Volatility forecasting: a new GARCH-type model for fuzzy sets-valued time series

In recent years, academia's attention has gradually shifted toward non-point-valued time series volatility forecasting models in the finance big data environment. This paper uses random set theory to define the random fuzzy sets-valued assets returns and propose a new Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-type model named the Set-GARCH model, which describes the evolution of sets-valued returns time series volatility. We conceptualize such a model in both cases of correlated and uncorrelated returns. We discuss the subtraction operation rule, the model specification, and the maximum likelihood estimation method for the Set-GARCH model and its derivative model. We also define how to convert the volatility of fuzzy sets-valued returns to the volatility of real returns. Using long timespan daily/weekly/monthly oil, S &P500, and gold returns data, both in-sample and out-of-sample empirical applications demonstrate that the volatility prediction ability of the Set-GARCH model and its derivative outperforms the point-valued GARCH-type models, conditional autoregressive range-type models, and two hotly debated interval-valued volatility models.