Estimation of flexible fuzzy GARCH models for conditional density estimation

In this work we introduce a new flexible fuzzy GARCH model for conditional density estimation. The model combines two different types of uncertainty, namely fuzziness or linguistic vagueness, and probabilistic uncertainty. The probabilistic uncertainty is modeled through a GARCH model while the fuzziness or linguistic vagueness is presented in the antecedent and combination of the rule base system. The fuzzy GARCH model under study allows for a linguistic interpretation of the gradual changes in the output density, providing a simple understanding of the process. Such a system can capture different properties of data, such as fat tails, skewness and multimodality in one single model. This type of models can be useful in many fields such as macroeconomic analysis, quantitative finance and risk management. The relation to existing similar models is discussed, while the properties, interpretation and estimation of the proposed are provided. The model performance is illustrated in simulated time series data exhibiting complex behavior and a real data application of volatility forecasting for the S&P 500 daily returns series.