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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.

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Bibliographic Info

Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2009-028.

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Length: 33 pages
Date of creation: May 2009
Date of revision:
Handle: RePEc:hum:wpaper:sfb649dp2009-028

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

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  1. Yoshihiko Nishiyama & Peter M Robinson, 2005. "The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives," STICERD - Econometrics Paper Series, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE /2005/483, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  2. Xia, Yingcun, 2006. "Asymptotic Distributions For Two Estimators Of The Single-Index Model," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 22(06), pages 1112-1137, December.
  3. Hardle, W. & Tsybakov, A.B., 1992. "How Sensitive are Average Derivatives?," Papers, Tilburg - Center for Economic Research 9208, Tilburg - Center for Economic Research.
  4. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, Cambridge University Press, number 9780521370905.
  5. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, Cambridge University Press, number 9780521424318.
  6. Y. Nishiyama & P. M. Robinson, 2000. "Edgeworth Expansions for Semiparametric Averaged Derivatives," Econometrica, Econometric Society, Econometric Society, vol. 68(4), pages 931-980, July.
  7. 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, Royal Statistical Society, vol. 64(3), pages 363-410.
  8. Ichimura, H., 1991. "Semiparametric Least Squares (sls) and Weighted SLS Estimation of Single- Index Models," Papers, Minnesota - Center for Economic Research 264, Minnesota - Center for Economic Research.
  9. Oliver Linton, 1993. "Second Order Approximation in the Partially Linear Regression Model," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1065, Cowles Foundation for Research in Economics, Yale University.
  10. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, Econometric Society, vol. 57(6), pages 1403-30, November.
  11. Hardle, W. & Hall, P. & Ichimura, H., 1991. "Optimal smoothing in single index models," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 1991007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  12. Powell, James L. & Stoker, Thomas M., 1996. "Optimal bandwidth choice for density-weighted averages," Journal of Econometrics, Elsevier, Elsevier, vol. 75(2), pages 291-316, December.
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