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Rate optimal estimation with the integration method in the presence of many covariates

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  • Hengartner, Nicolas W.
  • Sperlich, Stefan
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    Abstract

    For multivariate regressors, integrating the Nadaraya-Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 95 (2005)
    Issue (Month): 2 (August)
    Pages: 246-272

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    Handle: RePEc:eee:jmvana:v:95:y:2005:i:2:p:246-272

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    Related research

    Keywords: Separable models Partial linear models Marginal integration Nonparametric regression;

    References

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    1. Horowitz, Joel L, 2001. "Nonparametric Estimation of a Generalized Additive Model with an Unknown Link Function," Econometrica, Econometric Society, vol. 69(2), pages 499-513, March.
    2. Oliver Linton & E. Mammen & J. Nielsen, 1999. "The existence and asymptotic properties of a backfitting projection algorithm under weak conditions," LSE Research Online Documents on Economics 300, London School of Economics and Political Science, LSE Library.
    3. Sperlich, Stefan & Tj stheim, Dag & Yang, Lijian, 2002. "Nonparametric Estimation And Testing Of Interaction In Additive Models," Econometric Theory, Cambridge University Press, vol. 18(02), pages 197-251, April.
    4. Opsomer, Jan & Ruppert, David, 1997. "Fitting a Bivariate Additive Model by Local Polynomial Regression," Staff General Research Papers 1071, Iowa State University, Department of Economics.
    5. Li, Qi, 2000. "Efficient Estimation of Additive Partially Linear Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 41(4), pages 1073-92, November.
    6. Deaton,Angus & Muellbauer,John, 1980. "Economics and Consumer Behavior," Cambridge Books, Cambridge University Press, number 9780521296762, December.
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    Cited by:
    1. Kong, Efang & Linton, Oliver & Xia, Yingcun, 2010. "Uniform Bahadur Representation For Local Polynomial Estimates Of M-Regression And Its Application To The Additive Model," Econometric Theory, Cambridge University Press, vol. 26(05), pages 1529-1564, October.
    2. Qian, Junhui & Wang, Le, 2009. "Estimating Semiparametric Panel Data Models by Marginal Integration," MPRA Paper 18850, University Library of Munich, Germany.
    3. Manzan, sebastiano & Zerom, Dawit, 2008. "A Semiparametric Analysis of Gasoline Demand in the US: Reexamining The Impact of Price," MPRA Paper 14386, University Library of Munich, Germany.
    4. Moral-Arce, Ignacio & Rodríguez-Póo, Juan M. & Sperlich, Stefan, 2011. "Low dimensional semiparametric estimation in a censored regression model," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 118-129, January.
    5. Jorge Barrientos Marin, 2006. "Estimation And Testing An Additive Partially Linear Model In A System Of Engel Curves," Working Papers. Serie AD 2006-23, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    6. Holger Dette & Regine Scheder, 2011. "Estimation of additive quantile regression," Annals of the Institute of Statistical Mathematics, Springer, vol. 63(2), pages 245-265, April.
    7. Krishna Pendakur & Stefan Sperlich, 2010. "Semiparametric estimation of consumer demand systems in real expenditure," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(3), pages 420-457.

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