A novel probabilistic carbon price prediction model: Integrating the transformer framework with mixed-frequency modeling at different quartiles

Most of the previous carbon price forecasting studies focus on point prediction based on same-frequency data, which ignores the large amount of prediction information provided by mixed-frequency data, while point prediction fails to quantify the uncertainty of carbon price fluctuations. Therefore, to fill this research gap, and improve the accuracy of carbon price prediction, this study proposes a novel hybrid forecasting model by integrating quantile regression, deep learning, and mixed-frequency modeling. Firstly, this study introduces twenty-three variables from energy commodities, market indexes, macroeconomic indicators, and environmental indicators, and then the feature selection method is applied for data dimensionality reduction to obtain the input factors. Subsequently, this study innovatively integrates mixed-frequency data sampling regression (MIDAS) and quantile regression (QR) into the Transformer architecture to construct a hybrid forecasting model, i.e., the QRTransformer-MIDAS model, and achieves point and interval prediction of low-frequency carbon price using high-frequency input factors. Meanwhile, the probabilistic prediction is implemented using kernel density estimation (KDE). In the comparison experiments, the proposed hybrid forecasting model realize mean absolute percentage errors (MAPE) of 1.46 % and 1.33 % for point predictions in Shanghai and Hubei carbon price datasets, respectively, moreover, with 95 % confidence intervals, the coverage width criterion (CWC) achieves 0.35 and 0.38, respectively, which outperforms the benchmark models. These experimental results confirm the practicality and robustness of the hybrid forecasting model proposed in this study.