> Is it possible to know which model (garch(1,1), garch(2,1) or
> garch(1,2)) is best for modeling data upfront, so without running the
> actual models?? Should I look at autocorrelations or something like
> that?
> Thanks for your help. Daan Taks.
A relevant analogy is the arma (p,q), for which Box and Jenkins
(see the updated classic Box, Jenkins and Reinsel Time Series Analysis:
Forecasting and Control,
Third Edition) suggest the use of sample autocorrelations (and partial
autocorrelations) for the selection of p and q.
In the case of arch models, suppose that y(t), t = 1,....,T is a time
series, and that
the model is y(t) = a + b y(t-1) + u(t), where |b| < 1, and u(t) is the
error
of the form u(t) = h(t) e(t), where h(t) represents
conditional standard deviation of u(t), and e(t) is iid normal. Here,
h(t)*h(t) = E[u(t)*u(t) | I(t-1)],
with I(t-1) representing information available at time t-1. The arch(p)
model is
h(t)*h(t) = c + d1 u(t-1)*u(t-1) + d2 u(t-2)*u(t-2) + ... + dp
u(t-p)*u(t-p).
Equivalently,
u(t)*u(t) = c + d1 u(t-1)*u(t-1) + d2 u(t-2)*u(t-2) + ... + dp
u(t-p)*u(t-p) + w(t),
for some error series w(t) satisfying E[w(t)|
u(t-1)*u(t-1),...,u(t-p)*u(t-p)] = 0.
So the arch model, as defined above, is equivalently an ar(p) model for
the series
u(t)*u(t). You can then try a modified form of the Box-Jenkins
methodology on
the residual uhat(t) = y(t) - ahat - bhat y(t-1) , with ahat and bhat
some estimators
(such as least squares estimators).
(i) get the residuals uhat(t),
(ii) compute squares residuals uhat(t)*uhat(t)
(iii) compute autocorrelations of uhat(t)*uhat(t), and use them (via
Box-Jenkins) to pick p, in ar(p) model of u(t)*u(t).
The resulting p is an estimate of the arch(p) order.
Critique: Several challenges arise with this method. First, the
residuals uhat(t) differ from the actual
u(t), and this may affect your judgments. Second, the order of the ar
model y(t) = a + b y(t-1) + u(t)
above is order = 1, but in practice you want to estimate this order
consistently (In principle this can be done
prior to picking the arch order, via Box-Jenkins.)
Extensions: The (linear) GARCH model generalizes the ARCH model, and
typically implies an AR(p = infinity)
model for u(t)*u(t), so once again the autocorrelations computed from
steps (i)-(iii) above should (at least) be helpful
for choosing among these models. For non-linear models, such as E-GARCH,
statistics other than
autocorrelations would be called for, in order to select between ARCH,
GARCH, EGARCH, etc.
--
Scott Gilbert
Economics Department, MC4515
Southern Illinois University at Carbondale
Carbondale, IL 62901-4515
phone: (618) 453-5065
fax: (618) 453-2717
e-mail: [EMAIL PROTECTED]
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
