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Efficient estimation of a semiparametric partially linear varying coefficient model. (English) Zbl 1064.62043
Summary: We propose a general series method to estimate a semiparametric partially linear varying coefficient model. We establish the consistency and \(\sqrt n\)-normality property of the estimator of the finite-dimensional parameters of the model. We further show that, when the error is conditionally homoskedastic, this estimator is semiparametrically efficient in the sense that the inverse of the asymptotic variance of the estimator of the finite-dimensional parameter reaches the semiparametric efficiency bound of this model.
A small-scale simulation is reported to examine the finite sample performance of the proposed estimator, and an empirical application is presented to illustrate the usefulness of the proposed method in practice. We also discuss how to obtain an efficient estimation result when the error is conditional heteroskedastic.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C05 Monte Carlo methods
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References:
[1] Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71 1795–1843. · Zbl 1154.62323
[2] Andrews, D. W. K. (1991). Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 59 307–345. · Zbl 0727.62047
[3] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Inference for Semiparametric Models . Johns Hopkins Univ. Press. · Zbl 0786.62001
[4] Bickel, P. J. and Kwon, J. (2002). Inference for semiparametric models: Some current frontiers and an answer (with discussion). Statist. Sinica 11 863–960. · Zbl 0997.62028
[5] Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc. 95 888–902. · Zbl 0999.62052
[6] Cai, Z., Fan, J. and Yao, Q. (2000). Functional-coefficient regression models for nonlinear time series. J. Amer. Statist. Assoc. 95 941–956. · Zbl 0996.62078
[7] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489. · Zbl 0890.62053
[8] Chamberlain, G. (1992). Efficiency bounds for semiparametric regression. Econometrica 60 567–596. · Zbl 0774.62038
[9] Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. 88 298–308. · Zbl 0776.62066
[10] Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by generalized cross-validation. Numer. Math. 31 377–403. · Zbl 0377.65007
[11] Fan, J. and Huang, L.-S. (2001). Goodness-of-fit tests for parametric regression models. J. Amer. Statist. Assoc. 96 640–652. · Zbl 1017.62014
[12] Fan, J. and Huang, T. (2002). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Unpublished manuscript. · Zbl 1098.62077
[13] Fan, J., Yao, Q. and Cai, Z. (2003). Adaptive varying-coefficient linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 57–80. · Zbl 1063.62054
[14] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491–1518. · Zbl 0977.62039
[15] Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models : A Roughness Penalty Approach . Chapman and Hall, London. · Zbl 0832.62032
[16] Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models . Physica-Verlag, Heidelberg. · Zbl 0968.62006
[17] Hart, J. (1997). Nonparametric Smoothing and Lack-of-fit Tests . Springer, New York. · Zbl 0886.62043
[18] Hastie, T. and Tibshirani, R. (1993). Varying coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B 55 757–796. · Zbl 0796.62060
[19] Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L.-P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809–822. · Zbl 0921.62045
[20] Huang, J. Z. (1998). Projection estimation in multiple regression with application to functional ANOVA models. Ann. Statist. 26 242–272. · Zbl 0930.62042
[21] Huang, J. Z. (2003). Local asymptotics for polynomial spline regression. Ann. Statist. 31 1600–1635. · Zbl 1042.62035
[22] Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89 111–128. · Zbl 0998.62024
[23] Huang, J. Z., Wu, C. O. and Zhou, L. (2004). Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statist. Sinica 14 763–788. · Zbl 1073.62036
[24] Li, K. C. (1987). Asymptotic optimality for \(C_p\), \(C_L\), cross-validation and generalized cross-validation: Discrete index set. Ann. Statist. 15 958–975. JSTOR: · Zbl 0653.62037
[25] Li, Q., Huang, C. J., Li, D. and Fu, T.-T. (2002). Semiparametric smooth coefficient models. J. Bus. Econom. Statist. 20 412–422.
[26] Lorentz, G. G. (1966). Approximation of Functions . Holt, Rinehart and Winston, New York. · Zbl 0153.38901
[27] Mallows, C. L. (1973). Some comments on \(C_p\). Technometrics 15 661–675. · Zbl 0269.62061
[28] Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79 147–168. · Zbl 0873.62049
[29] Robinson, P. M. (1988). Root-N-consistent semiparametric regression. Econometrica 56 931–954. · Zbl 0647.62100
[30] Shen, X. (1997). On methods of sieves and penalization. Ann. Statist. 25 2555–2591. · Zbl 0895.62041
[31] Speckman, P. (1988). Kernel smoothing in partially linear models. J. Roy. Statist. Soc. Ser. B 50 413–436. · Zbl 0671.62045
[32] Stock, C. J. (1989). Nonparametric policy analysis. J. Amer. Statist. Assoc. 89 567–575.
[33] Xia, Y. C. and Li, W. K. (1999). On the estimation and testing of functional-coefficient linear models. Statist. Sinica 9 735–758. · Zbl 0958.62040
[34] Xia, Y. C. and Li, W. K. (2002). Asymptotic behavior of bandwidth selected by the cross-validation method for local polynomial fitting. J. Multivariate Anal. 83 265–287. · Zbl 1025.62016
[35] Zhang, W., Lee, S.-Y. and Song, X. (2002). Local polynomial fitting in semivarying coefficient models. J. Multivariate Anal. 82 166–188. · Zbl 0995.62038
[36] Zhou, S., Shen, X. and Wolfe, D. A. (1998). Local asymptotics for regression splines and confidence regions. Ann. Statist. 26 1760–1782. · Zbl 0929.62052
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