Probabilistic fuzzy systems for seasonality analysis and multiple horizon forecasts

Probabilistic fuzzy systems (PFS), a model which combines a linguistic description of the system behaviour with statistical properties of data, have been successfully applied to one day ahead Value at Risk (VaR) estimation for the stock market returns data. In this work, we propose a multi-covariate multi-output PFS model which provides the conditional density forecasts of returns for one day ahead and one month ahead periods. Such a multi-output PFS model was not considered in the literature. Furthermore, this model allows to analyze seasonal patterns in returns. The proposed model is applied to daily S&P500 stock returns. It is found that the proposed model indicates seasonal patterns in short and longer horizons as well as conservative VaR in long term forecasts. The model is shown to perform well in VaR estimation according to the unconditional coverage and independence tests.