Higher-order ARCH

The way in which equations (2.4) and (2.5) can be formulated is very flexible. For example, Yt can be written as an autoregressive process, e.g. YtóYtñ1òet , and/or
can include exogenous variables. More important is the way the conditional variance
ht can be expressed. The ARCH order of equation (2.5) can be increased to express
today’s conditional variance as a linear function of a greater amount of past information.
Thus the ARCH(q) can be written as:
htóuòa1e2
tñ1ò. . . òaqe2
tñq (2.6)
Such a model specification is generally preferred to a first-order ARCH since now the
conditional variance depends on a greater amount of information that goes as far as
q periods in the past. With an ARCH(1) model the estimated ht is highly unstable
since single large (small) surprises are allowed to drive ht to inadmissible extreme
values. With the higher-order ARCH of equation (2.6), the memory of the process is
spread over a larger number of past observations. As a result, the conditional
variance changes more slowly, which seems more plausible.