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On the Invertibility of EGARCH(p,q)

Author

Listed:
  • Guillaume Gaetan Martinet

    (ENSAE Paris Tech, France
    Department of Industrial Engineering and Operations Research Columbia University, USA)

  • Michael McAleer

    (Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam and Tinbergen Institute, The Netherlands, Department of Quantitative Economics, Complutense University of Madrid, and Institute of Economic Research, Kyoto University.)

Abstract

Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification can capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator (QMLE) of the EGARCH parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p,q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this paper that the EGARCH(p,q) model can be derived from a stochastic process, for which the invertibility conditions can be stated simply and explicitly. This will be useful in re-interpreting the existing properties of the QMLE of the EGARCH(p,q) parameters.

Suggested Citation

  • Guillaume Gaetan Martinet & Michael McAleer, 2015. "On the Invertibility of EGARCH(p,q)," Documentos de Trabajo del ICAE 2015-03, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
  • Handle: RePEc:ucm:doicae:1503
    Note: The authors are grateful to the Editor-in-Chief, Rob Taylor, an Associate Editor and two referees for very helpful comments and suggestions, and to Christian Hafner for insightful discussions. For financial support, the first author wishes to thank the National Science Council, Taiwan, and the second author is most grateful to the Australian Research Council and the National Science Council, Taiwan.
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    References listed on IDEAS

    as
    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Michael McAleer & Christian M. Hafner, 2014. "A One Line Derivation of EGARCH," Econometrics, MDPI, vol. 2(2), pages 1-6, June.
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    5. Michael McAleer, 2014. "Asymmetry and Leverage in Conditional Volatility Models," Econometrics, MDPI, vol. 2(3), pages 1-6, September.
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    Cited by:

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    6. Amélie Charles & Olivier Darné, 2019. "The accuracy of asymmetric GARCH model estimation," International Economics, CEPII research center, issue 157, pages 179-202.
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    8. Chang, Chia-Lin & McAleer, Michael & Wang, Yu-Ann, 2018. "Modelling volatility spillovers for bio-ethanol, sugarcane and corn spot and futures prices," Renewable and Sustainable Energy Reviews, Elsevier, vol. 81(P1), pages 1002-1018.
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    10. Chia-Lin Chang & Michael McAleer & Yu-Ann Wang, 2016. "Modelling volatility spillovers for bio-ethanol, sugarcane and corn," Documentos de Trabajo del ICAE 2016-03, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    11. Jeffrey Chu & Stephen Chan & Saralees Nadarajah & Joerg Osterrieder, 2017. "GARCH Modelling of Cryptocurrencies," JRFM, MDPI, vol. 10(4), pages 1-15, October.
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    15. Junru Zhang & Hadrian Geri Djajadikerta & Zhaoyong Zhang, 2018. "Does Sustainability Engagement Affect Stock Return Volatility? Evidence from the Chinese Financial Market," Sustainability, MDPI, vol. 10(10), pages 1-21, September.
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    More about this item

    Keywords

    Leverage; asymmetry; Existence; Stochastic process; Asymptotic properties; Invertibility.;
    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
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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