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Optimal smoothing for a computationally and statistically efficient single index estimator

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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. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
    2. 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.
    3. Linton, Oliver, 1995. "Second Order Approximation in the Partially Linear Regression Model," Econometrica, Econometric Society, vol. 63(5), pages 1079-1112, September.
    4. 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.
    5. 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, October.
    6. 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.
    7. Y. Nishiyama & P. M. Robinson, 2000. "Edgeworth Expansions for Semiparametric Averaged Derivatives," Econometrica, Econometric Society, vol. 68(4), pages 931-980, July.
    8. 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.
    9. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
    10. Hardle, W. & Hall, P. & Ichimura, H., 1991. "Optimal smoothing in single index models," LIDAM Discussion Papers CORE 1991007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. 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.
    12. 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, October.
    13. Xiangrong Yin & R. Dennis Cook, 2005. "Direction estimation in single-index regressions," Biometrika, Biometrika Trust, vol. 92(2), pages 371-384, June.
    14. 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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    Cited by:

    1. Strausz, Roland, 2009. "The political economy of regulatory risk," SFB 649 Discussion Papers 2009-040, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    2. Michał Grajek & Lars-Hendrik Röller, 2012. "Regulation and Investment in Network Industries: Evidence from European Telecoms," Journal of Law and Economics, University of Chicago Press, vol. 55(1), pages 189-216.
    3. Choroś, Barbara & Härdle, Wolfgang Karl & Okhrin, Ostap, 2009. "CDO and HAC," SFB 649 Discussion Papers 2009-038, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    4. repec:hum:wpaper:sfb649dp2009-040 is not listed on IDEAS
    5. Michał Grajek & Lars-Hendrik Röller, 2012. "Regulation and Investment in Network Industries: Evidence from European Telecoms," Journal of Law and Economics, University of Chicago Press, vol. 55(1), pages 189-216.
    6. repec:hum:wpaper:sfb649dp2009-041 is not listed on IDEAS
    7. Chuan Goh, 2009. "Bootstrap-based Bandwidth Selection for Semiparametric Generalized Regression Estimators," Working Papers tecipa-375, University of Toronto, Department of Economics.
    8. Grith, Maria & Härdle, Wolfgang Karl & Park, Juhyun, 2009. "Shape invariant modelling pricing kernels and risk aversion," SFB 649 Discussion Papers 2009-041, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    9. repec:hum:wpaper:sfb649dp2009-038 is not listed on IDEAS

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    More about this item

    Keywords

    ADE; Asymptotics; Bandwidth; MAVE method; Semi-parametric efficiency;
    All these keywords.

    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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