Engle's (1982) original ARCH model assumes
The basic idea of these models is to increase the order of the autoregressive polynomial described in (1.1).
For this purpose, we define
, so that the square error
is now
The derivation of the unconditional moments of the ARCH()
process is analogous to ARCH(1). The necessary and sufficient
condition for the existence of stationary variance is
When this condition is satisfied, the variance of the process is
Although the variance of conditioned on
changes
with the elements of the information set (it depends on the past
through the
most recent values of the squared innovation
process) , the ARCH process is unconditionally homoscedastic.
After the model is estimated, we plot a similar picture as in figure 6.7 to show how the volatility does not vanish so quickly as in the ARCH(1) model.
The log-likelihood function of the standard ARCH() model in
(6.27), conditoned on an initial observation, is given by
![]() |
(6.13) |
Let
and
so that
the conditional variance can be written as
.
The first order conditions then become simply