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The Correct Regularity Condition and Interpretation of Asymmetry in EGARCH

Author

Listed:
  • Chia-Lin Chang

    (National Chung Hsing University, Taiwan)

  • Michael McAleer

    (National Tsing Hua University, Taiwan; University of Sydney Business School; Erasmus University Rotterdam, The Netherlands, Complutense University of Madrid, Spain and Yokohama National University, Japan)

Abstract
In the class of univariate conditional volatility models, the three most popular are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992), and the exponential GARCH (or EGARCH) model of Nelson (1990, 1991). For purposes of deriving the mathematical regularity properties, including invertibility, to determine the likelihood function for estimation, and the statistical conditions to establish asymptotic properties, it is convenient to understand the stochastic properties underlying the three univariate models. The random coefficient autoregressive process was used to obtain GARCH by Tsay (1987), an extension of which was used by McAleer (2004) to obtain GJR. A random coefficient complex nonlinear moving average process was used by McAleer and Hafner (2014) to obtain EGARCH. These models can be used to capture asymmetry, which denotes the different effects on conditional volatility of positive and negative effects of equal magnitude, and possibly also leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility (see Black 1979). McAleer (2014) showed that asymmetry was possible for GJR, but not leverage. McAleer and Hafner showed that leverage was not possible for EGARCH. Surprisingly, the conditions for asymmetry in EGARCH seem to have been ignored in the literature, or have concentrated on the incorrect conditions, with no clear explanation, and hence with associated misleading interpretations. The purpose of the paper is to derive the regularity condition for asymmetry in EGARCH to provide the correct interpretation. It is shown that, in practice, EGARCH always displays asymmetry, though not leverage.

Suggested Citation

  • Chia-Lin Chang & Michael McAleer, 2017. "The Correct Regularity Condition and Interpretation of Asymmetry in EGARCH," Tinbergen Institute Discussion Papers 17-056/III, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20170056
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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. Guillaume Gaetan Martinet & Michael McAleer, 2018. "On the invertibility of EGARCH(p, q)," Econometric Reviews, Taylor & Francis Journals, vol. 37(8), pages 824-849, September.
    3. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(2), pages 280-310, April.
    4. Michael McAleer & Christian M. Hafner, 2014. "A One Line Derivation of EGARCH," Econometrics, MDPI, vol. 2(2), pages 1-6, June.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Robert Engle & Stephen Figlewski, 2015. "Modeling the Dynamics of Correlations among Implied Volatilities," Review of Finance, European Finance Association, vol. 19(3), pages 991-1018.
    7. Michael McAleer, 2014. "Asymmetry and Leverage in Conditional Volatility Models," Econometrics, MDPI, 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.
    9. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Pauwels, Laurent & Radchenko, Peter & Vasnev, Andrey, 2019. "Higher Moment Constraints for Predictive Density Combinations," Working Papers BAWP-2019-01, University of Sydney Business School, Discipline of Business Analytics.
    2. Chia-Lin Chang & Yiying Li & Michael McAleer, 2018. "Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice," Energies, MDPI, vol. 11(6), pages 1-19, June.
    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," IJFS, MDPI, vol. 6(1), pages 1-24, December.
    4. Najam Iqbal & Muhammad Saqib Manzoor & Muhammad Ishaq Bhatti, 2021. "Asymmetry and Leverage with News Impact Curve Perspective in Australian Stock Returns’ Volatility during COVID-19," JRFM, MDPI, vol. 14(7), pages 1-15, July.
    5. Chang, C-L. & Hsu, S.-H. & McAleer, M.J., 2018. "Asymmetric Risk Impacts of Chinese Tourists to Taiwan," Econometric Institute Research Papers EI2018-18, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    6. Amélie Charles & Olivier Darné, 2019. "The accuracy of asymmetric GARCH model estimation," Post-Print hal-01943883, HAL.
    7. Bentes, Sonia R., 2018. "Is stock market volatility asymmetric? A multi-period analysis for five countries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 258-265.
    8. Chang, C-L. & Hsu, S.-H. & McAleer, M.J., 2018. "Risk Spillovers in Returns for Chinese and International Tourists to Taiwan," Econometric Institute Research Papers 18-031/III, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    9. Chia-Lin Chang & Michael McAleer & Jiarong Tian, 2019. "Modeling and Testing Volatility Spillovers in Oil and Financial Markets for the USA, the UK, and China," Energies, MDPI, vol. 12(8), pages 1-24, April.
    10. Amélie Charles & Olivier Darné, 2019. "The accuracy of asymmetric GARCH model estimation," International Economics, CEPII research center, issue 157, pages 179-202.
    11. Cristina Chinazzo & Vahidin Jeleskovic, 2024. "Forecasting Bitcoin Volatility: A Comparative Analysis of Volatility Approaches," Papers 2401.02049, arXiv.org.
    12. Yuki Toyoshima, 2018. "Testing for Causality-In-Mean and Variance between the UK Housing and Stock Markets," JRFM, MDPI, vol. 11(2), pages 1-10, April.
    13. 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.
    14. Shadi Tehrani & Jesús Juan & Eduardo Caro, 2022. "Electricity Spot Price Modeling and Forecasting in European Markets," Energies, MDPI, vol. 15(16), pages 1-23, August.
    15. 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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    17. 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.
    18. Banerjee, Ameet Kumar & Sensoy, Ahmet & Rahman, Molla Ramizur & Palma, Alessia, 2024. "Commonality in volatility among green, brown, and sustainable energy indices," Finance Research Letters, Elsevier, vol. 64(C).
    19. You-How Go & Wee-Yeap Lau, 2020. "Does Trading Volume explain the Information Flow of Crude Palm Oil Futures Returns?," The Review of Finance and Banking, Academia de Studii Economice din Bucuresti, Romania / Facultatea de Finante, Asigurari, Banci si Burse de Valori / Catedra de Finante, vol. 12(2), pages 115-136, December.
    20. Dey, Asim K. & Hoque, G.M. Toufiqul & Das, Kumer P. & Panovska, Irina, 2022. "Impacts of COVID-19 local spread and Google search trend on the US stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 589(C).
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    22. Onur Özdemir, 2022. "Cue the volatility spillover in the cryptocurrency markets during the COVID-19 pandemic: evidence from DCC-GARCH and wavelet analysis," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-38, December.

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    More about this item

    Keywords

    Conditional volatility models; random coefficient complex nonlinear moving average process; EGARCH; asymmetry; leverage; regularity condition;
    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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