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Fourth Moment Structure of a Family of First-Order Exponential GARCH Models

Author

Listed:
  • Changli He
  • Timo Terasvirta

    (Department of Economics and Business Economics, Aarhus University)

  • Hans Malmsten

Abstract

In this paper we consider the fourth moment structure of a class of first-order Exponential GARCH models. This class contains as special cases both the standard Exponential GARCH model and the symmetric and asymmetric Logarithmic GARCH one. Conditions for the existence of any arbitrary moment are given. Furthermore, the expressions for the kurtosis and the autocorrelations of squared observations are derived. The properties of the autocorrelations of squared observations are derived. The properties of the autocorrelation structure are discussed and compared to those of the standard first-order GARCH process. In particular, it is seen that, contrary to the standard GARCH case, the decay rate of the autocorrelations is not constant and that the rate can be quite rapid in the beginning, depending on the parameters of the model.

Suggested Citation

  • Changli He & Timo Terasvirta & Hans Malmsten, 1999. "Fourth Moment Structure of a Family of First-Order Exponential GARCH Models," Research Paper Series 29, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:29
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    Cited by:

    1. Paolo Girardello & Orietta Nicolis & Giovanni Tondini, 2003. "Comparing Conditional Variance Models: Theory and Empirical Evidence," Multinational Finance Journal, Multinational Finance Journal, vol. 7(3-4), pages 177-206, September.
    2. Niklas Ahlgren & Alexander Back & Timo Terasvirta, 2024. "A new GARCH model with a deterministic time-varying intercept," Papers 2410.03239, arXiv.org, revised Oct 2024.
    3. Fernandes, Marcelo & Grammig, Joachim, 2006. "A family of autoregressive conditional duration models," Journal of Econometrics, Elsevier, vol. 130(1), pages 1-23, January.
    4. Menelaos Karanasos, "undated". "The Covariance Structure of Mixed ARMA Models," Discussion Papers 00/10, Department of Economics, University of York.
    5. P. Girardello & Orietta Nicolis & Giovanni Tondini, 2002. "Comparing conditional variance models: Theory and empirical evidence," Departmental Working Papers 2002-08, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
    6. Antonis Demos, 2002. "Moments and dynamic structure of a time-varying parameter stochastic volatility in mean model," Econometrics Journal, Royal Economic Society, vol. 5(2), pages 345-357, June.
    7. Pasquale Tridico & Riccardo Pariboni, 2017. "Structural Change, Aggregate Demand And The Decline Of Labour Productivity: A Comparative Perspective," Departmental Working Papers of Economics - University 'Roma Tre' 0221, Department of Economics - University Roma Tre.
    8. BAUWENS, Luc & GALLI, Fausto & GIOT, Pierre, 2003. "The moments of Log-ACD models," LIDAM Discussion Papers CORE 2003011, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Carnero, María Ángeles, 2001. "Outliers and conditional autoregressive heteroscedasticity in time series," DES - Working Papers. Statistics and Econometrics. WS ws010704, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. Shelton Peiris & Tim Swartz, 2020. "Revisiting the Kurtosis of Stationary Processes with Applications to Volatility Models," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 9(2), pages 1-1.
    11. Davide De Gaetano, 2017. "A Bootstrap Bias Correction Of Long Run Fourth Order Moment Estimation In The Cusum Of Squares Test," Departmental Working Papers of Economics - University 'Roma Tre' 0220, Department of Economics - University Roma Tre.
    12. Murat Midiliç, 2020. "Estimation of STAR–GARCH Models with Iteratively Weighted Least Squares," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 87-117, January.

    More about this item

    Keywords

    autocorrelation function of squared observations; conditional variance model; heavy tails; exponential GARCH; logarithmic GARCH;
    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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