Forecasting the European Union allowance price tail risk with the integrated deep belief and mixture density networks

The European Union Emissions Trading System (EU ETS) is the worlds largest carbon market, with the European Union Allowance (EUA) as its core trading unit. The significant price volatility of EUAs poses challenges for effective risk management. This study proposes a novel approach integrating Deep Belief Networks (DBN) and Mixture Density Networks (MDN) to predict the tail risk of EUA prices (EUATR), quantified using the Value-at-Risk (VaR) model. Utilizing daily data from August 1, 2012, to March 1, 2025, the DBN-MDN model effectively captures the nonlinear and heavy-tailed distribution of EUA prices. The model achieves superior predictive performance, with a Mean Absolute Error (MAE) of 0.0014, Root Mean Squared Error (RMSE) of 0.0019, Mean Absolute Percentage Error (MAPE) of 3.5441 %, and an R2 of 0.9332, significantly outperforming benchmark models including ARIMA, CNN, RF, SVM, XGBoost, LSTM, CNN-LSTM, and standalone MDN. Residual analysis and the Diebold-Mariano test confirm the model's robustness and statistical superiority. These findings provide market participants with precise tools for optimizing hedging strategies and offer policymakers insights for enhancing EU ETS market stability. This study underscores the potential of deep learning in carbon market risk forecasting, supporting the EU's carbon neutrality objectives.