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Optimal smoothing for a computationally and statistically efficient single index estimator

Author

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  • Xia, Yingcun
  • Härdle, Wolfgang Karl
  • Linton, Oliver

Abstract

In semiparametric models it is a common approach to under-smooth the nonparametric functions in order that estimators of the finite dimensional parameters can achieve root-n consistency. The requirement of under-smoothing may result as we show from inefficient estimation methods or technical difficulties. Based on local linear kernel smoother, we propose an estimation method to estimate the single-index model without under-smoothing. Under some conditions, our estimator of the single-index is asymptotically normal and most efficient in the semi-parametric sense. Moreover, we derive higher expansions for our estimator and use them to define an optimal bandwidth for the purposes of index estimation. As a result we obtain a practically more relevant method and we show its superior performance in a variety of applications.

Suggested Citation

  • Xia, Yingcun & Härdle, Wolfgang Karl & Linton, Oliver, 2009. "Optimal smoothing for a computationally and statistically efficient single index estimator," SFB 649 Discussion Papers 2009-028, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2009-028
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    References listed on IDEAS

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    1. Yoshihiko Nishiyama & Peter M. Robinson, 2005. "The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives," Econometrica, Econometric Society, vol. 73(3), pages 903-948, May.
    2. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    3. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521424318, January.
    4. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    5. Xiangrong Yin & R. Dennis Cook, 2005. "Direction estimation in single-index regressions," Biometrika, Biometrika Trust, vol. 92(2), pages 371-384, June.
    6. Y. Nishiyama & P. M. Robinson, 2000. "Edgeworth Expansions for Semiparametric Averaged Derivatives," Econometrica, Econometric Society, vol. 68(4), pages 931-980, July.
    7. Powell, James L. & Stoker, Thomas M., 1996. "Optimal bandwidth choice for density-weighted averages," Journal of Econometrics, Elsevier, vol. 75(2), pages 291-316, December.
    8. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
    9. Linton, Oliver, 1995. "Second Order Approximation in the Partially Linear Regression Model," Econometrica, Econometric Society, vol. 63(5), pages 1079-1112, September.
    10. Xia, Yingcun, 2006. "Asymptotic Distributions For Two Estimators Of The Single-Index Model," Econometric Theory, Cambridge University Press, vol. 22(6), pages 1112-1137, December.
    11. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521370905, January.
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    Cited by:

    1. Chuan Goh, 2009. "Bootstrap-based Bandwidth Selection for Semiparametric Generalized Regression Estimators," Working Papers tecipa-375, University of Toronto, Department of Economics.
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    3. repec:hum:wpaper:sfb649dp2009-038 is not listed on IDEAS
    4. repec:hum:wpaper:sfb649dp2009-040 is not listed on IDEAS
    5. repec:hum:wpaper:sfb649dp2009-041 is not listed on IDEAS

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    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • 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

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