Modelling Oil Price with Lie Algebras and Long Short-Term Memory Networks

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 S 2 manifolds that we consider, including the reference ones, employ matrix representations rather than differential operator representations of Lie algebras. Firstly, the performance of Lie NLS 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 S 2 manifold, namely Lie-LSTM OLS and Lie-LSTM NLS , are compared with those of the reference Lie OLS and Lie NLS models. The in-sample and out-of-sample results show that our proposed methods can achieve improved performance over Lie OLS and Lie NLS models in terms of RMSE and MAE metrics and hence can be more reliably used to assess volatility of time-series data.