Springer - Read

354 Chapter 10 Further Topics
Figure 10-10
The daily percentage
returns of the Dow
Jones Industrial Index
(E1032.TSM) from July 1,
1997, through April 9,
1999 (above), and the
estimates of σt ht for
the conditional Gaussian
GARCH(1,1) model
of Example 10.3.2.
-6 -4 -2 0 2 4
1.0 1.5 2.0 2.5 3.0
0 100 200 300 400
be advised to subtract the sample mean (unless you wish to assume that the parameter
a in (10.3.15) is zero). If you subtract the sample mean it will be used as the estimate of
a in the model (10.3.15). The GARCH Maximum Likelihood Estimation box will
then open. When you click on OK the optimization will proceed. Denoting by { ˜Zt}
the (possibly) mean-corrected observations, the GARCH coefficients are estimated
by numerically maximizing the likelihood of ˜Zp+1,..., ˜Zn conditional on the known
values ˜Z1,..., ˜Zp, and with assumed values 0 for each ˜Zt, t ≤ 0, and ˆσ 2 for each ht,
t ≤ 0, where ˆσ 2 is the sample variance of { ˜Z1,..., ˜Zn}. In other words the program
maximizes
L(α0,...,αp,β1,...,βq)
n
σt tp+1
1 ˜Zt
φ
σt
, (10.3.16)
with respect to the coefficients α0,...,αp and β1,...,βq, where φ denotes the standard
normal density, and the standard deviations σt √ ht,t ≥ 1, are computed
recursively from (10.3.11) with Zt replaced by ˜Zt, and with ˜Zt 0 and ht ˆσ 2 for
t ≤ 0. To find the minimum of −2ln(L) it is advisable to repeat the optimization by
clicking on the red MLE button and then on OK several times until the result stabilizes.
It is also useful to try other initial values for α0,...,αp, and β1,...,βq, to minimize
the chance of finding only a local minimum of −2ln(L). Note that the optimization
is constrained so that the estimated parameters are all non-negative with
ˆα1 +···+ ˆαp + ˆβ1 +···+ ˆβq < 1, (10.3.17)
and ˆα0 > 0. Condition (10.3.17) is necessary and sufficient for the corresponding
GARCH equations to have a causal weakly stationary solution.