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Volatility Modeling with a Generalized t-distribution

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  • Andrew Harvey
  • Rutger-Jan Lange

Abstract

Beta-t-EGARCH models in which the dynamics of the logarithm of scale are driven by the conditional score are known to exhibit attractive theoretical properties for the t-distribution and general error distribution (GED). The generalized-t includes both as special cases. We derive the information matrix for the generalized-t and show that, when parameterized with the inverse of the tail index, it remains positive definite as the tail index goes to infinity and the distribution becomes a GED. Hence it is possible to construct Lagrange multiplier tests of the null hypothesis of light tails against the alternative of fat tails. Analytic expressions may be obtained for the unconditional moments in the EGARCH model and the information matrix for the dynamic parameters obtained. The distribution may be extended by allowing for skewness and asymmetry in the shape parameters and the asymptotic theory for the associated EGARCH models may be correspondingly extended. For positive variables, the GB2 distribution may be parameterized so that it goes to the generalised gamma in the limit as the tail index goes to infinity. Again dynamic volatility may be introduced and properties of the model obtained. Overall the approach offers a unified, flexible, robust and practical treatment of dynamic scale.

Suggested Citation

  • Andrew Harvey & Rutger-Jan Lange, 2015. "Volatility Modeling with a Generalized t-distribution," Cambridge Working Papers in Economics 1517, Faculty of Economics, University of Cambridge.
    • Handle: RePEc:cam:camdae:1517
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    References listed on IDEAS

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    1. Arslan, Olcay, 2004. "Family of multivariate generalized t distributions," Journal of Multivariate Analysis, Elsevier, vol. 89(2), pages 329-337, May.
    2. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    3. Creal, Drew & Koopman, Siem Jan & Lucas, André, 2011. "A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 552-563.
    4. Panayiotis Theodossiou, 1998. "Financial Data and the Skewed Generalized T Distribution," Management Science, INFORMS, vol. 44(12-Part-1), pages 1650-1661, December.
    5. Kai-Li Wang & Christopher Fawson & Christopher B. Barrett & James B. McDonald, 2001. "A flexible parametric GARCH model with an application to exchange rates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(4), pages 521-536.
    6. Sean Holly & Ivan Petrella & Emiliano Santoro, 2013. "Aggregate fluctuations and the cross-sectional dynamics of firm growth," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 176(2), pages 459-479, February.
    7. Blasques, Francisco & van Brummelen, Janneke & Koopman, Siem Jan & Lucas, André, 2022. "Maximum likelihood estimation for score-driven models," Journal of Econometrics, Elsevier, vol. 227(2), pages 325-346.
    8. Harvey,Andrew C., 2013. "Dynamic Models for Volatility and Heavy Tails," Cambridge Books, Cambridge University Press, number 9781107630024.
    9. Jensen, Søren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(6), pages 1203-1226, December.
    10. Zhu, Dongming & Zinde-Walsh, Victoria, 2009. "Properties and estimation of asymmetric exponential power distribution," Journal of Econometrics, Elsevier, vol. 148(1), pages 86-99, January.
    11. Zhu, Dongming & Galbraith, John W., 2010. "A generalized asymmetric Student-t distribution with application to financial econometrics," Journal of Econometrics, Elsevier, vol. 157(2), pages 297-305, August.
    12. Giulio Bottazzi & Angelo Secchi, 2011. "A new class of asymmetric exponential power densities with applications to economics and finance," Industrial and Corporate Change, Oxford University Press and the Associazione ICC, vol. 20(4), pages 991-1030, August.
    13. Harvey, Andrew & Sucarrat, Genaro, 2014. "EGARCH models with fat tails, skewness and leverage," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 320-338.
    14. Bickel, David R., 2002. "Robust estimators of the mode and skewness of continuous data," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 153-163, April.
    15. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 66(1-2), pages 133-152.
    16. Zhu, Dongming & Galbraith, John W., 2011. "Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions," Journal of Empirical Finance, Elsevier, vol. 18(4), pages 765-778, September.
    17. Drew Creal & Siem Jan Koopman & André Lucas, 2013. "Generalized Autoregressive Score Models With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(5), pages 777-795, August.
    18. McDonald, James B. & Newey, Whitney K., 1988. "Partially Adaptive Estimation of Regression Models via the Generalized T Distribution," Econometric Theory, Cambridge University Press, vol. 4(3), pages 428-457, December.
    19. Brazauskas, Vytaras, 2002. "Fisher information matrix for the Feller-Pareto distribution," Statistics & Probability Letters, Elsevier, vol. 59(2), pages 159-167, September.
    20. Andrew Harvey & Rutger-Jan Lange, 2015. "Modeling the Interactions between Volatility and Returns," Cambridge Working Papers in Economics 1518, Faculty of Economics, University of Cambridge.
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    Keywords

    Asymmetric price transmission; cost pass-through; electricity markets; price theory; rockets and feathers;
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