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David Jennings wrote:
>
>As an aside, whilst they fit markets quite well, different parameters are
>required over different time frames. Thus the question to you guys is: if
>this is corect (and I believe it to be) it flatly ontradicts any fractal
>market hypothesis. Any assistance would be helpful.
>
My opinion is that it doesn't contradict any FMH. I'm cheating
here and looking at Edgar Peter's "Chaos and Order..."
specifically his Eq. (9.2). If you don't have the book,
this is an equation for a Pareto-Levy distribution, which has
4 characteristic parameters. [I would reproduce the Eq here,
but it would take some work!] Call the parameters A, B, C, and D.
Peters states that a normal dist'n results when A=2, B=0, C=1,
and D=1. A fat-tailed, skewed-to-the-right (i.e. leptokurtotic)
dist'n results when A is between 1 and 2, and B approaches 1.
It turns out A is actually the fractal dimension of the dist'n.
Now, what is the mean of this second distribution? It depends
on the time frame! For daily prices, if the mean is M, then
for 5-day returns the mean is 5*M. So, this is a case where
the distribution of the returns has a time-dependent parameter,
but the distribution is fractal.
So, to apply this to GARCH, it really comes down to this:
is the fractal dimension of the distribution independent of
the time frame? Any other parameters of the distribution
do not need to be independent of the time frame.
Jeff
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