6.7 Extensions of ARCH Models
The ARCH model has been extended to allow the conditional
variance to be a determinant of the mean, and under the name
ARCH-M was introduced by (,,). This is an
attractive form in financial applications, since it is natural to
suppose that the expected return on an asset is proportional to
the expected risk of the asset. The general model used in
financial series is an ARMA process in the mean and innovations
with time dependent conditional heteroskedasticity are represented
by an ARCH process. Its expression is given by
Note that the greatest complexity introduced in this model comes
from evaluating the derivatives of innovations and the
conditional variance with respect to the parameters in a non
obvious way and consequently, the explicit expression in the first
order likelihood conditions are not available in a such simple
form. Moreover, the information matrix is not block diagonal
between the parameters of the mean and variance, so the estimation
procedure must be carried out simultaneously (for more details,
see McCurdy and Morgan (1988)).
The conditions in a GARCH process to have a finite unconditional
variance are often unsatisfied in a high frequently sampled
financial series. Engel and Bollerslev (1986) introduced a new
variety of ARCH models, in which
, in the
formulation of GARCH(1,1) given in (6.28), named
Integrated GARCH model (IGARCH). It is easy to see in this model,
that
very similar to the conditional mean specification of a random
walk, however this model is strictly stationary unlike a random
walk. Consequently, the IGARCH model has some characteristics of
integrated processes. At this stage, could be interesting to go
further in understanding the persistence effect. In a few words,
if the shocks (imputs) in the conditional variance persist
indefinitely, the process is said to be persistent in variance.
It is clear from (6.6.1), that the effect of a shock
persists faraway but not very much only when
.
That is to say, the persistence disappears in terms of the past of
the process , i.e. in an unconditional
sense(for more details, see section 5.5 in Gouriéroux ; 1997).
Many empirical researchers in financial markets have found
evidence that bad news and good news have different behaviour in
the models, revealing an asymmetric behaviour in stock prices,
negative surprises give to increase the volatility more than
positive surprise, this effect is described as the leverage
effect. For this reason,Nelson (1991) proposes a new
model, in which the conditional variance is
where the
parameter allows this asymmetric effect.