Exploring volatility of crude oil intra-day return curves: a functional GARCH-X model

Crude oil intra-day return curves collected from the commodity futures market often appear to be serially uncorrelated and long-range dependent. Existing functional GARCH models, while able to accommodate short range conditional heteroscedasticity, are not designed to capture long-range dependence. We propose and study a new functional GARCH-X model for this purpose, where the covariate X is chosen to be weakly stationary and long-range dependent. Functional analogs of autocorrelation coefficients of squared processes for this model are derived, and compared to those estimated from crude oil return curves. The results show that the FGARCH-X model provides a significant correction to existing functional volatility models in terms of an in-sample fitting, while its out-of-sample performances do not appear to be more superior than those of the existing functional GARCH models.