Thanks very much Patrick! You reply makes a lot of sense.

With regard to equations [1], [2], [3], and [4], I did some more research
and found this quote from Mark Steele with regard to Engle's paper:"The
notation of this paper is not exactly the same as the notation we use for
compatibility with S-Plus, but one should not have any trouble making the
translation". Apparently according to Mark Steele, [1] & [2] can be
transformed into [3] & [4], i.e., they are just two different ways of
describing the same process.

I think Engle's m(t) notation means the mean of r(t) is m(t). If so, then
[1] and [3] are consistent with each other as far as both having the mean of
Y(t) as their constant. Engle stated in his paper that e(t) in his equation
is "the error term for the present period", and "the variance of e is one".
Zivot said "e(t) is iid with mean zero" (I think this means e(t) is the
standardized residuals having a mean zero but non-constant variance,
right?). So perhaps this is why the difference in notation between Engle and
Zivot, one uses the residuals while another uses standard residuals? I am
not sure why Engle said "the variance of e is one", I thought the variance
of the error term changes with time in GARCH.

I am still lost with regard to Stoffer's definition of GARCH(1,1) in
equation [5] and [6]. It seems that Stoffer was trying to describe a very
special case of GARCH(1,1) where the mean of Y(t) is assumed to be zero AND
the process is homoscedastic, i.e., constant variance. According to his book
(on page 281) e(t) in [5] is defined as: "e(t) ~ iid N(0,1)". So, I
think perhaps [5] should be Y(t) = e(t) instead, since sigma of e(t) is 1 by
definition. If that were true, then I guess the process he described is
actually more of an ARMA process than GARCH? Does that make sense?





On 2/3/08, Patrick Burns <patrick@burns-stat.com> wrote:
>
> Brian G. Peterson wrote:
>
> > [... skip (about some references) ...]
> >
> >>Eq's [1],[3],[5] in your list all refer to an AR(1) model for the
> >>returns, of the variance modified by a white-noise parameter.
> >>
> >>
>
> I don't think this is an accurate statement.  In Eq 1 the mean
> is modeled by 'c', that is a constant.  In Eq 3 the mean is modeled
> by 'm(t)' -- on the surface at least an arbitrary time series model
> that could be ARMA or whatever.  Eq 5 assumes a constant mean
> of zero.
>
> The other difference in the equations is whether or not 'e(t)' is the
> residuals or the standardized residuals.
>
> Patrick Burns
> patrick@burns-stat.com
> +44 (0)20 8525 0696
> http://www.burns-stat.com
> (home of S Poetry and "A Guide for the Unwilling S User")
>
> >>Eq's [2],[4],[6] in your list all describe the GARCH(1,1) "generalized"
> >>extension of the basic ARCH process, which does indeed utilize an ARMA
> >>process to model y^2(t).  See the discussion around and following
> >>Shumway and Stoffer's Eq. 5.45 or Zivot and Wang's Eq. 7.6
> >>
> >>Regards,
> >>
> >>   - Brian
> >>
> >>
> >>
> >>
> >>On 2/2/08, *Brian G. Peterson* <brian@braverock.com
> >><mailto: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
> >>
> >>
> >
> >_______________________________________________
> >R-SIG-Finance@stat.math.ethz.ch mailing list
> >https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> >-- Subscriber-posting only.
> >-- If you want to post, subscribe first.
> >
> >
> >
> >
>



-- 
Tom

	[[alternative HTML version deleted]]

