Electronic Journal of Statistics

Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data

Ingrid Van Keilegom and Lan Wang

Full-text: Open access

Abstract

We consider the problem of modeling heteroscedasticity in semiparametric regression analysis of cross-sectional data. Existing work in this setting is rather limited and mostly adopts a fully nonparametric variance structure. This approach is hampered by curse of dimensionality in practical applications. Moreover, the corresponding asymptotic theory is largely restricted to estimators that minimize certain smooth objective functions. The asymptotic derivation thus excludes semiparametric quantile regression models. To overcome these drawbacks, we study a general class of location-dispersion regression models, in which both the location function and the dispersion function are semiparametrically modeled. We establish unified asymptotic theory which is valid for many commonly used semiparametric structures such as the partially linear structure and single-index structure. We provide easy to check sufficient conditions and illustrate them through examples. Our theory permits non-smooth location or dispersion functions, thus allows for semiparametric quantile heteroscedastic regression and robust estimation in semiparametric mean regression. Simulation studies indicate significant efficiency gain in estimating the parametric component of the location function. The results are applied to analyzing a data set on gasoline consumption.

Article information

Source
Electron. J. Statist., Volume 4 (2010), 133-160.

Dates
First available in Project Euclid: 1 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1265033306

Digital Object Identifier
doi:10.1214/09-EJS547

Mathematical Reviews number (MathSciNet)
MR2645480

Zentralblatt MATH identifier
1329.62203

Keywords
Dispersion function heteroscedasticity partially linear model quantile regression semiparametric regression single-index model variance function

Citation

Van Keilegom, Ingrid; Wang, Lan. Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data. Electron. J. Statist. 4 (2010), 133--160. doi:10.1214/09-EJS547. https://projecteuclid.org/euclid.ejs/1265033306


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References

  • Akritas, M. G. and Van Keilegom, I. (2001). Nonparametric estimation of the residual distribution., Scand. J. Statist., 28, 549–568.
  • Carroll, R. J. (2003). Variances are not always nuisance parameters., Biometrics, 59, 211–220.
  • Chen, X., Linton, O. B. and Van Keilegom, I. (2003). Estimation of semiparametric models when the criterion function is not smooth., Econometrica, 71, 1591–1608.
  • Chiou, J. M. and Müller, H. G. (2004). Quasi-likelihood regression with multiple indices and smooth link and variance functions., Scand. J. Statist., 31, 367–386.
  • Davidian, M. and Carroll, R. J. (1987). Variance function estimation., J. Amer. Statist. Assoc., 82, 1079–1091.
  • Davidian, M., Carroll, R. and Smith, W. (1988). Variance functions and the minimum detectable concentration in assays., Biometrika, 75, 549–556.
  • Delecroix, M., Hristache, M. and Patilea, V. (2006). On semiparametric M-estimation in single-index regression., J. Statist. Plann. Infer., 136, 730–769.
  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity and estimates of the variance of UK inflation., Econometrica, 50, 987–1008.
  • Greene, W. H. (2002), Econometric Analysis, (5th Edition). Prentice Hall, New Jersey.
  • Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models., Ann. Statist., 21, 157–178.
  • Härdle, W., Liang, H. and Gao, J. (2000)., Partially Linear Models. Physica-Verlag, Heidelberg.
  • He, X. and Liang, H. (2000). Quantile regression estimates for a class of linear and partially linear errors-in-variables models., Statist. Sinica, 10, 129–140.
  • Horowitz, J. L. and Lee, S. (2005). Nonparametric estimation of an additive quantile regression model., J. Amer. Statist. Assoc., 100, 1238–1249.
  • Koenker, R. and Zhao, Q. (1994). L-estimation for linear heteroscedastic models., J. Nonpar. Statist., 3, 223–235.
  • Lee, S. (2003). Efficient semiparametric estimation of a partially linear quantile regression model., Econometric Theory, 19, 1–31.
  • Liang, H., Härdle, W. and Carroll, R. (1999). Estimation in a semiparametric partially linear errors-in-variables model., Ann. Statist., 27, 1519–1535.
  • Ma, Y., Chiou, J.-M. and Wang, N. (2006). Efficient semiparametric estimator for heteroscedastic partially linear models., Biometrika, 93, 75–84.
  • Müller, H.-G. and Zhao, P.-L. (1995). On a semiparametric variance function model and a test for heteroscedasticity., Ann. Statist., 23, 946–967.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003)., Semiparametric Regression. Cambridge University Press.
  • Ruppert, D., Wand, M. P., Host, U. and Hössjer, O. (1997). Local polynomial variance-function estimation., Technometrics, 39, 262–273.
  • Schick, A. (1996). Weighted least squares estimates in partly linear regression models., Statist. Probab. Letters, 27, 281–287.
  • Van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer-Verlag, New York.
  • Van Keilegom, I. and Carroll, R. J. (2007). Backfitting versus profiling in general criterion functions., Statist. Sinica, 17, 797–816.
  • Xia, Y., Tong, H. and Li, W. K. (2002). Single-index volatility models and estimation., Statist. Sin., 12, 785–799.
  • Yatchew, A. (2003)., Semiparametric Regression for the Applied Econometrician. Cambridge University Press.
  • Yu, K. and Jones, M. C. (1998). Local linear quantile regression., J. Amer. Statist. Assoc., 94, 228–237.
  • Zhao, Q. (2001). Asymptotically efficient median regression in the presence of heteroscedasticity of unknown form., Econometric Theory, 17, 765–784.