Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T08:45:41.542Z Has data issue: false hasContentIssue false
  • English
  • Français

SMOOTH VARYING-COEFFICIENT ESTIMATION AND INFERENCE FOR QUALITATIVE AND QUANTITATIVE DATA

Published online by Cambridge University Press:  17 March 2010

Qi Li  and
Jeffrey S. Racine
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a semiparametric varying-coefficient estimator that admits both qualitative and quantitative covariates along with a test for correct specification of parametric varying-coefficient models. The proposed estimator is exceedingly flexible and has a wide range of potential applications including hierarchical (mixed) settings, small area estimation, etc. A data-driven cross-validatory bandwidth selection method is proposed that can handle both the qualitative and quantitative covariates and that can also handle the presence of potentially irrelevant covariates, each of which can result in finite-sample efficiency gains relative to the conventional frequency (sample-splitting) estimator that is often found in such settings. Theoretical underpinnings including rates of convergence and asymptotic normality are provided. Monte Carlo simulations are undertaken to assess the proposed estimator’s finite-sample performance relative to the conventional semiparametric frequency estimator and to assess the finite-sample performance of the proposed test for correct parametric specification.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 6 , December 2010 , pp. 1607 - 1637
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank a co-editor and two referees for their valuable comments that led to improvements in the paper. Li’s research is partially supported by the Private Enterprise Research Center, Texas A&M University, and the National Science Foundation of China (project 70773005). Racine gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC: www.nserc.ca), the Social Sciences and Humanities Research Council of Canada (SSHRC: www.sshrc.ca), and the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca). Racine thanks Tristen Hayfield for his exemplary research assistance.

References

REFERENCES

Aitchison, J. & Aitken, C.G.G. (1976) Multivariate binary discrimination by the kernel method. Biometrika 63, 413–420.CrossRefGoogle Scholar
Bhattacharya, R.N. & Rao, R.R. (1986) Normal Approximations and Asymptotic Expansions. Krieger.Google Scholar
Bierens, H.J. (1982) Consistent model specification tests. Journal of Econometrics 20, 105–134.CrossRefGoogle Scholar
Bierens, H.J. (1990) A consistent conditional moment test of functional form. Econometrica 58, 1433–1458.CrossRefGoogle Scholar
Bierens, H.J. & Ploberger, W. (1997) Asymptotic theory of integrated conditional moment tests. Econometrica 65, 1129–1151.10.2307/2171881CrossRefGoogle Scholar
Cai, Z., Das, M., Xiong, H., & Wu, X. (2006) Functional coefficient instrumental variables models. Journal of Econometrics 133, 207–241.CrossRefGoogle Scholar
Cai, Z., Fan, J., & Yao, Q.W. (2000) Functional-coefficient regression models for nonlinear time series. Journal of the American Statistical Association 95, 941–956.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2000) Bootstrap tests: How many bootstraps? Econometric Reviews 19, 55–68.CrossRefGoogle Scholar
de Jong, P. (1987), A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields 75, 261–277.CrossRefGoogle Scholar
Escanciano, J.C. & Jacho-Chávez, D. (2009) Uniform in bandwidth consistency of smooth varying coefficient estimators. Economics Bulletin 29, 1892–1898.Google Scholar
Fan, J., Yao, Q.W., & Cai, Z. (2003) Adaptive varying-coefficient linear models. Journal of the Royal Statistical Society, Series B 65, 57–80.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (2000) Consistent model specification tests: Kernel-based tests versus Bierens’ ICM tests. Econometric Theory 16, 1016–1041.CrossRefGoogle Scholar
Gu, C. & Ma, P. (2005) Generalized nonparametric mixed-effect models: Computation and smoothing parameter selection. Journal of Computational and Graphical Statistics 14, 485–504.10.1198/106186005X47651CrossRefGoogle Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.CrossRefGoogle Scholar
Hall, P., Li, Q., & Racine, J.S. (2007) Nonparametric estimation of regression functions in the presence of irrelevant regressors, Review of Economics and Statistics 89, 784–789.CrossRefGoogle Scholar
Hall, P., Racine, J.S., & Li, Q. (2004) Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association 99, 1015–1026.CrossRefGoogle Scholar
Härdle, W., Hall, P., & Marron, J.S. (1988) How far are automatically chosen regression smoothing parameters from their optimum? Journal of The American Statistical Association 83, 86–101.Google Scholar
Härdle, W., Hall, P., & Marron, J.S. (1992) Regression smoothing parameters that are not far from their optimum. Journal of the American Statistical Association 87, 227–233.Google Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Härdle, W. & Marron, J. (1985) Optimal bandwidth selection in nonparametric regression function estimation. Annals of Statistics 13, 1465–1481.CrossRefGoogle Scholar
Hsiao, C., Li, Q., & Racine, J.S. (2007) A consistent model specification test with mixed categorical and continuous data. Journal of Econometrics 140, 802–826.CrossRefGoogle Scholar
Journal of Multivariate Analysis (2004) Vol. 91. Special issue on semiparametric and nonparametric mixed models.Google Scholar
Lee, J. (1990), U-Statistics: Theory and Practice. Marcel Dekker.Google Scholar
Li, Q., Huang, C.J., Li, D., & Fu, T.T. (2002) Semiparametric smooth coefficient models. Journal of Business & Economic Statistics 20, 412–422.CrossRefGoogle Scholar
Li, Q. & Racine, J. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Li, Q. & Zhou, J. (2005) The uniqueness of cross-validation selected smoothing parameters in kernel estimation of nonparametric models. Econometric Theory 21, 1017–1025.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate regression estimation: Local polynomial fitting for time series. Stochastic Processes and Their Applications 65, 81–101.CrossRefGoogle Scholar
Rosenthal, H.P. (1970) On the subspace of lp (p ≥ 1) spanned by sequences of independent random variables. Israel Journal of Mathematics 8, 273–303.CrossRefGoogle Scholar
Zeger, S.L. & Diggle, P.J. (1994) Semiparametric models for longitudinal data with application to cd4 cell numbers in HIV seroconverters. Biometrics 50, 689–699.CrossRefGoogle ScholarPubMed
Zhang, D., Lin, X., Raz, J., & Sowers, M. (1998) Semiparametric stochastic mixed models for longitudinal data. Journal of the American Statistical Association 93, 710–719.CrossRefGoogle Scholar