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Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality

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  • Andrews, Donald W.K.
  • Whang, Yoon-Jae

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 non-parametric 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 that AIR models do circumvent the curse of dimensionality. A lower bound on the rate of convergence 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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  • Andrews, Donald W.K. & Whang, Yoon-Jae, 1990. "Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality," Econometric Theory, Cambridge University Press, vol. 6(4), pages 466-479, December.
    • Handle: RePEc:cup:etheor:v:6:y:1990:i:04:p:466-479_00
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    References listed on IDEAS

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    1. Andrews, Donald W K, 1991. "Asymptotic Normality of Series Estimators for Nonparametric and Semiparametric Regression Models," Econometrica, Econometric Society, vol. 59(2), pages 307-345, March.
    2. 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.
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    1. Dong, Chaohua & Linton, Oliver, 2018. "Additive nonparametric models with time variable and both stationary and nonstationary regressors," Journal of Econometrics, Elsevier, vol. 207(1), pages 212-236.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Sukjin Han, 2020. "Nonparametric estimation of triangular simultaneous equations models under weak identification," Quantitative Economics, Econometric Society, vol. 11(1), pages 161-202, January.
    7. Härdle, Wolfgang & Huet, Sylvie & Mammen, Enno & Sperlich, Stefan, 1998. "Semiparametric additive indices for binary response and generalized additive models," SFB 373 Discussion Papers 1998,95, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    8. Jing Wang & Lijian Yang, 2009. "Efficient and fast spline-backfitted kernel smoothing of additive models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 663-690, September.
    9. 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;The Institute of Statistical Mathematics, vol. 58(1), pages 131-151, March.
    10. Rodriguez Poo, Juan M. & Vieu, Philippe, 2000. "Semiparametric estimation of weak and strong separable models," DES - Working Papers. Statistics and Econometrics. WS 10064, Universidad Carlos III de Madrid. Departamento de Estadística.
    11. 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.
    12. Damian Kozbur, 2013. "Inference in additively separable models with a high-dimensional set of conditioning variables," ECON - Working Papers 284, Department of Economics - University of Zurich, revised Apr 2018.
    13. Das, M., 2003. "Identification and sequential estimation of panel data models with insufficient exclusion restrictions," Journal of Econometrics, Elsevier, vol. 114(2), pages 297-328, June.
    14. Cui, Xia & Zhao, Weihua & Lian, Heng & Liang, Hua, 2019. "Pursuit of dynamic structure in quantile additive models with longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 130(C), pages 42-60.
    15. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    16. Sungwon Lee & Joon H. Ro, 2020. "Nonparametric Tests for Conditional Quantile Independence with Duration Outcomes," Working Papers 2013, Nam Duck-Woo Economic Research Institute, Sogang University (Former Research Institute for Market Economy).
    17. Yang, Lijian & Hardle, Wolfgang, 2000. "Derivative estimation and testing in generalized additive models," DES - Working Papers. Statistics and Econometrics. WS 10084, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Mitali Das, 2000. "Instrumental Variables Estimation of Nonparametric Models with Discrete Endogenous Regressors," Econometric Society World Congress 2000 Contributed Papers 1008, Econometric Society.
    19. 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.
    20. 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.
    21. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    22. Donald, Stephen G., 1995. "Two-step estimation of heteroskedastic sample selection models," Journal of Econometrics, Elsevier, vol. 65(2), pages 347-380, February.
    23. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76, Elsevier.
    24. 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.
    25. Lai, Peng & Meng, Jie & Lian, Heng, 2015. "Polynomial spline approach for variable selection and estimation in varying coefficient models for time series data," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 21-27.

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