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Published online by Cambridge University Press:  12 January 2010

Rong Liu*
University of Toledo
Lijian Yang*
Michigan State University
*Address correspondence to Rong Liu, Department of Mathematics, University of Toledo, Toledo, OH 43606, USA; e-mail:
Lijian Yang, Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA; e-mail:


Additive coefficient model (Xue and Yang, 2006a, 2006b) is a flexible regression and autoregression tool that circumvents the “curse of dimensionality.” We propose spline-backfitted kernel (SBK) and spline-backfitted local linear (SBLL) estimators for the component functions in the additive coefficient model that are both (i) computationally expedient so they are usable for analyzing high dimensional data, and (ii) theoretically reliable so inference can be made on the component functions with confidence. In addition, they are (iii) intuitively appealing and easy to use for practitioners. The SBLL procedure is applied to a varying coefficient extension of the Cobb-Douglas model for the U.S. GDP that allows nonneutral effects of the R&D on capital and labor as well as in total factor productivity (TFP).

Copyright © Cambridge University Press 2010

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Arnold, R. (2005) R&D and Productivity Growth: A Background Paper. Congressional Budget Office.Google Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes: Estimation and Prediction. Springer-Verlag.Google Scholar
Cai, Z., Fan, J., & Yao, Q.W. (2000) Functional-coefficient regression models for nonlinear time series. Journal of the American Statistical Association 95, 941956.CrossRefGoogle Scholar
Chen, R. & Tsay, R.S. (1993a) Nonlinear additive ARX models. Journal of the American Statistical Association 88, 955967.CrossRefGoogle Scholar
Chen, R. & Tsay, R.S. (1993b) Functional-coefficient autoregressive models. Journal of the American Statistical Association 88, 298308.Google Scholar
Cobb, C.W. & Douglas, P.H. (1928) A theory of production. American Economic Review 18, 139165.Google Scholar
Culpepper, W.L. (2004) High R&D Spending Fuels Revenue Growth Not Profits. Available for downloading at Scholar
de Boor, C. (2001) A Practical Guide to Splines. Springer-Verlag.Google Scholar
Hastie, T.J. & Tibshirani, R.J. (1990) Generalized Additive Models. Chapman and Hall.Google Scholar
Hastie, T.J. & Tibshirani, R.J. (1993) Varying-coefficient models. Journal of the Royal Statistical Society Series B 55, 757796.Google Scholar
Hengartner, N.W. & Sperlich, S. (2005) Rate optimal estimation with the integration method in the presence of many covariates. Journal of Multivariate Analysis 95, 246272.Google Scholar
Horowitz, J. & Mammen, E. (2004) Nonparametric estimation of an additive model with a link function. Annals of Statistics 32, 24122443.Google Scholar
Huang, J.Z. (1998a) Projection estimation in multiple regression with application to functional ANOVA models. Annals of Statistics 26, 242272.Google Scholar
Huang, J.Z. (1998b) Functional ANOVA models for generalized regression. Journal of Multivariate Analysis 67, 4971.Google Scholar
Huang, J.Z. & Shen, H. (2004) Functional coefficient regression models for non-linear time series: A polynomial spline approach. Scandinavian Journal of Statistics 31, 515534.Google Scholar
Huang, J.Z., Wu, C.O., & Zhou, L. (2002) Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89, 111128.Google Scholar
Huang, J.Z. & Yang, L. (2004) Identification of nonlinear additive autoregressive models. Journal of the Royal Statistical Society Series B 66, 463477.Google Scholar
Li, Q. & Racine, J.S. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Linton, O.B. (1997) Efficient estimation of additive nonparametric regression models. Biometrika 84, 469473.CrossRefGoogle Scholar
Liu, R. & Yang, L. (2008) Spline-backfitted kernel smoothing of additive coefficient model. Available at Scholar
Mammen, E., Linton, O.B., & Nielsen, J.P. (1999) The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics 27, 14431490.Google Scholar
Nielsen, J.P. & Sperlich, S. (2005) Smooth backfitting in practice. Journal of the Royal Statistical Society Series B 67, 4361.CrossRefGoogle Scholar
Rodríguez-Póo, J.M., Sperlich, S., & Vieu, P. (2003) Semiparametric estimation of separable models with possibly limited dependent variables. Econometric Theory 19, 10081039.Google Scholar
Solow, R.M. (1957) Technical change and the aggregate production function. The Review of Economics and Statistics 39, 312320.CrossRefGoogle Scholar
Sperlich, S., Tjøstheim, D., & Yang, L. (2002) Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18, 197251.CrossRefGoogle Scholar
Stone, C. J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13, 689705.Google Scholar
Tokic, D. (2003) How efficient were R&D and advertising investments for internet firms before the bubble burst? A DEA approach. Credit and Financial Management Review 9, 3951.Google Scholar
Wang, L. & Yang, L. (2007) Spline-backfitted kernel smoothing of nonlinear additive autoregression model. Annals of Statistics 35, 24742503.Google Scholar
Xue, L. & Yang, L. (2006a) Estimation of semiparametric additive coefficient model. Journal of Statistical Planning and Inference 136, 25062534.Google Scholar
Xue, L. & Yang, L. (2006b) Additive coefficient modelling via polynomial spline. Statistica Sinica 16, 14231446.Google Scholar
Yang, L., Härdle, W., & Nielsen, J.P. (1999) Nonparametric autoregression with multiplicative volatility and additive mean. Journal of Time Series Analysis 20, 579604.Google Scholar
Yang, L., Park, B.U., Xue, L., & Härdle, W. (2006) Estimation and testing of varying coefficients in additive models with marginal integration. Journal of the American Statistical Association 101, 12121227.CrossRefGoogle Scholar
Yang, L., Sperlich, S., & Härdle, W. (2003) Derivative estimation and testing in generalized additive models. Journal of Statistical Planning and Inference 115, 521542.Google Scholar