Crude oil intraday return curves collected from commodity futures markets often appear to be serially uncorrelated and long-range conditionally heteroscedastic. We model this stylised feature with a newly proposed functional GARCH-X model and use it to forecast crude oil intraday volatility. The predicted intraday volatility provides important economic implications in crude oil commodity futures markets in both intraday risk management and utility benefits improvements. The functional GARCH-X model provides a remarkable correction to modelling crude oil volatility in terms of an in-sample fitting, although its out-of-sample performances in forecasting intraday risk measures do not appear to be significantly superior to that of the existing functional GARCH(1,1) model. However, the FGARCH-X model, with its flexibility to capture long-range dependence and potential seasonality, does confer substantial economic benefits by embedding inter-daily volatility forecasts. Methodologically, we show that the new model has a well-behaved stationary solution, and we also address the inherent and critical issues associated with the estimation of functional volatility models by introducing novel data-driven, non-negative and predictive basis functions in the estimation process.

从商品期货市场收集的原油日内回报曲线通常表现为序列不相关和长期条件异方差。我们使用新提出的功能性 GARCH-X 模型对这种风格化特征进行建模，并用它来预测原油日内波动性。预测的日内波动对原油商品期货市场的日内风险管理和公用事业效益改善产生了重要的经济影响。函数式 GARCH-X 模型在样本内拟合方面对原油波动性建模提供了显着的修正，尽管其在预测日内风险度量方面的样本外性能似乎并不明显优于现有函数式模型。 GARCH(1,1) 模型。然而，FGARCH-X 模型，凭借其捕获长期依赖性和潜在季节性的灵活性，确实通过嵌入日间波动预测带来了巨大的经济效益。在方法上，我们表明新模型具有良好的平稳解决方案，并且我们还通过在估计中引入新颖的数据驱动、非负和预测基函数来解决与函数波动率模型估计相关的固有和关键问题过程。