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The Mean Variance Mixing GARCH (1,1) model

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  • Lars Forsberg
  • Anders Eriksson
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    Abstract

    Here we present a general framework for a GARCH (1,1) type of process with innovations with a probability law of the mean- variance mixing type, therefore we call the process in question the mean variance mixing GARCH \ (1,1) or MVM GARCH\(1,1). One implication is a GARCH\ model with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compensate for time dependence that affects the mean dynamics. From a probabilistic viewpoint the idea is straightforward. We just construct our stochastic process from the desired behavior of the cumulants. Further we provide explicit expressions for the unconditional second to fourth cumulants for the process in question. In the paper we present a specification of the MVM-GARCH process where the mixing variable is of the inverse Gaussian type. On the basis on this assumption we can formulate a maximum likelihood based approach for estimating the process closely related to the approach used to estimate an ordinary GARCH (1,1). Under the distributional assumption that the mixing random process is an inverse Gaussian i.i.d process the MVM-GARCH process is then estimated on log return data from the Standard and Poor 500 index. An analysis for the conditional skewness and kurtosis implied by the process is also presented in the paper

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    Bibliographic Info

    Paper provided by Econometric Society in its series Econometric Society 2004 Australasian Meetings with number 323.

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    Date of creation: 11 Aug 2004
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    Handle: RePEc:ecm:ausm04:323

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    Keywords: GARCH Skewness Conditional Skewness;

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    1. Drost, F.C. & Nijman, T.E., 1992. "Temporal aggregation of GARCH processes," Discussion Paper, Tilburg University, Center for Economic Research 1992-40, Tilburg University, Center for Economic Research.
    2. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
    3. Drost, F.C. & Werker, B.J.M., 1994. "Closing the GARCH gap: Continuous time GARCH modeling," Discussion Paper, Tilburg University, Center for Economic Research 1994-2, Tilburg University, Center for Economic Research.
    4. Andersson, Jonas, 2001. "On the Normal Inverse Gaussian Stochastic Volatility Model," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 19(1), pages 44-54, January.
    5. Harvey, Campbell R. & Siddique, Akhtar, 1999. "Autoregressive Conditional Skewness," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 34(04), pages 465-487, December.
    6. Badrinath, S G & Chatterjee, Sangit, 1988. "On Measuring Skewness and Elongation in Common Stock Return Distributions: The Case of the Market Index," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 61(4), pages 451-72, October.
    7. Kraus, Alan & Litzenberger, Robert H, 1976. "Skewness Preference and the Valuation of Risk Assets," Journal of Finance, American Finance Association, American Finance Association, vol. 31(4), pages 1085-1100, September.
    8. Lawrence R. Glosten & Ravi Jagannathan & David E. Runkle, 1993. "On the relation between the expected value and the volatility of the nominal excess return on stocks," Staff Report, Federal Reserve Bank of Minneapolis 157, Federal Reserve Bank of Minneapolis.
    9. Lee, Tom K Y & Tse, Y K, 1991. "Term Structure of Interest Rates in the Singapore Asian Dollar Market," Journal of Applied Econometrics, John Wiley & Sons, Ltd., John Wiley & Sons, Ltd., vol. 6(2), pages 143-52, April-Jun.
    10. Engle, Robert F & Lilien, David M & Robins, Russell P, 1987. "Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model," Econometrica, Econometric Society, Econometric Society, vol. 55(2), pages 391-407, March.
    11. Lars Forsberg & Tim Bollerslev, 2002. "Bridging the gap between the distribution of realized (ECU) volatility and ARCH modelling (of the Euro): the GARCH-NIG model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., John Wiley & Sons, Ltd., vol. 17(5), pages 535-548.
    12. Hansen, B.E., 1992. "Autoregressive Conditional Density Estimation," RCER Working Papers 322, University of Rochester - Center for Economic Research (RCER).
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