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

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  • 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.
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  • Dunker , Fabian & Florens, Jean-Pierre & Hohage, Thorsten & Johannes, Jan & Mammen, Enno, 2013. "Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression," LIDAM Discussion Papers ISBA 2013041, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2013041
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    1. 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.
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    12. 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.
    13. Victor Chernozhukov & Christian Hansen, 2005. "An IV Model of Quantile Treatment Effects," Econometrica, Econometric Society, vol. 73(1), pages 245-261, January.
    14. 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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    Cited by:

    1. Beyhum, Jad & Lapenta, Elia & Lavergne, Pascal, 2023. "One-step nonparametric instrumental regression using smoothing splines," TSE Working Papers 23-1467, Toulouse School of Economics (TSE).
    2. Andrii Babii & Jean-Pierre Florens, 2017. "Are Unobservables Separable?," Papers 1705.01654, arXiv.org, revised Mar 2021.
    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. Dunker, Fabian & Hoderlein, Stefan & Kaido, Hiroaki, 2014. "Nonparametric Identification of Endogenous and Heterogeneous Aggregate Demand Models: Complements, Bundles and the Market Level," Economics Series 307, Institute for Advanced Studies.
    5. Carolina Caetano & Juan Carlos Escaniano, 2015. "Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity," CAEPR Working Papers 2015-009, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    6. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2020. "Robust frontier estimation from noisy data: A Tikhonov regularization approach," Econometrics and Statistics, Elsevier, vol. 14(C), pages 1-23.
    7. Loh, Isaac, 2023. "Nonparametric identification and estimation with discrete instruments and regressors," Journal of Econometrics, Elsevier, vol. 235(2), pages 1257-1279.
    8. Fève, Frédérique & Florens, Jean-Pierre, 2014. "Iterative algorithm for non parametric estimation of the instrumental variables quantiles," Economics Letters, Elsevier, vol. 123(3), pages 300-304.
    9. Jad Beyhum & Jean-Pierre FLorens & Ingrid Van Keilegom, 2020. "Nonparametric instrumental regression with right censored duration outcomes," Papers 2011.10423, arXiv.org.
    10. Beyhum, Jad & Florens, Jean-Pierre & Van Keilegom, Ingrid, 2020. "Nonparametric Instrumental Regression with Right Censored Duration Outcomes," TSE Working Papers 20-1164, Toulouse School of Economics (TSE).
    11. 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.
    12. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2017. "Nonparametric identification of random coefficients in endogenous and heterogeneous aggregate demand models," CeMMAP working papers 11/17, Institute for Fiscal Studies.
    13. Jad Beyhum & Elia Lapenta & Pascal Lavergne, 2023. "One-step smoothing splines instrumental regression," Papers 2307.14867, arXiv.org, revised Apr 2024.
    14. Christoph Breunig & Stephan Martin, 2020. "Nonclassical Measurement Error in the Outcome Variable," Papers 2009.12665, arXiv.org, revised May 2021.
    15. Feve, Frederique & Florens, Jean-Pierre & Van Keilegom, Ingrid, 2012. "Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models," LIDAM Discussion Papers ISBA 2012036, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Fabian Dunker & Thorsten Hohage, 2014. "On parameter identification in stochastic differential equations by penalized maximum likelihood," Papers 1404.0651, arXiv.org.
    17. Fabian Dunker, 2015. "Adaptive estimation for some nonparametric instrumental variable models," Papers 1511.03977, arXiv.org, revised Aug 2021.
    18. Krief, Jerome M., 2017. "Direct instrumental nonparametric estimation of inverse regression functions," Journal of Econometrics, Elsevier, vol. 201(1), pages 95-107.
    19. Asin, Nicolas & Johannes, Jan, 2016. "Adaptive non-parametric instrumental regression in the presence of dependence," LIDAM Discussion Papers ISBA 2016015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. Poirier, Alexandre, 2017. "Efficient estimation in models with independence restrictions," Journal of Econometrics, Elsevier, vol. 196(1), pages 1-22.

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    More about this item

    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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