6.6 GARCH(p,q) Model
The ARCH model is based on an autoregressive representation of the
conditional variance. One may also add a moving average part.
The GARCH(
,
) process (Generalised
AutoRegressive Conditionally Heteroscedastic) is thus obtained.
The model is defined by
where
are imposed
to ensure that the conditional variance is strictly positive.
The conditional variance can be expressed as
where
and
are polynomials in the
backshift operator B. If the roots of
lie outside the
unit circle, we can rewrite the conditional variance as
Hence, this expression reveals that a GARCH(
,
) process can
be viewed as an ARCH(
) with a rational lag structure
imposed on the coefficients.
The GARCH(
,
) model may be rewritten in an alternative form,
as an ARMA model on squared perturbations.
For this purpose, let us introduce
. Replacing
by
in the GARCH
representation yields
that is to say
with
for
and
for
.
It is an ARMA (
,
) representation for the process
but with an error term, which is a white noise process
that does not necessarily have a constant variance.
6.6.1 GARCH(1,1) Model
The most used heteroscedastic model in financial time series is a
GARCH(1,1), (see Bera and Higgins (1993) for a very complete
revision).
This particular model parameterises the conditional variance as
 |
(6.28) |
Using the law of iterated expectations
Assuming the process began infinitely far in the past with a
finite initial variance, the sequence of the variances converge to
a constant

if
therefore, the GARCH process is unconditionally homoscedastic.
The
parameter indicates the contributions to
conditional variance of the most recent news, and the
parameter corresponds to the moving average part in the
conditional variance, that is to say, the recent level of
volatility. In this model, it could be convenient to define a
measure, in the forecasting context, about the impact of present
news in the future volatility. To carry out a study of this
impact, we calculate the expected volatility
-steps ahead, that
is
Therefore, the persistence depends on the
sum.
If
, the shocks have a decaying impact on
future volatility.
In this example we generate several time
series following a GARCH(1,1) model, with the same value of
, but different values of each parameter to
reflect the impact of
in the model.
Table 6.2:
Estimates and t-ratios of both models
with the same
value
|
 |
 |
 |
Model A |
0.05 |
0.85 |
0.10 |
estimates |
0.0411 |
0.8572 |
0.0930 |
t-ratio |
(1.708) |
(18.341) |
(10.534) |
Model B |
0.05 |
0.10 |
0.85 |
estimates |
0.0653 |
0.1005 |
0.8480 |
t-ratio |
1.8913 |
3.5159 |
18.439 |
|
Figure 6.9:
Simulated GARCH(1,1) data with
and
.
|
Figure 6.10:
Estimated volatility of the simulated GARCH(1,1) data
with
and
.
|
Figure 6.11:
Simulated GARCH(1,1) data with
and
.
|
Figure 6.12:
Estimated volatility of the simulated GARCH(1,1) data
and
.
|
All estimates are significant in both models. The estimated
parameters
and
indicate that the
conditional variance is time-varying and strongly persistent
. But the conditional
variance is very different in both models as we can see in figures
6.10 and 6.12.