I thought I should add a brief note to point out that the
Kolmogorov-Smirnov test for the GSCI daily changes is actually quite a
remarkable result.
I've used the K-S test quite a lot, and always found it a little unsatisfactory,
because it is a very powerful test and real financial data seldom matches a
parameterized p.d.f. very well. i.e. In my experience the K-S test always
rejects the null hypothesis (that the proposed distributional shape is
correct). In fact, when I started writing the post, my expectation was that the
thrust of the piece would be to assert:
the K-S shows that neither
p.d.f. is the true one; so we have leeway in choosing the most
analytically convenient of the candidates.
However, the actual result we ended up with is the opposite of this. A p-Value
of 20% essentially means that you cannot reject the null.
The modelled
p.d.f. of GARCH(1,1) in daily changes with GED innovations is actually a
reasonable description of the data.
One seldom sees such a good fit in finance, so let's add a few words of caution:
this is an in-sample result — the statistic is computed from the data
fitted to. So the procedure has already sought out and eliminated some of the
realizations that are problematic for the parameterization. The out-of-sample
future data may not match quite so well.