierre
  • Hohage, Thorsten
  • Johannes, Jan
  • Mammen, Enno

Abstract

This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model. This is illustrated in computed examples with simulated data.

Suggested Citation

  • Dunker, Fabian & Florens, Jean-Pierre & Hohage, Thorsten & Johannes, Jan & Mammen, Enno, 2014. "Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression," Journal of Econometrics, Elsevier, vol. 178(P3), pages 444-455.
  • Handle: RePEc:eee:econom:v:178:y:2014:i:p3:p:444-455
    DOI: 10.1016/j.jeconom.2013.06.001
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    References listed on IDEAS

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    1. 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.
    2. Xiaohong Chen & Victor Chernozhukov & Sokbae Lee & Whitney K. Newey, 2014. "Local Identification of Nonparametric and Semiparametric Models," Econometrica, Econometric Society, vol. 82(2), pages 785-809, March.
    3. 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, January.
    4. Florens, Jean-Pierre & Sbaï, Erwann, 2010. "Local Identification In Empirical Games Of Incomplete Information," Econometric Theory, Cambridge University Press, vol. 26(06), pages 1638-1662, December.
    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. Victor Chernozhukov & Christian Hansen, 2005. "An IV Model of Quantile Treatment Effects," Econometrica, Econometric Society, vol. 73(1), pages 245-261, January.
    7. 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.
    8. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, September.
    9. Joel L. Horowitz & Sokbae Lee, 2007. "Nonparametric Instrumental Variables Estimation of a Quantile Regression Model," Econometrica, Econometric Society, vol. 75(4), pages 1191-1208, July.
    10. Chernozhukov, Victor & Imbens, Guido W. & Newey, Whitney K., 2007. "Instrumental variable estimation of nonseparable models," Journal of Econometrics, Elsevier, vol. 139(1), pages 4-14, July.
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    Citations

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    Cited by:

    1. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2014. "Nonparametric identification of endogenous and heterogeneous aggregate demand models: complements, bundles and the market level," CeMMAP working papers CWP23/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Poirier, Alexandre, 2017. "Efficient estimation in models with independence restrictions," Journal of Econometrics, Elsevier, vol. 196(1), pages 1-22.
    3. Xiaohong Chen & Victor Chernozhukov & Sokbae Lee & Whitney K. Newey, 2014. "Local Identification of Nonparametric and Semiparametric Models," Econometrica, Econometric Society, vol. 82(2), pages 785-809, March.
    4. Fabian Dunker & Thorsten Hohage, 2014. "On parameter identification in stochastic differential equations by penalized maximum likelihood," Papers 1404.0651, arXiv.org.
    5. Fabian Dunker, 2015. "Convergence of the risk for nonparametric IV quantile regression and nonparametric IV regression with full independence," Papers 1511.03977, arXiv.org.
    6. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2016. "Robust frontier estimation from noisy data: a Tikhonov regularization approach," TSE Working Papers 16-665, Toulouse School of Economics (TSE), revised Jul 2018.
    7. repec:eee:econom:v:201:y:2017:i:1:p:95-107 is not listed on IDEAS
    8. Cazals, Catherine & Fève, Frédérique & Florens, Jean-Pierre & Simar, Léopold, 2016. "Nonparametric instrumental variables estimation for efficiency frontier," Journal of Econometrics, Elsevier, vol. 190(2), pages 349-359.
    9. Babii, Andrii & Florens, Jean-Pierre, 2017. "Distribution of residuals in the nonparametric IV model with application to separability testing," TSE Working Papers 17-802, Toulouse School of Economics (TSE).
    10. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2017. "Nonparametric Identification of Random Coefficients in Endogenous and Heterogeneous Aggregate Demand," Courant Research Centre: Poverty, Equity and Growth - Discussion Papers 224, Courant Research Centre PEG.
    11. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2017. "Nonparametric identification of random coefficients in endogenous and heterogeneous aggregate demand models," CeMMAP working papers CWP11/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item

    Keywords

    Nonparametric regression; Nonlinear inverse problems; Iterative regularization; Instrumental regression;

    JEL classification:

    • 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
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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