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Regular variation of GARCH processes

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  • Basrak, Bojan
  • Davis, Richard A.
  • Mikosch, Thomas
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    Abstract

    We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the second moment does not exist, the sample autocorrelations have non-degenerate limit distributions.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 99 (2002)
    Issue (Month): 1 (May)
    Pages: 95-115

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    Handle: RePEc:eee:spapps:v:99:y:2002:i:1:p:95-115

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    Related research

    Keywords: Point process Vague convergence Multivariate regular variation Mixing condition Stationary process Heavy tail Sample autocovariance Sample autocorrelation GARCH Finance Markov chain;

    References

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    1. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    2. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(03), pages 318-334, September.
    3. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
    4. de Haan, L. & Resnick, S. I., 1981. "On the observation closest to the origin," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 301-308, August.
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