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Adaptive nonparametric instrumental variables estimation: empirical choice of the regularisation parameter

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  • Joel Horowitz

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    (Institute for Fiscal Studies and Northwestern University)

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    Abstract

    In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularised (that is, modified) to make consistent estimation possible. The amount of modification is contolled by a regularisation parameter. The optimal value of this parameter depends on unknown population characteristics and cannot be calculated in applications. Theoretically justified methods for choosing the regularisatoin parameter empirically in applications are not yet available. This paper presents such a method for use in series estimation, where the regularisation parameter is the number of terms in a series approximation to g. The method does not required knowledge of the smoothness of g or of other unknown functions. It adapts to their unknown smoothness. The estimator of g based on the empirically selected regularisation parameter converges in probabillity at a rate that is at least as fast as the asymptotically optimal rate multiplied by (logn)1/2, where n is the sample size. The asymptotic integrated mean-square error (AIMSE) of the estimator is within a specified factor of the optimal AIMSE.

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    Bibliographic Info

    Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP30/13.

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    Date of creation: Jul 2013
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    Handle: RePEc:ifs:cemmap:30/13

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    Related research

    Keywords: ill-posed inverse problem. regularisatoin; sieve estimation; series estimation; nonparametric estimation;

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    References

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    1. Joel Horowitz & Sokbae 'Simon' Lee, 2010. "Uniform confidence bands for functions estimated nonparametrically with instrumental variables," CeMMAP working papers CWP19/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Xiaohong Chen & Demian Pouzo, 2009. "Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals," CeMMAP working papers CWP20/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Richard Blundell & Joel L. Horowitz, 2007. "A Non-Parametric Test of Exogeneity," Review of Economic Studies, Oxford University Press, vol. 74(4), pages 1035-1058.
    4. Xiaohong Chen & Demian Pouzo, 2008. "Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals," Cowles Foundation Discussion Papers 1640R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2009.
    5. Serge Darolles & Jean-Pierre Florens & Eric Renault, 2000. "Nonparametric Instrumental Regression," Working Papers 2000-17, Centre de Recherche en Economie et Statistique.
    6. 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.
    7. Whitney K. Newey & James L. Powell & Francis Vella, 1999. "Nonparametric Estimation of Triangular Simultaneous Equations Models," Econometrica, Econometric Society, vol. 67(3), pages 565-604, May.
    8. Horowitz, Joel L., 2012. "Specification testing in nonparametric instrumental variable estimation," Journal of Econometrics, Elsevier, vol. 167(2), pages 383-396.
    9. 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.
    10. Joel L. Horowitz & Sokbae Lee, 2007. "Nonparametric Instrumental Variables Estimation of a Quantile Regression Model," Econometrica, Econometric Society, vol. 75(4), pages 1191-1208, 07.
    11. Chen, Xiaohong & Pouzo, Demian, 2009. "Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals," Journal of Econometrics, Elsevier, vol. 152(1), pages 46-60, September.
    12. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
    13. Patrick Gagliardini & Olivier Scaillet, 2012. "Nonparametric Instrumental Variable Estimation of Structural Quantile Effects," Econometrica, Econometric Society, vol. 80(4), pages 1533-1562, 07.
    14. Joel L. Horowitz, 2007. "Asymptotic Normality Of A Nonparametric Instrumental Variables Estimator," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 48(4), pages 1329-1349, November.
    15. Blundell, Richard & Pashardes, Panos & Weber, Guglielmo, 1993. "What Do We Learn About Consumer Demand Patterns from Micro Data?," American Economic Review, American Economic Association, vol. 83(3), pages 570-97, June.
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