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ESTIMATION OF THE MAXIMAL MOMENT EXPONENT OF A GARCH(1,1) SEQUENCE

Published online by Cambridge University Press:  06 June 2003

István Berkes
Affiliation:
Hungarian Academy of Sciences
Lajos Horváth
Affiliation:
University of Utah
Piotr Kokoszka
Affiliation:
Utah State University

Abstract

We propose an estimator for the maximal moment exponent of a GARCH(1,1) sequence. We establish its consistency asymptotic normality with rate n−1/2. Finite sample properties are investigated by means of a small simulation study.The research for this paper was partially supported by NSF grant INT-0223262. István Berkes and Lajos Horváth were supported by the Hungarian National Foundation for Scientific Research, grant T 29621. Piotr Kokoszka and Lajos Horváth were supported by NATO grant PST.CLG.977607.

Type
Research Article
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Berkes, I. & L. Horváth, (2003) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics, forthcoming.Google Scholar
Berkes, I., L. Horváth, & P. Kokoszka (2002) GARCH processes: Structure and estimation. Bernoulli, forthcoming. Preprint available at http://math.usu.edu/∼piotr/research.html.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
Csörgő, S. (1982) The empirical moment generating function. In B.V. Gnedenko, M.L. Puri, & I. Vincze, Nonparametric Statistical Inference (Colloquia Mathematica Societatis Janos Bolyai 32), pp. 139159. Amsterdam: North-Holland.
Csörgő, S. & J.L. Teugels (1990) Empirical Laplace transform and approximation of compound distribution. Journal of Applied Probability 27, 88101.CrossRefGoogle Scholar
Dempster, M. (ed.) (2002) Risk Management: Value at Risk and Beyond. Cambridge University Press.
Embrechts, P., C. Klüppelberg, & T. Mikosch (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.
Feuerverger, A. & P. Hall (1999) Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics 27, 760781.Google Scholar
Goldie, C.M. & R.L. Smith (1987) Slow variation with remainder: Theory and applications Quarterly Journal of Mathematics, Oxford 38, 4571.Google Scholar
Horváth, L., P.S. Kokoszka, & G. Teyssiére (2001) Empirical process of the squared residuals of an ARCH sequence. Annals of Statistics 29, 445469.Google Scholar
Hsing, T. (1991) On tail index estimation using dependent data. Annals of Statistics 19, 15471569.CrossRefGoogle Scholar
Hull, J.C. (2000) Options, Futures, and Other Derivatives. Upper Saddle River, NJ: Prentice-Hall.
Leadbetter, M.R. (1991) On a basis for “peaks over threshold” modelling. Statistics and Probability Letters 12, 357362.CrossRefGoogle Scholar
Lee, S.W. & B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 2952.CrossRefGoogle Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575596.CrossRefGoogle Scholar
McNeil, A.J. & R. Frey (2000) Estimation of tail-related risk measures for heteroskedastic financial time series: An extreme value approach. Journal of Empirical Finance 7, 271300. Available at http://www.math.ethz.ch/∼mcneil/.Google Scholar
Mikosch, T. & C. Stărică (2000) Limit theory for the sample autocorrelation and extremes of a GARCH(1,1) process. Annals of Statistics 28, 14271451.Google Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318334.CrossRefGoogle Scholar
Novak, S.Y. (2000) Confidence intervals for a tail index estimator. In J. Franke, W. Haerdle, & G. Stahl (eds.), Proceedings of the Conference “Measuring Risk in Complex Stochastic Systems, pp. 229236. Berlin: Springer-Verlag.
Pitts, S.M., P. Embrechts, & R. Grübel (1996) Confidence bounds for the adjustment coefficient. Advanced Applied Probability 28, 802827.CrossRefGoogle Scholar
Quintos, C., Z. Fan, & P.C.B. Phillips (2001). Structural change tests in tail behaviour and the Asian crisis. Review of Economic Studies 68 633663.CrossRefGoogle Scholar
Resnick, S.I. & C. Stărică (1995) Consistency of Hill's estimator for dependent data. Journal of Applied Probability 32, 139167.CrossRefGoogle Scholar
Resnick, S.I. & C. Stărică (1997) Asymptotic behaviour of Hill's estimator for autoregressive data. Communications in Statistics: Stochastic Models 13, 703721.Google Scholar
Resnick, S.I. & C. Stărică (1998) Tail estimation for dependent data. Annals of Applied Probability 8, 11561183.Google Scholar