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ASYMPTOTICS FOR GARCH SQUARED RESIDUAL CORRELATIONS

Published online by Cambridge University Press:  06 June 2003

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

We develop an asymptotic theory for quadratic forms of the autocorrelations of squared residuals from a GARCH(p,q) model. Denoting by , k ≥ 1, these autocorrelations computed from a realization of length n, we show that the statistic is a matrix computed from the data, converges to the chi-square distribution with K degrees of freedom for any 1 ≤ i1 < ··· < iK. Our results are valid under weak assumptions on the innovations and model coefficients that admit that arbitrary low-order moments of the observations can be infinite. The matrix and its asymptotic limit D depend on the distribution of the innovations. A small simulation study illustrates the theory and shows, in particular, that using the matrix D computed under the assumption of normal innovations may lead to incorrect conclusions if the innovations have a different distribution.We thank the two referees for their comments and Professor Bruce E. Hansen, the co-editor in charge, for his sound advice on how to improve the paper. The work of István Berkes was supported by the Hungarian National Foundation for Scientific Research, grant T 29621. The work of Lajos Horváth and Piotr Kokoszka was supported by NATO grant PST.CLG.977607.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 4 , August 2003 , pp. 515 - 540
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Berkes, I. & L. Horváth (2001) The Efficiency of the Estimators of the Parameters in GARCH Processes. Preprint.
Berkes, I. & L. Horváth (2002) Limit Results for the Empirical Process of Squared Residuals in GARCH Models. Preprint.
Berkes, I., L. Horváth, & P. Kokoszka (2001) GARCH processes: Structure and estimation. Bernoulli. Forthcoming.
Bierens, H.J. & W. Ploberger (1997) Asymptotic theory of integrated conditional moment tests. Econometrica 65, 1129–1151.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
Bougerol, P. & N. Picard (1992a) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1730.Google Scholar
Bougerol, P. & N. Picard (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Box, G.E.P, G.M. Jenkins, & G.C. Reinsel (1994) Time Series Analysis: Forecasting and Control, 3rd ed. Englewood Cliffs, New Jersey: Prentice Hall.
de Jong, R.M. (1996) The Bierens test under data dependence. Journal of Econometrics 72, 1–32.CrossRefGoogle Scholar
Guillaume, D.M., M.M. Dacorogna, R.D. Dave, U.A. Müller, R.B. Olsen, & O.V. Pictet (1997) From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets. Finance and Stochastics 1, 95–129.CrossRefGoogle Scholar
Hong, Y. (1996) Consistent testing for serial correlation of unknown form. Econometrica 64, 837–864.CrossRefGoogle Scholar
Horváth, L. & P. Kokoszka (2001) Large sample distribution of weighted sums of ARCH(p) square residual correlations. Econometric Theory 17, 283–295.CrossRefGoogle Scholar
Horváth, L., P. Kokoszka, & G. Teyssiére (2001) Empirical process of the squared residuals of an ARCH sequence. Annals of Statistics 29, 445–469.Google Scholar
Horváth, L., P. Kokoszka, & G. Teyssiére (2002) Bootstrap Specification Tests for ARCH Based on the Empirical Process of the Squared Residuals. Preprint.
Hull, J.C. (2000) Options, Futures, and Other Derivatives. Upper Saddle River, New Jersey: Prentice Hall.
Lee, S.-W. & B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Li, W.K. (1992) On the asymptotic distribution of residual autocorrelations in nonlinear time series modelling. Biometrika 79, 435–437.CrossRefGoogle Scholar
Li, W.K. & T.K. Mak (1994) On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15, 627–636.CrossRefGoogle Scholar
Ling, S. & W.K. Li (1997) On fractionally integrated autoregressive moving–average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194.CrossRefGoogle Scholar
Ljung, G. & G. Box (1978) On a measure of lack of fit in time series models. Biometrika 66, 67–72.CrossRefGoogle Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi–maximum likelihood estimator in IGARCH (1,1) and covariance stationary GARCH (1,1) models. Econometrica 6, 575–596.CrossRefGoogle Scholar
McNeill, 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, 271–300.CrossRefGoogle Scholar
Nelson, D.B. (1990) Stationarity and persistence in GARCH(1, 1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Stout, W.F. (1974) Almost Sure Convergence. New York: Academic Press.