Zivot vs. Engle vs. Stoffer - help with the meaning of different GARCH notations, please!

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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.
   

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 at braverock.com 
<mailto:brian at 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