[R-SIG-Finance] Zivot vs. Engle vs. Stoffer - help with the meaning of different GARCH notations, please!

[Patrick Burns]

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 at 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 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
>>    
>>
>
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