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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

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.

Suggested Citation

  • Hengartner, Nicolas W. & Sperlich, Stefan, 2005. "Rate optimal estimation with the integration method in the presence of many covariates," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 246-272, August.
  • Handle: RePEc:eee:jmvana:v:95:y:2005:i:2:p:246-272
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    References listed on IDEAS

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    4. Stefan Sperlich & Oliver Linton & Wolfgang Härdle, 1999. "Integration and backfitting methods in additive models-finite sample properties and comparison," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 419-458, December.
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    Citations

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    Cited by:

    1. Qian, Junhui & Wang, Le, 2012. "Estimating semiparametric panel data models by marginal integration," Journal of Econometrics, Elsevier, vol. 167(2), pages 483-493.
    2. Nicolai Bissantz & Holger Dette & Thimo Hildebrandt & Kathrin Bissantz, 2016. "Smooth backfitting in additive inverse regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(4), pages 827-853, August.
    3. 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.
    4. repec:spr:testjl:v:26:y:2017:i:2:d:10.1007_s11749-016-0508-0 is not listed on IDEAS
    5. 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.
    6. Samuele Centorrino & Jean-Pierre Florens, 2014. "Nonparametric Instrumental Variable Estimation of Binary Response Models," Department of Economics Working Papers 14-07, Stony Brook University, Department of Economics.
    7. 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.
    8. Stefan Sperlich, 2014. "On the choice of regularization parameters in specification testing: a critical discussion," Empirical Economics, Springer, vol. 47(2), pages 427-450, September.
    9. Holger Dette & Regine Scheder, 2011. "Estimation of additive quantile regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(2), pages 245-265, April.
    10. 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).
    11. Holger Dette & Matthias Guhlich & Natalie Neumeyer, 2015. "Testing for additivity in nonparametric quantile regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 437-477, June.
    12. Li, Shu & Ernest, Jan & Bühlmann, Peter, 2017. "Nonparametric causal inference from observational time series through marginal integration," Econometrics and Statistics, Elsevier, vol. 2(C), pages 81-105.
    13. 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.

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