Assessing the stability of Gaussian mixture models for monthly returns of the S&P 500 index

The study analyses the unconditional distribution of monthly S&P 500 stock index returns for the long-run time period 1871-2004. The return distribution can be adequately described by a mixture of two Gaussian normal distributions. However, when analysing sub-samples of this long-time horizon, substantial deviations between the empirical and the estimated two-component distribution become evident. Formal tests clearly reject the hypothesis of random draws from the estimated distribution. A comprehensive analysis of ten-year windows within the framework of a rolling window strategy reveals that window-specific estimated two-component mixtures can adequately describe the empirical distributions in almost all windows. Nevertheless, the substantial variation in the weight of the mixtures as well as in the parameters of the mixed distributions suggests that there are severe difficulties involved in maintaining the notion of an underlying distribution being constant to a certain degree.