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Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator

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  • Yingcun Xia
  • Wolfgang Härdle
  • Oliver Linton

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

  • Yingcun Xia & Wolfgang Härdle & Oliver Linton, 2009. "Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator," SFB 649 Discussion Papers SFB649DP2009-028, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2009-028
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    References listed on IDEAS

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    1. Hardle, W. & Hall, P. & Ichimura, H., 1991. "Optimal smoothing in single index models," CORE Discussion Papers 1991007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
    3. 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.
    4. Linton, Oliver, 1995. "Second Order Approximation in the Partially Linear Regression Model," Econometrica, Econometric Society, vol. 63(5), pages 1079-1112, September.
    5. Xia, Yingcun, 2006. "Asymptotic Distributions For Two Estimators Of The Single-Index Model," Econometric Theory, Cambridge University Press, vol. 22(06), pages 1112-1137, December.
    6. 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.
    7. 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.
    8. Nishiyama, Yoshihiko & Robinson, Peter M., 2005. "The bootstrap and the Edgeworth correction for semiparametric averaged derivatives," LSE Research Online Documents on Economics 2297, London School of Economics and Political Science, LSE Library.
    9. 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.
    10. Xiangrong Yin & R. Dennis Cook, 2005. "Direction estimation in single-index regressions," Biometrika, Biometrika Trust, vol. 92(2), pages 371-384, June.
    11. Y. Nishiyama & P. M. Robinson, 2000. "Edgeworth Expansions for Semiparametric Averaged Derivatives," Econometrica, Econometric Society, vol. 68(4), pages 931-980, July.
    12. 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.
    13. Ichimura, H., 1991. "Semiparametric Least Squares (sls) and Weighted SLS Estimation of Single- Index Models," Papers 264, Minnesota - Center for Economic Research.
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    More about this item

    Keywords

    ADE; Asymptotics; Bandwidth; MAVE method; Semi-parametric efficiency;

    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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