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Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

Part of:Inference from stochastic processesLimit theoremsApplicationsStochastic processes

Published online by Cambridge University Press:  04 January 2016

Rafał Kulik and
Philippe Soulier
Rafał Kulik*
Affiliation:
University of Ottawa
Philippe Soulier*
Affiliation:
Université Paris Ouest-Nanterre
*
∗ Postal address: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa ON, K1N 6N5, Canada. Email address: rkulik@uottawa.ca
∗∗ Postal address: Département de Mathématiques, Université Paris Ouest-Nanterre, 200 Avenue de la République, 92000 Nanterre Cedex, France. Email address: philippe.soulier@u-paris10.fr
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Abstract

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In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Keywords

Heavy taillong-range dependencesample autocovariancestochastic volatility

MSC classification

Primary:60G55: Point processes
Secondary:60F05: Central limit and other weak theorems62M10: Time series, auto-correlation, regression, etc.62P05: Applications to actuarial sciences and financial mathematics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012, pp. 1113 - 1141
Copyright
© Applied Probability Trust 

References

Anderson, T. W. (1971). The Statistical Analysis of Time Series. John Wiley, New York.Google Scholar
Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Prob. 22, 2242–2274.Google Scholar
Baillie, R. T., Bollerslev, T. and Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econometrics 74, 3–30.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
Bollerslev, T. and Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. J. Econometrics 73, 151–184.Google Scholar
Breidt, F. J. and Davis, R. A. (1998). Extremes of stochastic volatility models. Ann. Appl. Prob. 8, 664–675.Google Scholar
Breidt, F. J., Crato, N. and de Lima, P. (1998). The detection and estimation of long memory in stochastic volatility. J. Econometrics 83, 325–348.Google Scholar
Davis, R. A. (1983). Stable limits for partial sums of dependent random variables. Ann. Prob. 11, 262–269.CrossRefGoogle Scholar
Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917.Google Scholar
Davis, R. A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelation. In Probability, Statistics and Seismology (J. Appl. Prob. Spec. Vol. 38A), ed. Daley, D. J., Applied Probability Trust, Sheffield, pp. 93–104.Google Scholar
Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Davis, R. and Resnick, S. (1985). 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 of moving averages. Ann. Statist. 14, 533–558.CrossRefGoogle Scholar
Douc, R., Roueff, F. and Soulier, P. (2008). On the existence of some ARCH(∞) processes. Stoch. Process. Appl. 118, 755–761.Google Scholar
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google 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., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002–1024.CrossRefGoogle 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
Horváth, L. and Kokoszka, P. (2008). Sample autocovariances of long-memory time series. Bernoulli 14, 405–418.Google Scholar
Hosking, J. R. M. (1996). Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series. J. Econometrics 73, 261–284.CrossRefGoogle Scholar
Jach, A., McElroy, T. and Politis, D. N. (2012). Subsampling inference for the mean of heavy-tailed long-memory time series. J. Time Ser. Anal. 33, 96–111.Google Scholar
Kokoszka, P. S. and Taqqu, M. S. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist. 24, 1880–1913.Google Scholar
McElroy, T. and Politis, D. (2007). Self-normalization for heavy-tailed time series with long memory. Statistica Sinica 17, 199–220.Google Scholar
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
Robinson, P. M. (2001). The memory of stochastic volatility models. J. Econometrics 101, 195–218.Google Scholar
Robinson, P. M. and Zaffaroni, P. (1997). Modelling nonlinearity and long memory in time series. In Nonlinear Dynamics and Time Series (Montreal, PQ, 1995; Fields Inst. Commun. 11), American Mathematical Society, Providence, RI, pp. 161–170.Google Scholar
Robinson, P. M. and Zaffaroni, P. (1998). Nonlinear time series with long memory: a model for stochastic volatility. J. Statist. Planning. Infer. 68, 359–371.Google Scholar
Surgailis, D. (2008). A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares. Adv. Appl. Prob. 40, 1198–1222.Google Scholar
Surgailis, D. and Viano, M.-C. (2002). Long memory properties and covariance structure of the EGARCH model. ESAIM Prob. Statist. 6, 311–329 (electronic).CrossRefGoogle Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.Google Scholar
Wu, W. B., Huang, Y. and Zheng, W. (2010). Covariances estimation for long-memory processes. Adv. Appl. Prob. 42, 137–157.Google Scholar