When talking about the SPX data, I glibly asserted that the data was evidently
not I.I.D. normal. I then proceeded to show how the
Generalized Error Distribution can be used to describe the data quite
well and to reject the hypothesis that the data is I.I.D. Normal with
a reasonable degree of confidence.
It occurred to me that some readers of this blog might be a little less familiar
with eyeballing financial data sets, so it might be interesting to
generate such a sample path in a
Monte-Carlo simulation of the process.
The above plot has four panes. In the upper left we show the aggregation of
about 2500 IID Normal draws (this is approximately the number of business days
in a decade). There is no drift and no heterskedasticity in this data set, it
is a pure random walk. Below that is the time series of estimated GARCH
volatility. You can see that the process essentially discovers the population
variance of 1, but it is occasionally kicked away from that value by outlying
innovations. In the upper right is the time series of innovations. In my normal
analysis this is after
standardizing by dividing each innovation by the standard deviation
forecast, using the GARCH model, from the prior data. For this particular dataset,
the standardization has no particular impact. The series of well defined fuzz
essentially reperesents the time series of a homoskedastic dataset, and its visual
appearance is in stark contrast to the of a heteroskedastic series,
such as that presented in the earlier post about the GED.
At the bottom, I have replaced the histogram of the standardized innovations with a plot
illustrating the
empirical distribution function and comparing it to the
cumulative distribution function. The maximum distance between these two curves, after scaling,
is the test statistic used in the
Kolmogorov-Smirnov test, which is a powerful, distribution fee, bin free, test for univariate
distribution identification. We see a p-Value of 21% for this data, which represents the probability
of finding an maximum distance at least as large as the sample, which clearly cannot be used to
reject the null hypothesis in this case.