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On the Invertibility of EGARCH(p,q)

Author

Listed:
  • Guillaume Gaetan Martinet

    (Paris Tech, France, Columbia University, USA)

  • Michael McAleer

    (National Tsing Hua University, Taiwan, Erasmus University Rotterdam, the Netherlands, Complutense University of Madrid, Spain.)

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)," Tinbergen Institute Discussion Papers 15-022/III, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:20150022
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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, Open Access Journal, vol. 2(2), pages 1-6, June.
    3. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    4. Olivier Wintenberger, 2013. "Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 846-867, December.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. McAleer, Michael & Chan, Felix & Marinova, Dora, 2007. "An econometric analysis of asymmetric volatility: Theory and application to patents," Journal of Econometrics, Elsevier, vol. 139(2), pages 259-284, August.
    7. Michael McAleer, 2014. "Asymmetry and Leverage in Conditional Volatility Models," Econometrics, MDPI, Open Access Journal, vol. 2(3), pages 1-6, September.
    8. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Citations

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    Cited by:

    1. Manabu Asai & Chia-Lin Chang & Michael McAleer, 2016. "Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Spillovers," Documentos de Trabajo del ICAE 2016-15, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    2. Chang, Chia-Lin & McAleer, Michael, 2017. "The correct regularity condition and interpretation of asymmetry in EGARCH," Economics Letters, Elsevier, vol. 161(C), pages 52-55.
    3. Chia-Lin Chang & Michael McAleer & Chien-Hsun Wang, 2017. "An Econometric Analysis of ETF and ETF Futures in Financial and Energy Markets Using Generated Regressors," International Journal of Financial Studies, MDPI, Open Access Journal, vol. 6(1), pages 1-24, December.
    4. Chia-Lin Chang & Yiying Li & Michael McAleer, 2018. "Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice," Energies, MDPI, Open Access Journal, vol. 11(6), pages 1-19, June.
    5. Urom, Christian & Onwuka, Kevin O. & Uma, Kalu E. & Yuni, Denis N., 2020. "Regime dependent effects and cyclical volatility spillover between crude oil price movements and stock returns," International Economics, Elsevier, vol. 161(C), pages 10-29.
    6. Manabu Asai & Michael McAleer, 2016. "Asymptotic Theory for Extended Asymmetric Multivariate GARCH Processes," Documentos de Trabajo del ICAE 2016-14, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    7. Xiaoyi Shen & Albert K. Tsui & Zhaoyong Zhang, 2019. "Volatility Timing in CPF Investment Funds in Singapore: Do They Outperform Non-CPF Funds?," Risks, MDPI, Open Access Journal, vol. 7(4), pages 1-16, October.
    8. Junru Zhang & Hadrian Geri Djajadikerta & Zhaoyong Zhang, 2018. "Does Sustainability Engagement Affect Stock Return Volatility? Evidence from the Chinese Financial Market," Sustainability, MDPI, Open Access Journal, vol. 10(10), pages 1-21, September.
    9. 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.
    10. Chang, C-L. & McAleer, M.J. & Wang, Y-A., 2016. "Modelling Volatility Spillovers for Bio-ethanol, Sugarcane and Corn," Econometric Institute Research Papers EI2016-15, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    11. Amélie Charles & Olivier Darné, 2019. "The accuracy of asymmetric GARCH model estimation," International Economics, CEPII research center, issue 157, pages 179-202.
    12. Rangan Gupta & Chi Keung Marco Lau & Seong-Min Yoon, 2019. "OPEC News Announcement Effect on Volatility in the Crude Oil Market: A Reconsideration," Advances in Decision Sciences, Asia University, Taiwan, vol. 23(4), pages 1-23, December.
    13. Jeffrey Chu & Stephen Chan & Saralees Nadarajah & Joerg Osterrieder, 2017. "GARCH Modelling of Cryptocurrencies," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 10(4), pages 1-15, October.

    More about this item

    Keywords

    Leverage; asymmetry; existence; stochastic process; asymptotic properties; invertibility;

    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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