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Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality

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Abstract

This paper considers series estimators of additive interactive regression (AIR) models. AIR models are nonparametric regression models that generalize additive regression models by allowing interactions between different regressor variables. They place more restrictions on the regression function, however, than do fully nonparametric regression models. By doing so, they attempt to circumvent the curse of dimensionality that afflicts the estimation of fully nonparametric regression models. In this paper, we present a finite sample bound and asymptotic rate of convergence results for the mean average squared error of series estimators that show the AIR models do circumvent the curse of dimensionality. The rate of convergency of these estimators is shown to depend on the order of the AIR model and the smoothness of the regression function, but not on the dimension of the regressor vector. Series estimators with fixed and data-dependent truncation parameters are considered.

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File URL: http://cowles.econ.yale.edu/P/cd/d09a/d0925.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 925.

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Length: 20 pages
Date of creation: Sep 1989
Date of revision:
Publication status: Published in Econometric Theory (1990), 6: 455-479
Handle: RePEc:cwl:cwldpp:925

Note: CFP 771.
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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Additive interactive regression model; cross-validation; curse of dimensionality; generalized cross-validation; mean average squared error; nonparametric estimation; nonparametric regression; series estimator;

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  1. Gallant, A. Ronald, 1981. "On the bias in flexible functional forms and an essentially unbiased form : The fourier flexible form," Journal of Econometrics, Elsevier, vol. 15(2), pages 211-245, February.
  2. Andrews, Donald W K, 1991. "Asymptotic Normality of Series Estimators for Nonparametric and Semiparametric Regression Models," Econometrica, Econometric Society, vol. 59(2), pages 307-45, March.
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Cited by:
  1. Cai, Zongwu & Das, Mitali & Xiong, Huaiyu & Wu, Xizhi, 2006. "Functional coefficient instrumental variables models," Journal of Econometrics, Elsevier, vol. 133(1), pages 207-241, July.
  2. Li, Qi & Hsiao, Cheng & Zinn, Joel, 2003. "Consistent specification tests for semiparametric/nonparametric models based on series estimation methods," Journal of Econometrics, Elsevier, vol. 112(2), pages 295-325, February.
  3. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
  4. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
  5. Michael Hamers & Michael Kohler, 2006. "Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer, vol. 58(1), pages 131-151, March.
  6. Oliver Linton & Yoon-Jae Whang, 2000. "Nonparametric Estimation with Aggregated Data," STICERD - Econometrics Paper Series /2000/397, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  7. Gordon B. Dahl, 2002. "Mobility and the Return to Education: Testing a Roy Model with Multiple Markets," Econometrica, Econometric Society, vol. 70(6), pages 2367-2420, November.
  8. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
  9. Donald, Stephen G., 1995. "Two-step estimation of heteroskedastic sample selection models," Journal of Econometrics, Elsevier, vol. 65(2), pages 347-380, February.
  10. Donald W.K. Andrews, 1989. "Asymptotic Optimality of Generalized C_{L}, Cross-Validation, and Generalized Cross-Validation in Regression with Heteroskedastic Errors," Cowles Foundation Discussion Papers 906, Cowles Foundation for Research in Economics, Yale University.
  11. Gozalo, Pedro L. & Linton, Oliver B., 2001. "Testing additivity in generalized nonparametric regression models with estimated parameters," Journal of Econometrics, Elsevier, vol. 104(1), pages 1-48, August.
  12. Mitali Das, 2000. "Instrumental Variables Estimation of Nonparametric Models with Discrete Endogenous Regressors," Econometric Society World Congress 2000 Contributed Papers 1008, Econometric Society.
  13. Jing Wang & Lijian Yang, 2009. "Efficient and fast spline-backfitted kernel smoothing of additive models," Annals of the Institute of Statistical Mathematics, Springer, vol. 61(3), pages 663-690, September.
  14. Camlong-Viot, Christine & Rodríguez-Póo, Juan M. & Vieu, Philippe, 2003. "Nonparametric and Semiparametric Estimation of Additive Models with both Discrete and Continuous Variables under Dependence," SFB 373 Discussion Papers 2003,38, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.

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