Abstract
One of the most popular univariate asymmetric conditional volatility models is the exponential GARCH (or EGARCH) specification. In addition to asymmetry, which captures the different effects on conditional volatility of positive and negative effects of equal magnitude, EGARCH can also accommodate leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator of the EGARCH parameters are not available under general conditions, but rather only for special cases under highly restrictive and unverifiable conditions. It is often argued heuristically that the reason for the lack of general statistical properties arises from the presence in the model of an absolute value of a function of the parameters, which does not permit analytical derivatives, and hence does not permit (quasi-) maximum likelihood estimation. It is shown in this paper for the non-leverage case that: (1) the EGARCH model can be derived from a random coefficient complex nonlinear moving average (RCCNMA) process; and (2) the reason for the lack of statistical properties of the estimators of EGARCH under general conditions is that the stationarity and invertibility conditions for the RCCNMA process are not known.
Highlights
A One Line Derivation of EGARCHDepartment of Quantitative Finance, National Tsing Hua University, Taichung 402, Taiwan Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam 3000, The Netherlands Department of Quantitative Economics, Complutense University of Madrid, Madrid 28040, Spain Received: 16 June 2014; in revised form: 19 June 2014 / Accepted: 20 June 2014 /
In the world of univariate conditional volatility models, the ARCH model of Engle [1] and the generalization to the GARCH model by Bollerslev [2] are the two most widely estimated symmetric models of time-varying conditional volatility, where symmetry refers to the identical effects on volatility of positive and negative shocks of equal magnitude.The asymmetric effects on conditional volatility of positive and negative shocks of equal magnitude can be captured in different ways by the exponential GARCH model of Nelson [3,4], and the GJR model of Glosten, Jagannathan and Runkle [5].These are the two most widely estimated asymmetric univariate models of conditional volatility.A special case of asymmetry is that of leverage
It is often argued heuristically that the reason for the lack of general statistical properties arises from the presence in the model of an absolute value of a function of the parameters, which does not permit analytical derivatives, and does not permit
Summary
Department of Quantitative Finance, National Tsing Hua University, Taichung 402, Taiwan Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam 3000, The Netherlands Department of Quantitative Economics, Complutense University of Madrid, Madrid 28040, Spain Received: 16 June 2014; in revised form: 19 June 2014 / Accepted: 20 June 2014 /
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