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The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives

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  • Yoshihiko Nishiyama
  • Peter M Robinson

Abstract

In a number of semiparametric models, smoothing seems necessary in order to obtain estimates of the parametric component which are asymptotically normal and converge at parametric rate. However, smoothing can inflate the error in the normal approximation, so that refined approximations are of interest, especially in sample sizes that are not enormous. We show that a bootstrap distribution achieves a valid Edgeworth correction in case of density-weighted averaged derivative estimates of semiparametric index models. Approaches to bias-reduction are discussed. We also develop a higher order expansion, to show that the bootstrap achieves a further reduction in size distortion in case of two-sided testing. The finite sample performance of the methods is investigated by means of Monte Carlo simulations from a Tobit model.

Suggested Citation

  • Yoshihiko Nishiyama & Peter M Robinson, 2005. "The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives," STICERD - Econometrics Paper Series 483, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    • Handle: RePEc:cep:stiecm:483
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    Cited by:

    1. Matias D. Cattaneo & Max H. Farrell & Michael Jansson & Ricardo Masini, 2022. "Higher-order Refinements of Small Bandwidth Asymptotics for Density-Weighted Average Derivative Estimators," Papers 2301.00277, arXiv.org, revised Feb 2024.
    2. Marcia M Schafgans & Victoria Zinde-Walshyz, 2008. "Smoothness Adaptive AverageDerivative Estimation," STICERD - Econometrics Paper Series 529, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    3. 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.
    4. Chuan Goh, 2009. "Bootstrap-based Bandwidth Selection for Semiparametric Generalized Regression Estimators," Working Papers tecipa-375, University of Toronto, Department of Economics.
    5. Cattaneo, Matias D. & Crump, Richard K. & Jansson, Michael, 2014. "Small Bandwidth Asymptotics For Density-Weighted Average Derivatives," Econometric Theory, Cambridge University Press, vol. 30(1), pages 176-200, February.
    6. Ai, Chunrong & Chen, Xiaohong, 2007. "Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables," Journal of Econometrics, Elsevier, vol. 141(1), pages 5-43, November.
    7. Ichimura, Hidehiko & Todd, Petra E., 2007. "Implementing Nonparametric and Semiparametric Estimators," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 74, Elsevier.
    8. Hardle, Wolfgang & Xia, Yingcun & Linton, Oliver, 2009. "Optimal smoothing for a computationally and statistically efficient single index estimator," LSE Research Online Documents on Economics 58173, London School of Economics and Political Science, LSE Library.
    9. Victoria Zinde-Walsh & Marcia M.A. Schafgans, 2007. "Robust Average Derivative Estimation," Departmental Working Papers 2007-12, McGill University, Department of Economics.
    10. Nishiyama, Y., 2004. "Minimum normal approximation error bandwidth selection for averaged derivatives," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(1), pages 53-61.
    11. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    12. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76, Elsevier.
    13. Christopher Withers & Saralees Nadarajah, 2013. "Density estimates of low bias," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(3), pages 357-379, April.
    14. Subbotin, Viktor, 2008. "Essays on the econometric theory of rank regressions," MPRA Paper 14086, University Library of Munich, Germany.
    15. Subbotin, Viktor, 2007. "Asymptotic and bootstrap properties of rank regressions," MPRA Paper 9030, University Library of Munich, Germany, revised 20 Mar 2008.
    16. Gao, Jiti & Gijbels, Irene, 2005. "Bandwidth selection for nonparametric kernel testing," MPRA Paper 11982, University Library of Munich, Germany, revised Jun 2007.

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    More about this item

    Keywords

    Bootstrap; Edgeworth correction; semiparametric averaged derivatives;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models

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