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

[Zeno Adams]

I think the equations can be regarded as equivalent. As Patrick pointed
out the main difference is if you write the mean equation in
standardized residuals or not.

<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)

e(t) can be written as sigma(t)*v(t) with v(t)~N(0,1) (often also
student's-t or GED)

<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)

here you have the more detailed notation with sqrt(h(t)) = sigma(t) and
e(t) ~N(0,1) =  v(t) 
and h(t)*e^2(t) = e^2(t)from Zivot's book
m(t)probably indicates that the mean is calculated as a rolling mean
and so thus changes over time.

<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)

here the constant is neglected in the mean equation as it is very small
for most daily data. Since Y(t) = sigma(t)*e(t) it now shows up as
a1*Y^2(t-1) in the variance equation.


Zeno