3.8 The Exponential GARCH Model
To overcome some weaknesses of the GARCH model in handling financial time series, Nelson (1991) proposes the exponential GARCH (EGARCH) model. In particular, to allow for asymmetric effects between positive and negative asset returns, he considered the weighted innovation
where θ and γ are real constants. Both ϵt and |ϵt| − E(|ϵt|) are zero-mean iid sequences with continuous distributions. Therefore, E[g(ϵt)] = 0. The asymmetry of g(ϵt) can easily be seen by rewriting it as
For the standard Gaussian random variable ϵt, . For the standardized Student-t distribution in Eq. (3.7), we have
An EGARCH(m, s) model can be written as
where α0 is a constant, B is the back-shift (or lag) operator such that Bg(ϵt) = g(ϵt−1), and 1 + β1B + ⋯ + βs−1Bs−1 and 1 − α1B − ⋯ − αmBm are polynomials with zeros outside the unit circle and have no common factors. By outside the unit circle we mean that absolute values of the zeros are greater than 1. ...