question about GARCH - newbie question

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question about GARCH - newbie question

oliver wee
hello,
I have been looking at multiple websites on GARCH and
have looked at some books and I am getting
contradictory models given for GARCH.

If I use the GARCH function to fit my model, I am
confused as to what the coefficents given refer to.

For example if I fit a GARCH(1,1) model, GARCH will
give me three coefficients Ao, Ai, and Bi

I know Ao refers to the constant of the model.
But what about Ai and Bi?

One website I looked at says the model for GARCH is:
1) S^2t = Ao + Ai S^2 t-1 + Bi E^2 t-1 + Et
(sigma squared sub t = a sub 0 + a sub 1 sigma squared
sub t-1 + b sub 1 epsilon squared sub t-1 + epsilon
sub t)
while a book I was reading says the general model is
2) S^2t = Ao + Ai Y^2 t-1 + Bi S^2 t-1
(sigma squared sub t = a sub 0 + a sub 1 Y squared sub
t-1 + b sub 1 sigma squared sub t-1).

where Y refers to the data points of th time series
S refers to the variance and E refers to the white
noise component.

I am very confused by this.

Also, two other questions regarding GARCH,
1) How exactly do I forecast future values of my
fitted GARCH model using R and how do I graph them?
predict() won't work since it won't accept the n.ahead
argument. (Assuming I already have fitted my original
data set to an ARIMA model and have predicted future
values using the predict() command).  

2) This is not about R, but about GARCH in general. I
just want to check if I understood GARCH correctly.
For example, I have already found an ARIMA model to
fit my original data set. In order for me to find the
appropriate GARCH model to fit my data set, I have to
get the acf of the squared values of my original data
set (assuming the data set is stationary, that is, its
acf show no significant correlations at all lags, but
the acf of its squared values is not stationary). Once
I get the acf of the squared values, I just compare
this to the acf of arma(p,q) models- similar to how I
was able to find my fitted ARIMA model? Is this
correct? Also, once I have found the appropriate GARCH
model, I can use this model to predict BOTH the future
values of the data set and its variance? and these
future values will be far different than the one given
my my fitted ARIMA model?

I am sorry for the abundance of questions and I am
pretty sure some of these questions sound pretty dumb.
Its that I'm doing my study of time series mostly
through self-study and I got really confused about
GARCH.

Thank you very much for your help.

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