Exponential ARCH (EGARCH)

To address some of the limitations an exponential ARCH parameterization or
EGARCH has been proposed. The variance of the residual error term for the
EGARCH(1, 1) is given by
ln(ht )óuòbln(htñ1)òcttñ1ò{(Dttñ1Dñ(2/n)1/2) (2.17)
where ttóet/ht (standardized residual). Hence, the logarithm of the conditional
variance ln(ht ) at period t is a function of the logarithm of the conditional period
variance, lagged one period back ln(htñ1), the standarized value of the last residual
error, ttñ1, and the deviation of the absolute value of ttñ1 from the expected absolute
value of the standardized normal variate, (2/n)1/2. The parameter c measures the
impact ‘asymmetries’ on the last period’s shocks have on current volatility. Thus, if
c\0 then negative past errors have a greater impact on the conditional variance
ln(ht ) than positive errors. The conditional variance ht is expressed as a function of
both the size and sign of lagged errors.