Predicting Daily Probability Distributions of S&P500 Returns

Most approaches in forecasting merely try to predict the next value of the time series.In contrast, this paper presents a framework to predict the full probability distribution. Itis expressed as a mixture model: the dynamics of the individual states is modeled with so-calledquot;expertsquot; (potentially nonlinear neural networks), and the dynamics between the states is modeledusing a hidden Markov approach. The full density predictions are obtained by a weighted superpositionof the individual densities of each expert. This model class is called quot;hidden Markov expertsquot;.Results are presented for daily Samp;P500 data. While the predictive accuracy of the mean doesnot improve over simpler models, evaluating the prediction of the full density shows a clear out-of-sampleimprovement both over a simple GARCH(1,l) model (which assumes Gaussian distributedreturns) and over a quot;gated expertsquot; model (which expresses the weighting for each state non-recursivelyas a function of external inputs). Several interpretations are given: the blending ofsupervised and unsupervised learning, the discovery of hidden states, the combination of forecasts,the specialization of experts, the removal of outliers, and the persistence of volatility