On the Invertibility of EGARCH

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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 estimators (QMLE) of the EGARCH parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable conditions. A limitation in the development of asymptotic properties of the QMLE for EGARCH is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this paper that the EGARCH 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 parameters.
The authors are grateful to Christian Hafner for very helpful 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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Black, F. (1976), Studies of stock market volatility changes, 1976 Proceedings of the American Statistical Association, Business and Economic Statistics Section, pp. 177-181. Bollerslev, T. (1986), Generalised autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327. Demos, A. and D. Kyriakopoulou (2014), Asymptotic normality of the QMLEs in the EGARCH(1,1) model. Available online: (accessed on 14 June 2014). Engle, R.F. (1982), Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007. Glosten, L., R. Jagannathan and D. Runkle (1992), On the relation between the expected value and volatility of nominal excess return on stocks, Journal of Finance, 46, 1779-1801. Marek, T. (2005), On invertibility of a random coefficient moving average model, Kybernetika, 41(6), 743-756. McAleer, M., F. Chan and D. Marinova (2007), An econometric analysis of asymmetric volatility: Theory and application to patents, Journal of Econometrics, 139, 259-284. McAleer, M. and C. Hafner (2014), A one line derivation of EGARCH, Econometrics, 2(2), 92-97. Nelson, D.B. (1990), ARCH models as diffusion approximations, Journal of Econometrics, 45, 7-38. Nelson, D.B. (1991), Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59, 347-370. Tsay, R.S. (1987), Conditional heteroscedastic time series models, Journal of the American Statistical Association, 82, 590-604. Straumann, D. and T. Mikosch (2006), Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equation approach, Annals of Statistics, 34, 2449–2495. Wintenberger, O. (2013), Continuous invertibility and stable QML estimation of the EGARCH(1,1) model, Scandinavian Journal of Statistics, 40, 846–867.