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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.

In the world of univariate conditional volatility models, the ARCH model of Engle [

The asymmetric effects on conditional volatility of positive and negative shocks of equal magnitude can be captured in different ways by the exponential GARCH (or EGARCH) model of Nelson [

A special case of asymmetry is that of leverage. As defined by Black [

Although the GJR model can capture asymmetry, leverage is not possible, unless the short run persistence effect (that is, the ARCH parameter) is negative. Such a restriction is not consistent with the standard sufficient condition for conditional volatility to be positive.

The univariate GARCH model has been extended to its multivariate counterpart in, for example, the BEKK model of Baba

However, the EGARCH model has not yet been developed formally for multivariate processes, with appropriate regularity conditions. Kawakatsu [

Although it is not essential, leverage is an attractive feature of the EGARCH model. Since EGARCH is derived as a discrete-time approximation to a continuous-time stochastic volatility process, and is expressed in logarithms, the exponential operator is required to obtain conditional volatility, which is guaranteed to be positive. Therefore, no restrictions are required for conditional volatility to be positive.

However, the statistical properties for 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.

Some specific statistical results are available under highly restrictive assumptions that cannot readily be verified. For example, Straumann and Mikosch [

Wintenberger [

Demos and Kyriakopoulou [

It is shown in this paper for the non-leverage case that the EGARCH model can be derived from a random coefficient complex nonlinear moving average (RCCNMA) process, and that 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.

The remainder of the paper is organized as follows. In

Let the conditional mean of financial returns be given as:
_{t}_{t}_{t−1}) + _{t}_{t}_{t}_{t}_{t−1} is the information set at time _{t}

The EGARCH specification of Nelson [_{t}_{t−1}| + _{t−1} + _{t−1}, |_{t}_{t} ~ iid_{t−1} is included in the model. Asymmetry exists if

As _{t−1} is a function of both _{t}_{t−1}| is not differentiable with respect to the parameters. Moreover, invertibility of EGARCH is also problematic because of the presence of the logarithmic transformation as well as the absolute value function.

Consider a random coefficient complex nonlinear moving average (RCCNMA) process given by:
_{t−1}, _{t} ~ iid_{t} ~ iid

If _{t}_{t}_{t}_{t}

As the RCCNMA process given in Equation (3) is not in the class of random coefficient linear moving average models examined in Marek [

It follows from Equation (3) that:

The use of an infinite lag for the RCCNMA process in Equation (3) would yield the EGARCH model in Equation (2).

It is worth noting that the transformation of _{t}_{t}_{t}_{t}_{t}_{t}

The interpretation of leverage, whereby

More importantly, in terms of interpreting the model, the derivation of EGARCH, albeit without the logarithmic transformation, in Equation (4) shows that the statistical properties of the (quasi-) maximum likelihood estimator of the EGARCH parameters do not exist under general conditions because the model is based on a RCCNMA process, for which the stationarity and invertibility conditions are not known.

The paper was concerned with one of the most popular univariate asymmetric conditional volatility models, namely the exponential GARCH (or EGARCH) specification. The EGARCH model is popular, among other reasons, because it can capture both asymmetry, namely the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility.

The statistical properties for 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 was shown in the paper for the non-leverage case that the EGARCH model could be derived from a random coefficient complex nonlinear moving average (RCCNMA) process, and that 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.

The authors are grateful to the Editor, Kerry Patterson, Alistair Freeland, and three referees for helpful comments and suggestions. For financial support, the first author wishes to acknowledge the Australian Research Council and the National Science Council, Taiwan.

The authors contributed jointly to the paper.

The authors declare no conflict of interest.