In the previous post
we illustrated the evident abnormality of financial data by examining the longitudenal returns of the
S&P 500 Index.
I used the Generalized Error Distribution as it possesses the ability to be smoothly transformed from a Normal Distribution into a leptokurtotic distribution and that allowed me to use the Maximum Likelihood Ratio Test to distinguish between the null hypothesis (that the data is I.I.D. Normal) and the alternate hypothesis (that it is not).
I subscribe to the theory that if something is right you should be able to draw the same conclusions via various methods and data sets. So I am going to look again at the likely models for the innovations of financial data (we're taking a GARCH(1,1) model as given); but, this time, I decided to look at the S&P Goldman Sachs Commodity Index and to use a test based on Pearson's χ² Test. (In the following the data is actually based on the first deliverable contract on the GSCI traded at the CME.)
Before that, however, we should discuss what the possible options are the for the PDF of the process innovations. The candidates are:
- The Normal Distribution
- Levy Flight
- The Generalized Error Distribution
- Student's t Distribution
- something else…
Benoît Mandelbrot discussed the long tails in the distribution of the changes in commodity prices in his early papers on finance (these are collected in Fractals and Scaling In Finance
). He found that the tails appeared to exhibit a scaling property and avocated the Levy Flight as it is a stable distribution with this scaling property. He work was decades pre-GARCH and the distribution has many undesirable properties — such as a divergent variance. I will not model it here.
Clearly we view the Normal Distribution as a non-starter. The GED and the Student's t Distribution both provide excellent matches to data and can provide quite leptokurtotic forms. Student's t has some difficulties for other parts of finance theory — only moments of order less than the degrees of freedom parameter exist which makes it difficult to use the negative exponential utility. GED does not have problems with negative exponential utility.
After a long winded discussion, we will advance rapidly through the results. A GARCH(1,1) model was fitted to the daily changesof the first deliverable GSCI future (combined to produce a synthetic series). n.b. I haven't reserved an out-of-sample period for the data as I view the utility of empirical GARCH as so well established that I don't think we really need to test for it. The analysis includes a histogram of the daily innovations extracted from the data. This histogram is fitted to a p.d.f. form using a maximum likelihood method and Poisson statistics for the bin counts. After the fit, a d.o.f. corrected χ² statistic is computed.
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The results are: for the Normal Distribution, χ²/dof = 766.024145/98 with a p-Value of less than 0.00000001; for the GED, 126.231861/97 with a p-Value of 0.02468288; and for the Student-t, 101.748435/97 with a p-Value of 0.35078429. Remember that for a χ² distribution with n degrees of freedom the mean is n and the variance is 2n. The Normal Distribution is convincingly rejected; the Student t is the best fit and the GED is ok — neither can be rejected at a reasonable confidence level such as 0.001.