GARCH

An alternative and more flexible lag structure of the ARCH(q) model is provided by
the GARCH(p, q), or Generalized ARCH model:
htóuò; p
ió1
aie2
tñiò; q
jó1
bjhtñj (2.8)
with ió1, . . . , p and jó1, . . . , q. In equation (2.8) the conditional variance ht is a
function of both past innovations and lagged values conditional variance, i.e. htñ1, . . . , htñp. The lagged conditional variance is often referred to as old news because
it is defined as
htñjóuò; p
ió1
ae2
tñiñ1ò; q
jó1
bjhtñjñ1 (2.9)
In other words, if i, jó1, then htñj in equation (2.8) and formulated in equation (2.9)
is explainable by past information and the conditional variance at time tñ2 or lagged
two periods back. In order for a GARCH(p, q) model to make sense the next condition
must be satisfied:
1PG; aiò;bj H[0
In this situation the GARCH(p, q) corresponds to an infinite-order ARCH process with
exponentially decaying weights for longer lags. Researchers have suggested that loworder
GARCH(p, q) processes may have properties similar to high-order ARCH but
with the advantage that they have significantly fewer parameters to estimate. Empirical
evidence also exists that a low-order GARCH model fits as well or even better
than a higher-order ARCH model with linearly declining weights. A large number of
empirical studies has found that a GARCH(1, 1) is adequate for most financial time
series.