Estimation in Financial Models - RiskLab

Theorem 6 Depending on the value of we have the following asymptotic
behaviour for the normalized deviation q I n ()(^ n , ):
q
lim P I
n,!1 n ()(^ n , ) z =
8
><
>:
(z); jj < 1;
H (z); jj =1;
Ch(z); jj > 1;
where (z) is the standard normal distribution, Ch(z) is the Cauchy distribution
and H (z) is the distribution of the random variable
2 3 2
where W denotes a Wiener process.
W 2 (1) , 1
R 1
0
W 2 (s)ds ;
Hence, in the stationary case jj < 1 the normalized deviation q I n ()(^ n ,)
is asymptotically normally distributed, whereas in the other cases it has
as limit distribution the Cauchy distribution or the quite unexpected H
distribution. The question arises wether we can reduce the number of limit
distributions by modifying the normalizing factor q I n (). Indeed, instead of
using the Fisher information I n () =EhMi n , choosing the stochastic Fisher
information hMi n as normalizing factor we obtain
Theorem 7
lim
n,!1 P
q
(
(z); jj 6=1;
hMi n (^ n , ) z =
H (z); jj =1:
Summarizing: the MLE ^ n is (strongly) consistent and the normalized deviations
q I n ()(^ n , ) and q hMi n (^ n , ) are asymptotically distributed as
shown in both theorems.
3.2.2 ARCH and GARCH models
In the following we concentrate on ARCH and GARCH models and especially
on estimation in these models. We refer to the fundamental papers by
Engle [22] and Bollerslev [12] and to Bollerslev, Chou and Kroner [13] and
Bollerslev, Engle and Nelson [14].
Conventional econometric time series models assume constantvariance. However,
over a decade ago risk and uncertainty considerations lead to the development
of new econometric time series models that allow for the modeling
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