GARCH versus exponential smoothing (ES)

In many respects the GARCH(1, 1) representation shares many features of the popular
exponential smoothing to which can be added the interpretation that the level of
current volatility is a function of the previous period’s volatility and the square of the
previous period’s returns. These two models have many similarities, i.e. today’s
volatility is estimated conditionally upon the information set available at each period.
Both the GARCH(1, 1) model in equation (2.8) and the (ES) model in equation (2.7)
use the last period’s returns to determine current levels of volatility. Subsequently,
it follows that today’s volatility is forecastable immediately after yesterday’s market
closure.2 Since the latest available information set is used, it can be shown that both
models will provide more accurate estimators of volatility than the use of historical
volatility.
However, there are several differences in the operational characteristics of the two
models. The GARCH model, for example, uses two independent coefficients to estimate
the impact the variables have in determining current volatility, while the ES
model uses only one coefficient and forces the variables e2
tñ1 and htñ1 to have a unit
effect on current period volatility. Thus, a large shock will have longer lasting impact
on volatility using the GARCH model of equation (2.8) than the ES model of (2.3)
The terms a and b in GARCH do not need to sum to unity and one parameter is
not the complement of the other. Hence, it avoids the potential for simultaneity bias
in the conditional variance. Their estimation is achieved by maximizing the likelihood
function.3 This is a very important point since the values of a and b are critical in
determining the current levels of volatility. Incorrect selection of the parameter
values will adversely affect the estimation of volatility. The assumption that a and b
sum to unity is, however, very strong and presents an hypothesis that can be tested
rather than a condition to be imposed. Acceptance of the hypothesis that a and b
sum to unity indicates the existence of an Integrated GARCH process or I-GARCH.
This is a specification that characterizes the conditional variance ht as exhibiting a nonstationary component. The implication of this is that shocks in the lagged
squared error term e2
tñi will have a permanent effect on the conditional variance.
Furthermore, the GARCH model has an additional parameter, u, that acts as a
floor and prevents the volatility dropping to below that level. In the extreme case
where a and bó0, volatility is constant and equal to u. The value of u is estimated
together with a and b using maximum likelihood estimation and the hypothesis uó0
can be tested easily. The absence of the u parameter in the ES model allows volatility,
after a few quiet trading days, to drop to very low levels.