| Summary: | In this paper, we propose hybrid models for modelling the daily oil price during the period from 2 January 1986 to 5 April 2021. The models on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> manifolds that we consider, including the reference ones, employ matrix representations rather than differential operator representations of Lie algebras. Firstly, the performance of Lie<sub>NLS</sub> model is examined in comparison to the Lie-OLS model. Then, both of these reference models are improved by integrating them with a recurrent neural network model used in deep learning. Thirdly, the forecasting performance of these two proposed hybrid models on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> manifold, namely Lie-LSTM<sub>OLS</sub> and Lie-LSTM<sub>NLS</sub>, are compared with those of the reference Lie<sub>OLS</sub> and Lie<sub>NLS</sub> models. The in-sample and out-of-sample results show that our proposed methods can achieve improved performance over Lie<sub>OLS</sub> and Lie<sub>NLS</sub> models in terms of RMSE and MAE metrics and hence can be more reliably used to assess volatility of time-series data.
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