Abstract
This paper is concerned with inference of panel data varying-coefficient partially linear models with a one-way error structure. The model is a natural extension of the well-known panel data linear model (due to Baltagi 1995) to the setting of semiparametric regressions. The authors propose a weighted profile least squares estimator (WPLSE) and a weighted local polynomial estimator (WLPE) for the parametric and nonparametric components, respectively. It is shown that the WPLSE is asymptotically more efficient than the usual profile least squares estimator (PLSE), and that the WLPE is also asymptotically more efficient than the usual local polynomial estimator (LPE). The latter is an interesting result. According to Ruckstuhl, Welsh and Carroll (2000) and Lin and Carroll (2000), ignoring the correlation structure entirely and “pretending” that the data are really independent will result in more efficient estimators when estimating nonparametric regression with longitudinal or panel data. The result in this paper shows that this is not true when the design points of the nonparametric component have a closeness property within groups. The asymptotic properties of the proposed weighted estimators are derived. In addition, a block bootstrap test is proposed for the goodness of fit of models, which can accommodate the correlations within groups. Some simulation studies are conducted to illustrate the finite sample performances of the proposed procedures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baltagi, B. H., Econometric Analysis of Panel Data, John Wiley & Sons, New York, 1995.
Cai, Z., Fan, J. and Yao, Q., Functional-coefficient regression models for nonlinear time series, J. Amer. Statist. Assoc., 95, 2000, 941–956.
Carroll, R. J., Ruppert, D. and Welsh, A. H., Nonparametric estimation via local estimating equations, J. Amer. Statist. Assoc., 93, 1998, 214–227.
Fan, J. Q. and Huang, T., Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 11(6), 2005, 1031–1057.
Fan, J. Q. and Zhang, W., Statistical estimation in varying coefficient models, Ann. Statist., 27, 1999, 1491–1518.
Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L. P., Nonparametric smoothing estimates of time-varying coeffiocient models with longitudinal data, Biometrika, 85, 1998, 809–822.
Huang, J. Z., Wu, C. O. and Zhou, L., Polynomial spline estimation and inference for varying coefficient models with longitudinal data, Statist. Sinica, 14, 2004, 763–788.
Kniesner, T. and Li, Q., Semiparametric panel data models with dynamic adjustment: Theoretical considerations and an application to labor supply, University of Guelph, manuscript, 1994.
Li, Q., Huang, C. J., Li, D. and Fu, T. T., Semiparametric smooth coefficient models, J. Business and Econ. Statist., 20, 2002, 412–422.
Lin, X. H. and Carroll, R. J., Nonparametric function estimation for clustered data when the predictor is measured without/with error, J. Amer. Statist. Assoc., 95, 2000, 520–534.
Müller, H. G. and Zhao, P., On a semiparametric variance function model and a test for heteroscedasticity, Ann. Statist., 23, 1995, 946–967.
Ruckstuhl, A. F., Welsh, A. H. and Carroll, R. J., Nonparametric function estimation of the relationship between two repeatedly measured variables, Statist. Sinica, 10, 2000, 51–71.
Shi, J. and Lau, T. S., Empirical likelihood for partially linear models, J. Multivariate Anal., 72, 2000, 132–149.
Stout, W. F., Almost Sure Convergence, Academic Press, New York, 1974.
Verbeke, G. and Molenberghs, G., Linear Mixed Models for Longitudinal Data, Springer Series in Statistics, Springer-Verlag, New York, 2000.
Wu, C. O., Chiang, C. T. and Hoover, D. R., Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data, J. Amer. Statist. Assoc, 93, 1998, 1388–1402.
Xia, Y. and Li, W. K., On the estimation and testing of functional-coefficient linear models, Statistica Sinica, 9, 1999, 737–757.
You, J. H., Xu, Q. F. and Zhou, B., Statistical inference for partially linear regression models with measurement errors, Chin. Ann. Math., 29B(2), 2008, 207–222.
Zeger, S. L. and Diggle, P. J., Semiparametric model for longitudinal data with application to CD4 cell numbers in HIV seroconvertiers, Biometrics, 50, 1994, 689–699.
Zhang, W., Lee, S. Y. and Song, X., Local polynomial fitting in semivarying coefficient models, J. Multivariate Anal., 82, 2002, 166–188.
Zhou, Y. and You, J. H., Strong convergence rates of several estimators in varying-coefficient partially linear models, University of North Carolina at Chapel Hill, manuscript, 2003.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Leading Academic Discipline Program, 211 Project for Shanghai University of Finance and Economics (the 3rd phase) (No. B803) and the Shanghai Leading Academic Discipline Project (No. B210).
Rights and permissions
About this article
Cite this article
Zhou, B., You, J., Xu, Q. et al. Weighted profile least squares estimation for a panel data varying-coefficient partially linear model. Chin. Ann. Math. Ser. B 31, 247–272 (2010). https://doi.org/10.1007/s11401-008-0153-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-008-0153-3