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

[Brian G. Peterson]

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