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A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Donatas Surgailis
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, 08663 Vilnius, Lithuania. Email address: [email protected]
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Abstract

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.

Keywords

Linear ARCH modelquadratic ARCH modellong memorypartial sumLévy stable processfractional Brownian motion

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G18: Self-similar processes 60G52: Stable processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1198 - 1222
Copyright
Copyright © Applied Probability Trust 2008 

References

Beran, J. (2006). On location estimation for LARCH processes. J. Multivariate Anal. 97, 1766–1782.Google Scholar
Berkes, I. and Horváth, L. (2003). Asymptotic results for long memory LARCH sequences. Ann. Appl. Prob. 13, 641–668.Google Scholar
Black, F. (1976). Studies in stock price volatility changes. In Proc. 1976 Meeting Amer. Statist. Assoc., Business Econom. Session, American Statistical Association, Alexandria, VA, pp. 177–181.Google Scholar
Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049–2080.CrossRefGoogle Scholar
Davis, R. A. and Mikosch, T. (2000). The sample autocorrelations of financial time series models. In Nonlinear and Nonstationary Signal Processing, eds Fitzgerald, W. J. et al., Cambridge University Press, pp. 247–274.Google Scholar
Davis, R. and Resnick, S. (1985a). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.CrossRefGoogle Scholar
Davis, R. and Resnick, S. (1985b). More limit theory for the sample correlation function of moving averages. Stoch. Process. Appl. 20, 257–279.Google Scholar
Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions moving averages. Ann. Statist. 14, 533–558.Google Scholar
Davis, R. A. and Resnick, S. I. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Prob. 6, 1191–1210.Google Scholar
Doukhan, P., Teyssière, G. and Winant, P. (2006). A LARCH(∞) vector valued process. In Dependence in Probability and Statistics (Lecture Notes Statist. 187), eds. Bertail, P. et al., Springer, Berlin, pp. 245–258.CrossRefGoogle Scholar
Giraitis, L. and Surgailis, D. (2002). ARCH-type bilinear models with double long memory. Stoch. Process. Appl. 100, 275–300.Google Scholar
Giraitis, L., Leipus, R. and Surgailis, D. (2008). ARCH(∞) models and long memory properties. To appear in Handbook of Financial Time Series.Google Scholar
Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002–1024.Google Scholar
Giraitis, L., Leipus, R., Robinson, P. M. and Surgailis, D. (2004). LARCH, leverage and long memory. J. Financial Econometrics 2, 177–210.Google Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters–Noordhoff, Groningen.Google Scholar
Jakubowski, A. (1986). Principle of conditioning in limit theorems for sums of random variables. Ann. Prob. 14, 902–913.Google Scholar
Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.Google Scholar
Mikosch, T. (2003). Modelling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications and the Environment, eds Finkelstädt, B. and Rootzén, H., Chapman & Hall/CRC, Boca Raton, FL, pp. 185–286.Google Scholar
Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47, 67–84.Google Scholar
Sentana, E. (1995). Quadratic ARCH models. Rev. Econom. Stud. 3, 77–102.Google Scholar
Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.Google Scholar
Vaičiulis, M. (2003). Convergence of sums of Appell polynomials with infinite variance. Lithuanian Math. J. 43, 67–82.Google Scholar
Von Bahr, B. and Esséen, C.-G. (1965). Inequalities for the rth absolute moment of a sum of random variables, 1≤ r≤ 2. Ann. Math. Statist. 36, 299–303.Google Scholar