Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-18T04:15:25.220Z Has data issue: false hasContentIssue false
  • English
  • Français

ADAPTIVE NONPARAMETRIC REGRESSION WITH CONDITIONAL HETEROSKEDASTICITY

Published online by Cambridge University Press:  08 September 2014

Sainan Jin ,
Liangjun Su  and
Zhijie Xiao
Sainan Jin*
Affiliation:
Singapore Management University
Liangjun Su
Affiliation:
Singapore Management University
Zhijie Xiao
Affiliation:
Boston College
*
*Address correspondence to Sainan Jin, School of Economics, Singapore Management University, 90 Stamford Road, Singapore 178903; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study adaptive nonparametric regression estimation in the presence of conditional heteroskedastic error terms. We demonstrate that both the conditional mean and conditional variance functions in a nonparametric regression model can be estimated adaptively based on the local profile likelihood principle. Both the one-step Newton–Raphson estimator and the local profile likelihood estimator are investigated. We show that the proposed estimators are asymptotically equivalent to the infeasible local likelihood estimators [e.g., Aerts and Claeskens (1997) Journal of the American Statistical Association 92, 1536–1545], which require knowledge of the error distribution. Simulation evidence suggests that when the distribution of the error term is different from Gaussian, the adaptive estimators of both conditional mean and variance can often achieve significant efficiency over the conventional local polynomial estimators.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 6 , December 2015 , pp. 1153 - 1191
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

REFERENCES

Aerts, M. & Claeskens, G. (1997) Local polynomial estimation in multiparameter likelihood models. Journal of the American Statistical Association 92, 1536–1545.Google Scholar
Ai, C. (1997) A semiparametric maximum likelihood estimator. Econometrica 65, 933–963.Google Scholar
Akritas, M. & Van Keilegom, I. (2001) Nonparametric estimation of the residual distribution. Scandinavian Journal of Statistics 28, 549–568.Google Scholar
Andrews, D.W.K. (1994a) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 43–72.Google Scholar
Andrews, D.W.K. (1994b) Empirical process methods in econometrics. In Engle, R.F. & McFadden, D. (eds.), Handbook of Econometrics, Vol. IV. North-Holland.Google Scholar
Andrews, D.W.K. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–596.CrossRefGoogle Scholar
Bernstein, D.S. (2005) Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press.Google Scholar
Bickel, P.J. (1982) On adaptive estimation. Annals of Statistics 10, 647–671.Google Scholar
Claeskens, G. & Aerts, M. (2000) On local estimating equations in additive multiparameter models. Statistics and Probability Letters 49, 139–148.CrossRefGoogle Scholar
Claeskens, G. & Van Keilegom, I. (2003) Bootstrap confidence bands for regression curves and their derivatives. Annals of Statistics 31, 1852–1884.Google Scholar
Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 998–1004.Google Scholar
Fan, J. (1993) Local linear regression smoothers and their minimax efficiencies. Annals of Statistics 21, 196–216.Google Scholar
Fan, J., Farmen, M., & Gijbels, I. (1998) Local maximum likelihood estimation and inference. Journal of the Royal Statistics Society, Series B 60, 591–608.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman and Hall.Google Scholar
Fan, J., Heckman, N.E., & Wand, M.P. (1995) Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. Journal of the American Statistical Association 90, 141–150.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.Google Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel regression. Econometric Theory 24, 726–748.Google Scholar
Härdle, W. & Tsybakov, A. (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 233–242.Google Scholar
Kreiss, J. (1987) On adaptive estimation in stationary ARMA process. Annals of Statistics 15, 112–133.Google Scholar
Lee, A.J. (1990) U-statistics: Theory and Practice. Marcel Dekker.Google Scholar
Linton, O. & Xiao, Z. (2007) A nonparametric regression estimator that adapts to error distribution of unknown form. Econometric Theory 23, 371–413.Google Scholar
Masry, E. (1996a) Multivariate regression estimation: Local polynomial fitting for time series. Stochastic Processes and Their Applications 65, 81–101.Google Scholar
Masry, E. (1996b) Multivariate local polynomial regression for time series: Uniform strong consistency rates. Journal of Time Series Analysis 17, 571–599.Google Scholar
Rudin, W. (1976) Principles of Mathematical Analysis, 3rd ed. McGraw-Hill.Google Scholar
Ruppert, D., Wand, M.P., Holst, U., & Hössjer, O. (1997) Local polynomial variance function estimation. Technometrics 39, 262–273.CrossRefGoogle Scholar
Staniswalis, J.E. (1989) On the kernel estimate of a regression function in likelihood-based models. Journal of the American Statistical Association 84, 276–283.Google Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Cambridge University Press.Google Scholar
Ziegelmann, F. (2002) Nonparametric estimation of volatility functions: The local exponential estimator. Econometric Theory 18, 985–991.CrossRefGoogle Scholar
Supplementary material: File

Jin et al. supplementary material

Supplementary data

Download Jin et al. supplementary material(File)
File 135.1 KB
Supplementary material: PDF

Jin et al. supplementary material

Supplementary material

Download Jin et al. supplementary material(PDF)
PDF 350.3 KB