Thanks Brian. Sorry about the reference being not clear. Here is my 2nd try:

Zivot: MFTSWS, 2003 edition. [1] page 212, equation 7.1, and [2] page 216,
equation 7.5
Engle: Garch101, middle of page 160.
Stoffer: you had it exactly right. page 286, equation 5.30 and 5.44.

So my follow up question to you is why do "these models all appear
equivalent" to you? Take equation [6] for example, it says to me that one
could model the variance part of the process using garch and only garch. If
these notations were all the same, then garch alone should also be able to
model [2], right? But it seems to me that that's not the case, i.e., arma +
garch is needed to model [2]. Do you know what I mean? Does that make sense?

Thanks!

On 2/2/08, Brian G. Peterson <brian@braverock.com> wrote:
>
> tom soyer wrote:
> > Hi,
> >
> > I have a question with regard to different GARCH notations I found in
> the
> > literature, and I am wondering if anyone knows how to reconcile these
> > differences. Below are three different notations that supposedly all
> define
> > the GARCH(1,1) process:
> >
> > In Zivot's book, MFTSWS, the GARCH(1,1) process is defined as:
> > [1]: Y(t) = c + e(t), and
> > [2]: sigma^2(t) = a0 + a1*e^2(t-1) + b1*sigma^2(t-1)
>
> I just looked in my current copy of Zivot and Wang MFTSwS+ (2006), p.
> 230, Eqs 7.4 and following, and your notation here doesn't match what's
> in the reference (your Eq [2] appears equivalent to Eq. 7.5).  perhaps
> next time you can be more specific in your reference (pages and Eq.
> numbers?)
>
>
> > In Engle's paper, the GARCH(1,1) process is defined (in financial
> notation),
> > like this:
> >  [3]: r(t) = m(t) + sqrt(h(t))*e(t), and
> > [4]: h(t+1) = a0 + a1*h(t)*e^2(t) + b1*h(t)
>
> I don't know which Engle paper you're referring to.  With the possible
> exception of m(t) in your Eq[3] and the use of t+1 as the target in
> Eq[4] (thus specifying the prediction), Eq [4] is equivalent to Eq [2]
> and Eq [6]
>
> > In Stoffer's book, the GARCH(1,1) is define as:
> > [5]: Y(t) = sigma(t)*e(t), and
> > [6]: sigma^2(t) = a0 + a1*Y^2(t-1) + b1*sigma^2(t-1)
>
> Shumway and Stoffer "Time Series Analysis and Its Applications, 2nd
> Ed."(2006), p. 286 Eqs. 5.30 and 5.44 match your Eq [5] and [6] and
> match Zivot&Wang's representation.
>
> Note that Shumway and Stoffer also has several fairly extensive examples
> of working with GARCH models in R.
>
> > Does anyone know if all three above are just different ways of saying
> the
> > same thing, or are they drastically different with respect to the
> > specification of the GARCH model to be fitted?
>
> Notation is always a real pain to sort out as you are reading various
> papers and books.  It is not uncommon to find errors in the references,
> which is usually cleared up only via looking further back in time to
> more primary sources.
>
> So, without precise references, I can only give you a qualified "these
> models all appear equivalent".
>
> Regards,
>
>   - Brian
>



-- 
Tom

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