Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T18:17:32.706Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENT REGRESSIONS VIA OPTIMALLY COMBINING QUANTILE INFORMATION

Published online by Cambridge University Press:  06 June 2014

Zhibiao Zhao  and
Zhijie Xiao
Zhibiao Zhao
Affiliation:
Penn State University
Zhijie Xiao*
Affiliation:
Boston College
*
*Address correspondence to Zhijie Xiao, Department of Economics, Boston College, Chestnut Hill, MA 02467, USA and Shandong University, China; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a generally applicable framework for constructing efficient estimators of regression models via quantile regressions. The proposed method is based on optimally combining information over multiple quantiles and can be applied to a broad range of parametric and nonparametric settings. When combining information over a fixed number of quantiles, we derive an upper bound on the distance between the efficiency of the proposed estimator and the Fisher information. As the number of quantiles increases, this upper bound decreases and the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound under appropriate conditions. In the case of nonregular statistical estimation, the proposed estimator leads to super-efficient estimation. We illustrate the proposed method for several widely used regression models. Both asymptotic theory and Monte Carlo experiments show the superior performance over existing methods.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 6 , December 2014 , pp. 1272 - 1314
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

Akahira, M. & Takeuchi, K. (1995) Non-Regular Statistical Estimation. Springer.CrossRefGoogle Scholar
Beran, R. (1974) Asymptotically efficient adaptive rank estimates in location models. Annals of Statistics 2, 248–266.CrossRefGoogle Scholar
Bickel, P. (1982) On adaptive estimation. Annals of Statistics 10, 647–671.CrossRefGoogle Scholar
Black, F. (1972) Capital market equilibrium with restricted borrowing. Journal of Business 45, 444–454.CrossRefGoogle Scholar
Bradic, J., Fan, J., & Wang, W. (2011) Penalized composite quasi-likelihood for ultrahigh dimensional variable selection. Journal of the Royal Statistical Society, Series B 73, 325–349.Google Scholar
Chamberlain, G. (1994) Quantile regression, censoring and the structure of wages. Advances in Econometrics, Sixth World Congress, vol. 1, pp. 171–210.Google Scholar
Chen, X., Linton, O., & Jacho-Chavez, D. (2011) Averaging of Moment Condition Estimators. Working paper.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1007.Google Scholar
Fan, J., Farmen, M., & Gijbels, I. (1998) Local maximum likelihood estimation and inference. Journal of the Royal Statistical Society, Series B 60, 591–608.CrossRefGoogle Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman and Hall.Google Scholar
He, X. & Shao, Q.M. (1996) A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs. Annals of Statistics 24, 2608–2630.Google Scholar
Jurečková, J. & Procházka, B. (1994) Regression quantiles and trimmed least squares estimator in nonlinear regression model. Journal of Nonparametric Statistics 3, 201–222.Google Scholar
Kai, B., Li, R., & Zou, H. (2010) Local composite quantile regression smoothing: An efficient and safe alternative to local polynomial regression. Journal of the Royal Statistical Society, Series B 72, 49–69.Google Scholar
Knight, K. (1989) Limit Theory for Autoregressive-parameter estimates in an infinite-variance random walk. The Canadian Journal of Statistics 17, 261–278.Google Scholar
Koenker, R. (1984) A note on L-estimates for linear models. Statistics & Probability Letters 2, 323–325.Google Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.Google Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–49.CrossRefGoogle Scholar
Koenker, R. & Zhao, Q. (1994) L-estimation for linear heteroscedastic models. Journal of Nonparametric Statistics 3, 223–235.Google Scholar
Koenker, R. & Zhao, Q. (1996) Conditional quantile estimation and inference for ARCH models. Econometric Theory 12, 793–813.CrossRefGoogle Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.CrossRefGoogle Scholar
Portnoy, S. & Koenker, R. (1989) Adaptive L-estimation of linear models. Annals of Statistics 17, 362–381.Google Scholar
Ruppert, D., Sheather, S.J., & Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association 90, 1257–1270.Google Scholar
Sharpe, W.F. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425–442.Google Scholar
Silverman, B.W. (1986) Density Estimation. Chapman and Hall.Google Scholar
Stone, C. (1975) Adaptive maximum likelihood estimation of a location parameter. Annals of Statistics 3, 267–284.Google Scholar
Stout, W.F. (1974) Almost Sure Convergence. Academic Press.Google Scholar
Wang, H., Zhu, Z., & Zhou, J. (2009) Quantile regression in partially linear varying coefficient models. Annals of Statistics 37, 3841–3866.Google Scholar
Xiao, Z. & Koenker, R. (2009) Conditional quantile estimation and inference for GARCH models. Journal of the American Statistical Association 104, 1696–1712.CrossRefGoogle Scholar
Yu, K. & Jones, M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228–237.Google Scholar
Zhao, Q. (2001) Asymptotically efficient median regression in the presence of heteroskedasticity of unknown form. Econometric Theory 17, 765–784.Google Scholar
Zou, H. & Yuan, M. (2008) Composite quantile regression and the oracle model selection theory. Annals of Statistics 36, 1108–1126.Google Scholar