Problems

As with the ARCH(1), for the variance to be computable, the sum of a1, . . . , aq in
equation (2.6) must be less than one. Generally, this is not a problem with financial
return series. However, a problem that often arises with the higher-order ARCH is
that not every one of the a1, . . . , aq coefficients is positive, even if the conditional
variance computed is positive at all times. This fact cannot be easily explained in
economic terms, since it implies that a single large residual return could drive the
conditional variance negative.
The model in equation (2.6) has an additional disadvantage. The number of
parameters increases with the ARCH order and makes the estimation process
formidable. One way of attempting to overcome this problem is to express past errors
in an ad hoc linear declining way. Equation (2.6) can then be written as:
htóuòa;wke2
tñk (2.7)
where wk, kó1, . . . , q and &wkó1 are the constant linearly declining weights. While
the conditional variance is expressed as a linear function of past information, this
model attaches greater importance to more recent shocks in accounting for the most
of the ht changes. This model was adopted by Engle which has been found to give a
good description of the conditional variance in his 1982 study and a later version
published in 1983. The restrictions for the variance equation parameters remain as
in ARCH(1).