Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T05:19:30.704Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC INFERENCE FOR CONDITIONAL QUANTILES OF TIME SERIES

Published online by Cambridge University Press:  16 January 2013

Ke-Li Xu
Ke-Li Xu*
Affiliation:
Texas A&M University
*
*Address correspondence to Ke-Li Xu, 3063 Allen, 4228 TAMU, Department of Economics, Texas A&M University, College Station, TX 77843-4228, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers model-free hypothesis testing and confidence interval construction for conditional quantiles of time series. A new method, which is based on inversion of the smoothed empirical likelihood of the conditional distribution function around the local polynomial estimator, is proposed. The associated inferential procedures do not require variance estimation, and the confidence intervals are automatically shaped by data. We also construct the bias-corrected empirical likelihood, which does not require undersmoothing. Limit theories are developed for mixing time series. Simulations show that the proposed methods work well in finite samples and outperform the normal confidence interval. An empirical application to inference of the conditional value-at-risk of stock returns is also provided.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 4 , August 2013 , pp. 673 - 698
Copyright
Copyright © Cambridge University Press 2013 

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

Basrak, B., Davis, R.A., & Mikosch, T. (2002) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.CrossRefGoogle Scholar
Buchinsky, M. (1998) Recent advances in quantile regression models. Journal of Human Resources 33, 88–126.CrossRefGoogle Scholar
Cai, Z. (2002) Regression quantiles for time series data. Econometric Theory 18, 169–192.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Chan, N.H., Deng, S.J., Peng, L., & Xia, Z. (2007) Interval estimation for the conditional value-at-risk based on GARCH models with heavy tailed innovations. Journal of Econometrics 137, 556–576.CrossRefGoogle Scholar
Chaudhuri, P. (1991) Nonparametric estimates of regression quantiles and their local Bahadur representation. Annals of Statistics 19, 760–777.CrossRefGoogle Scholar
Chen, S.X. & Hall, P. (1993) Smoothed empirical likelihood confidence intervals for quantiles. Annals of Statistics 21, 1166–1181.CrossRefGoogle Scholar
Chen, S.X. & Qin, Y.S. (2000) Empirical likelihood confidence intervals for local linear smoothers. Biometrika 87, 946–953.CrossRefGoogle Scholar
Christoffersen, P. & Gonçalves, S. (2005) Estimation risk in financial risk management. Journal of Risk 7, 1–28.CrossRefGoogle Scholar
Cook, R.D. & Weisberg, S. (1990) Confidence curves in nonlinear regression. Journal of American Statistical Association 85, 544–551.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory: An Introduction for Econometricians. Oxford University Press.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J. & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer.CrossRefGoogle Scholar
Fan, J., Yao, Q., & Tong, H. (1996) Estimation of conditional densities and sensitivity measures. Biometrika 83, 189–206.CrossRefGoogle Scholar
Gao, F. & Song, F. (2008) Estimation risk in GARCH VaR and ES estimates. Econometric Theory 24, 1404–1428.CrossRefGoogle Scholar
Gourieroux, C. & Jasiak, J. (2010) Local likelihood density estimation and value at risk. Journal of Probability and Statistics 2010, article 754251.Google Scholar
Hall, P., Racine, J., & Li, Q. (2004) Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association 99, 1015–1026.CrossRefGoogle Scholar
Hall, P., Wolff, R.C.L., & Yao, Q. (1999) Methods for estimating a conditional distribution function. Journal of the American Statistical Association 94, 154–163.CrossRefGoogle Scholar
Hansen, B. (2004) Nonparametric Estimation of Smooth Conditional Distributions. Technical report, University of Wisconsin.Google Scholar
Hart, J.D. (1996) Some automated methods of smoothing time-dependent data. Journal of Nonparametric Statistics 6, 115–142.CrossRefGoogle Scholar
Jorion, P. (2000) Value-at-Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill.Google Scholar
Kitamura, Y. (2006) Empirical likelihood methods in econometrics: Theory and practice. In Blundell, R., Torsten, P. & Newey, W.K. (eds.), Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Cambridge University Press.Google Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Koenker, R., Portnoy, S., & Ng, P. (1992) Nonparametric estimation of conditional quantile functions. In Dodge, Y. (ed.), L1-Statistical Analysis and Related Methods, pp. 217–229. Elsevier.Google Scholar
Li, Q. & Racine, J.S. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Li, Q. & Racine, J.S. (2008) Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data. Journal of Business and Economic Statistics 26, 423–434.CrossRefGoogle Scholar
Masry, E. & Fan, J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 165–179.CrossRefGoogle Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Otsu, T. (2008) Conditional empirical likelihood estimation and inference for quantile regression models. Journal of Econometrics 142, 508–538.CrossRefGoogle Scholar
Owen, A. (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.CrossRefGoogle Scholar
Owen, A. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.CrossRefGoogle Scholar
Owen, A. (2001) Empirical Likelihood. Chapman and Hall/CRC.Google Scholar
Tsay, R.S. (2005) Analysis of Financial Time Series, 2nd Ed. Wiley-Interscience.CrossRefGoogle Scholar
Whang, Y.-J. (2006) Smoothed empirical likelihood methods for quantile regression models. Econometric Theory 22, 173–205.CrossRefGoogle Scholar
Yu, K. & Jones, M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228–237.CrossRefGoogle Scholar