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Stationarity and Memory of ARCH Models

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  • Paolo Zaffaroni

Abstract

Sufficient conditions for strict stationarity of ARCH(8) are established, without imposing covariance stationarity and for any specification of the conditional second moment coefficients. GARCH(p,q) as well as the case of hyperbolically decaying coefficients are included, such as the autoregressive coefficients of ARFIMA(p,d,q), once the non-negativity constraints are imposed. Second, we show the necessary and sufficient conditions for covariance stationarity of ARCH(8), both for the levels and the squares. These prove to be much stronger than the strict stationarity conditions. The covariance stationarity condition for the levels rules out long memory in the squares.

Suggested Citation

  • Paolo Zaffaroni, 2000. "Stationarity and Memory of ARCH Models," STICERD - Econometrics Paper Series 383, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    • Handle: RePEc:cep:stiecm:383
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    File URL: https://sticerd.lse.ac.uk/dps/em/em383.pdf
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    Cited by:

    1. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
    2. Zaffaroni, Paolo & d'Italia, Banca, 2003. "Gaussian inference on certain long-range dependent volatility models," Journal of Econometrics, Elsevier, vol. 115(2), pages 199-258, August.

    More about this item

    Keywords

    ARCH(8); GARCH(p; q); nonlinear moving average representation; strict and weak stationarity; memory.;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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