Fuzzy GARCH models

Time series data exhibits complex behavior including path-dependency and non-linearity. It is important to employ flexible methods, such as a mixture of GARCH models, to allow for possible changes in the nature of this complex behavior. This work proposes a novel flexible fuzzy GARCH model that attempts to capture complex data behavior. Issues of parameterization and estimation are discussed. Regarding the former, we extend a previously studied fuzzy GARCH model and propose a new model. This model can capture changes in the conditional variance over time and also more complex data features such as skewness and multimodality, making the proposed model more general than standard GARCH models and the previous fuzzy GARCH model. It is further shown that the proposed model can be related to advanced dynamic mixture models for conditional volatility of the data, as well as for data average. The dynamic mixture nature of the model is captured using fuzzy rules that establishrelations between the defined variables. The model parameters are obtained using the maximum likelihood approach. The results are illustrated using simulated data with different specifications, and real stock market data.