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Conditional Quantile Estimation and Inference for Arch Models

Published online by Cambridge University Press:  11 February 2009

Roger Koenker
Affiliation:
University of Illinois
Quanshui Zhao
Affiliation:
The City University of Hong Kong

Abstract

Quantile regression methods are suggested for a class of ARCH models. Because conditional quantiles are readily interpretable in semiparametric ARCH models and are inherendy easier to estimate robustly than population moments, they offer some advantages over more familiar methods based on Gaussian likelihoods. Related inference methods, including the construction of prediction intervals, are also briefly discussed.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Bera, A.K. & Higgihs, M.L. (1993) On ARCH models: Properties, estimation, and testing, Journal of Economic Survey 7, 305366.CrossRefGoogle Scholar
Bickel, P.J. (1975) One-step Huber estimates in linear models. Journal of the American Statistical Association 70, 428433.CrossRefGoogle Scholar
Bickel, P.J. (1978) Using residuals robustly. Annals of Statistics 6, 266291.CrossRefGoogle Scholar
Bickel, P.J. & Lehmann, E. (1976) Descriptive statistics for nonparametric models. Ill: Dispersion. Annals of Statistics 4, 11391158.CrossRefGoogle Scholar
Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542547.CrossRefGoogle Scholar
Bollerslev, T., Chou, R., Kroner, K. (1992) ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics 50, 559.CrossRefGoogle Scholar
Bollerslev, T., Engle, R.F., & Nelson, D. (1994) ARCH models. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4. Amsterdam: North-Holland.Google Scholar
Brown, B.M. (1971) Martingale central limit theorem. Annals of Mathematical Statistics 1, 5966.CrossRefGoogle Scholar
Carroll, R.J. & Ruppert, D. (1988) Transformation and Weighting in Regression. London: Chapman and Hall.CrossRefGoogle Scholar
Drost, F.C., Klaassen, C.A.J., & Werker, B.J.M. (1994) Adaptiveness in time series models. In Mandl, P. & Hušková, M. (eds.), Asymptotic Statistics: Proceedings of the 5th Prague Symposium. New York: Springer-Verlag.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 9871007.CrossRefGoogle Scholar
Geweke, Geweke. (1989) Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics 40, 6386.CrossRefGoogle Scholar
Granger, C.W.J., White, H., & Kamstra, M. (1989) Interval forecasting: An analysis based upon ARCH-quantile estimators. Journal of Econometrics 40, 8796.CrossRefGoogle Scholar
Gutenbrunner, C., Jurečková, J., Koenker, R., & Portnoy, S.L. (1994) Tests of linear hypotheses based on regression rank scores. Journal of Nonparametric Statistics 2, 307331.CrossRefGoogle Scholar
Hájek, Y. (1969) Nonparametric Statistics. Holden-Day.Google Scholar
Huber, P.J. (1967) Behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium 1, 221233.Google Scholar
Jurečkova, J. (1977) Asymptotic relations of M-estimates and R-estimates in linear regression models. Annals of Statistics 5, 464472.CrossRefGoogle Scholar
Jurečkova, J. & Prochazka, B. (1994) Regression quantiles and trimmed least squares estimators in the nonlinear regression model. Journal of Nonparametric Statistics 3, 201222.Google Scholar
Koenker, R. (1994) Confidence intervals for regression quantiles. In Mandl, P. and HuSkova, M. (eds.), Asymptotic Statistics: Proceedings of the 5th Prague Symposium. New York: Springer-Verlag.Google Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–30.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. (1982) Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50, 4361.CrossRefGoogle Scholar
Koenker, R. & d'Orey, V. (1987) Computing regression quantiles. Applied Statistics 36, 383393.CrossRefGoogle Scholar
Koenker, R. & Zhao, Q. (1994) L-estimation for linear heteroscedastic models. Journal of Nonparametric Statistics 3, 223235.CrossRefGoogle Scholar
Koul, H.L. & Mukherjee, (1994) Regression quantiles and related processes under long range dependence. Journal of Multivariate Statistics 51, 318337.CrossRefGoogle Scholar
Koul, H.L. & Saleh, E. (1992) Autoregression Quantiles and Related Rank-Scores Process. Technical report RM-527, Michigan State University.Google Scholar
Linton, O. (1994) Adaptive estimation of ARCH models. Econometric Theory 9, 539569.CrossRefGoogle Scholar
Nelson, D. (1991) Conditional heteroscedasticity in asset returns. Econometrica 59, 347370.CrossRefGoogle Scholar
Nelson, D. & Foster, D. (1994) Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 142.CrossRefGoogle Scholar
Newey, W. & Powell, J. (1987) Asymmetric least squares estimation and testing. Econometrica 55, 819847.CrossRefGoogle Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186199.CrossRefGoogle Scholar
Portnoy, S.L. (1991) Asymptotic behavior of regression quantiles in non-stationary, dependent cases. Journal of Multivariate Analysis 38, 100113.CrossRefGoogle Scholar
Portnoy, S.L. & Zhou, Q. (1994) A Direct Approach to Construct Confidence Intervals and Confidence Bands for Regression Quantiles. Preprint.Google Scholar
Ruppert, D. & Carroll, R.J. (1980) Trimmed least square estimation in the linear model. Journal of the American Statistical Association 75, 828838.CrossRefGoogle Scholar
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 44, 11151154.CrossRefGoogle Scholar
Stout, W.F. (1974) Almost Sure Convergence. New York: Wiley.Google Scholar
Taylor, S. (1986) Modeling Financial Time Series. New York: Wiley.Google Scholar
Wolak, F. (1987) An exact test for multiple inequality and equality constraints in the linear regression model. Journal of the American Statistical Association 82, 782793.CrossRefGoogle Scholar