In an earlier post
we looked at two candidate distributions for the GARCH innovations of GSCI data,
specifically the
Generalized Error Distribution and the
Student's t Distribution.
Using
Pearson's χ² Test on histograms of the GSCI GARCH innovations, we saw
that both the Student and GED distributions were acceptible descriptions of the
observed data — although the Student was closest.
Here, we follow up this analysis by performing the
Kolmogorov-Smirnov test on the GSCI innovations. This test is thought
of as a more modern test in that it is bin free, distribution
free, and quite powerful. However, it is a solely univariate test
whereas χ² can be applied to binned histograms in any number of dimensions.
%20Kolmogorov%20Test.jpg)
In the above document the standard GARCH workup with the K-S test analysis is
shown for the Student option. The test statistic, Dmax, was found to
be 0.03423 with a p-Value of 0.00268. Note that a total of 5 degrees
of freedom were used in fitting the distribution and the K-S test was designed
to be applied to directly measured data without estimation. This
introduces a bias into the test, so we should be cautious in using the results
(I have corrected the d.o.f. used in computing the normalized test statistic).
Below is the analysis for the GED parameterization. This gives a Dmax of
0.01877 with a em>p-Value of 0.27275. The same comment regarding bias
due to fitting applies, although since the number of fitted parameters is the
same, and we are using the test to distinguish between two choices, the bias is
probably not too critical.
%20Kolmogorov%20Test.jpg)
The p-Value for the normalized test statistic represents the probability of a
discrepancy at least as large as this one having arisen by chance. The measure
for the Student distribution is 100 × less likely than that for the
GED so, assuming that the correction for estimation bias affects these
probabilities in essentially the same manner, this test strongly prefers the
GED over the Student distribution — which is the opposite conclusion to that
from the Pearson χ² Test, which mildly prefers the Student distribution.
We now have two different answers to the problem of distributional choice for a
financial dataset. The issue of truth in financial data is different to that in
physical data. In physics, one can feel confident about making a staement as to
whether a models is true or false, whereas in finance models are merely representations
of reality that are more or less accurate. Both are probably
sufficiently empirically suitable.