GARCH question with Zivot's book and finmetrics

["Michael Sun" <mam3xs@gmail.com>]

Dear S+/R users,

By reading Zivot's book, the manual of Finmetrics and other papers, I got some questions on GARCH model. (hope it's not sily:-p)

Suppose a return series
r(t) = c + a(t), where a(t) is the innovation term.

Follow a simple GARCH (1, 1) model,

r(t) = c + a(t)
a(t) = sigma(t) * e(t)
sigma(t)^2 = alpha(0) + alpha(1)*a(t-1)^2 + beta*sigma(t-1)^2

So the residuals of the model is a(t), and the standarised residuals(some paper refered filtered residuals) is e(t)=a(t)/sigama(t).

As it is indicated in Zivot's book (and other literatures) "the error distributions could be followed student's t distribution, ged, etc..."
So confirming here, it is the residuals of the model, a(t), that is  following a student's t distribution! Am I correct?

After that, the standardrised residuals, e(t)= a(t)/sigma(t), can be fitted with EVT theory. This is to capture the asymetries for the financial markets.

a(t) ~ t(v)
e(t) ~ EVT

Is this making sense?

Appreciate for any comments.

Cheers.
Mam