Assessing fat-tailed sequential forecast distributions for the Dow-Jones index with logarithmic scoring rules

We use the logarithmic scoring rule for distributions to assess a variety of fat-tailed sequential forecasting distributions for the Dow-Jones industrial stock index from 1980 to the present. The methodology applies Bruno de Finetti''s contributions to understanding how to compare the quality of different coherent forecasting distributions for the same sequence of observations, using proper scoring rules. Four different forms of forecasting distributions are compared: a mixture Normal, a mixture of convex combinations of three Normal distributions, a mixture exponential power distribution, and a mixture of a convex combination of three exponential power distributions. The mixture linear combination of three EP distributions achieves the best score on a fairly regular basis, followed by the mixture EP and the mixture linear combination of three Normals. All three make marked regular improvements in assessing volatility phenomena (tail behaviour of the distributions) compared to the Normal distribution, with an especially noticeable improvement achieved after the information gained from the drastic fall in stock prices that occurred in October, 1987. The mixture EP distributions are designed to incorporate fat-tailed properties into the forecast probabilities. It is surprising that the mixture linear combination of three Normal distributions fairs as well as it does in the comparison of scores. Overall, the methodology provides a practical improvement in comparing the quality of statistical forecasts over the so-called ``model-fitting'''' procedures that have long been used in the statistical assessment of contentious economic issues. The results of the scored forecasting exercise are supported by an analysis of the expected scores via the entropies of the forecast distributions, and a quadratic score of these expectations, as well.