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Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

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    Abstract

    We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L^2-norm rates for severely ill-posed problems, and are power of log(n) slower than the optimal L^2-norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided.

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    File URL: http://cowles.econ.yale.edu/P/cd/d19a/d1923.pdf
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    Bibliographic Info

    Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1923.

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    Length: 36 pages
    Date of creation: Nov 2013
    Date of revision:
    Handle: RePEc:cwl:cwldpp:1923

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    Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
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    Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

    Related research

    Keywords: Nonparametric instrumental variables; Statistical ill-posed inverse problems; Optimal uniform convergence rates; Weak dependence; Random matrices; Splines; Wavelets;

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    1. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    2. 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.
    3. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(03), pages 726-748, June.
    4. Serge Darolles & Jean-Pierre Florens & Eric Renault, 2000. "Nonparametric Instrumental Regression," Working Papers 2000-17, Centre de Recherche en Economie et Statistique.
    5. Joel L. Horowitz, 2011. "Applied Nonparametric Instrumental Variables Estimation," Econometrica, Econometric Society, vol. 79(2), pages 347-394, 03.
    6. Gagliardini, Patrick & Scaillet, Olivier, 2012. "Tikhonov regularization for nonparametric instrumental variable estimators," Journal of Econometrics, Elsevier, vol. 167(1), pages 61-75.
    7. Cohen, Albert & Hoffmann, Marc & Reiß, Markus, 2002. "Adaptive wavelet Galerkin methods for linear inverse problems," SFB 373 Discussion Papers 2002,50, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    8. Cattaneo, Matias D. & Farrell, Max H., 2013. "Optimal convergence rates, Bahadur representation, and asymptotic normality of partitioning estimators," Journal of Econometrics, Elsevier, vol. 174(2), pages 127-143.
    9. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
    10. de Jong, Robert M., 2002. "A note on "Convergence rates and asymptotic normality for series estimators": uniform convergence rates," Journal of Econometrics, Elsevier, vol. 111(1), pages 1-9, November.
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