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Econometrics 2014, 2(2), 92-97; doi:10.3390/econometrics2020092

A One Line Derivation of EGARCH

1,2,3,4,*  and 5
1 Department of Quantitative Finance, National Tsing Hua University, Taichung 402, Taiwan 2 Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam 3000, The Netherlands 3 Tinbergen Institute, Rotterdam 3000, The Netherlands 4 Department of Quantitative Economics, Complutense University of Madrid, Madrid 28040, Spain 5 Institute of Statistics, Biostatistics and Actuarial Sciences, Université Catholique de Louvain, Louvain-la-Neuve 1348, Belgium
* Author to whom correspondence should be addressed.
Received: 16 June 2014 / Revised: 19 June 2014 / Accepted: 20 June 2014 / Published: 23 June 2014
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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.
Keywords: leverage; asymmetry; existence; random coefficient models; complex non-linear moving average process leverage; asymmetry; existence; random coefficient models; complex non-linear moving average process
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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McAleer, M.; Hafner, C.M. A One Line Derivation of EGARCH. Econometrics 2014, 2, 92-97.

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