Estimation in Financial Models - RiskLab

and by substituting the linear variance function, F bb is consistently estimated
by
^F bb = 1 T
2
X
t
4 x0 tx t
h t
+2 X j
2 j
3
" 2 t,j
x 0
h
t,jx 2 t,j
5 :
t
Finally we give a remark to the ML estimation for the GARCH regression
model
" t = y t , x 0 tb;
h t = v 0 t;
with v t and as above. In order to estimate the mean parameters b we
dierentiate with respect to b as shown for the ARCH regression model with
the single dierence
@h t
qX
pX
@b = ,2 @h t,j
j x t,j " t,j + j
@b :
j=1
Paying attention to this dierence, the part of the Fisher information matrix
corresponding to b is consistently estimated as in the ARCH regression model.
We denote by F , respectively F , the part of the Fisher information matrix
corresponding to , respectively , and the elements in the o-diagonal
block of the information matrix by F b , respectively by F b . The elements
F b , respectively F b , may be shown to be zero. Because of this asymptotic
independence , respectively , can be estimated without loss of asymptotic
eciency based on a consistent estimate of b and vice versa. Using this fact
Engle [22], x6, formulates a simple scoring algorithm for the ML estimation
of the parameters and b.
As for the properties of the estimators, Bollerslev [14] remarks that for the
general ARCH class of models the verication of sucient regularity conditions
for the MLE to be consistent and asymptotically normally distributed
is very dicult. A detailed proof is only worked out in a few cases. Normally
one assumes that these regularity conditions are satised, such that
the ML estimators ^, respectively ^, and ^b are consistent and asymptotically
normally distributed with limiting distribution
p
T (^ , ) ,! N(0; F
,1
); resp. p T (^ , ) ,! N(0; F ,1 );
j=1
and p T (^b , b) ,! N(0; F ,1
bb ): (3.76)
Closing our considerations about estimation in ARCH/GARCH models we
remark that Geweke [31] developes Bayesian inference procedures for ARCH
models by using Monte Carlo methods to determine the a posteriori distribution.
The Bayesian analysis is discussed in the next section.
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