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Stationarity and Memory of ARCH Models

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  • Paolo Zaffaroni
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    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.

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    File URL: http://sticerd.lse.ac.uk/dps/em/em383.pdf
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    Bibliographic Info

    Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2000/383.

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    Date of creation: Mar 2000
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    Handle: RePEc:cep:stiecm:/2000/383

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    Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

    Related research

    Keywords: ARCH(8); GARCH(p; q); nonlinear moving average representation; strict and weak stationarity; memory.;

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    1. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
    2. Karanasos, Menelaos, 1999. "The second moment and the autocovariance function of the squared errors of the GARCH model," Journal of Econometrics, Elsevier, vol. 90(1), pages 63-76, May.
    3. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    4. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(01), pages 29-52, March.
    5. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(01), pages 107-131, April.
    6. Nelson, Daniel B & Cao, Charles Q, 1992. "Inequality Constraints in the Univariate GARCH Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(2), pages 229-35, April.
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    Cited by:
    1. Paolo Zaffaroni, 2003. "Gaussian inference on certain long-range dependent volatility models," Temi di discussione (Economic working papers) 472, Bank of Italy, Economic Research and International Relations Area.
    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.

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