Estimation in Financial Models - RiskLab

Denoting u 0 t (1;" 2 t,1;:::;" 2 t,q) and h t t
2 we abbreviate (3.68) by
h t = u 0 t:
With this notation the rst order derivatives are
@l t
@ = 1 u t
2h t
" 2 t
h t
, 1
!
: (3.71)
In order to obtain the limiting distributions for the normalized deviations of
the estimators in (3.76) below, we have to estimate the Fisher information
matrix F, see x3.1.1, p.19, which is the negative expectation of the Hessian
averaged over all observations. Therefore we calculate the Hessian
@ 2 l t
@@ 0 = , 1
2h 2 t
u t u 0 t
! "
" 2 t "
2
+ t
, 1
h t h t
# @
@ 0 1
2h t
u t
: (3.72)
Since the conditional expectation of the factor " 2 t =h t is one and of the second
term in (3.72) is zero, the Fisher information matrix is given by
F = X t
and consistently estimated by
^F = 1 T
1
2T E " 1
h 2 t
X
t
" 1
2h 2 t
u t u 0 t
u t u 0 t
Later we will discuss the properties of the estimators, see p.49.
In many applications with the linear ARCH(q) model a large number of parameters
and a long lag length q are needed. These problems are avoided by an
alternative, more general parametrization of h t introduced by Bollerslev [12].
This more general model is called the Generalized ARCH, or GARCH(p; q),
model
2 t = 0 +
qX
i=1
i " 2 t,i +
pX
i=1
#
#
:
;
i 2 t,i;
where q > 0, p 0, 0 > 0, i 0 for i = 1;:::;q, and i 0 for
i = 1;:::;p. The generalization in the GARCH(p; q) model in comparison
to the ARCH(q) model is that beside past values of the process, also past
conditional variances enter.
Denote 0 ( 0 ; 1 ;:::; q ; 1 ;:::; p ), h t t
2 and vt 0 (1;" 2 t,1;:::; " 2 t,q;
h t,1 ;:::;h t,p ). As for the ARCH model we estimate by dierentiating the
log-likelihood with respect to
@l t
@ = 1 !
@h t " 2 t
, 1 :
2h t @ h t
47