Multi-Step-Ahead Carbon Price Forecasting Based on Variational Mode Decomposition and Fast Multi-Output Relevance Vector Regression Optimized by the Multi-Objective Whale Optimization Algorithm

The accurate and stable forecasting of carbon prices is vital for governors to make policies and essential for market participants to make investment decisions, which is important for promoting the development of carbon markets and reducing carbon emissions in China. However, it is challenging to improve the carbon price forecasting accuracy due to its non-linearity and non-stationary characteristics, especially in multi-step-ahead forecasting. In this paper, a hybrid multi-step-ahead forecasting model based on variational mode decomposition (VMD), fast multi-output relevance vector regression (FMRVR), and the multi-objective whale optimization algorithm (MOWOA) is proposed. VMD is employed to extract the primary mode for the carbon price. Then, FMRVR, which is used as the forecasting module, is built on the preprocessed data. To achieve high accuracy and stability, the MOWOA is utilized to optimize the kernel parameter and input the lag of the FMRVR. The proposed hybrid forecasting model is applied to carbon price series from three major regional carbon emission exchanges in China. Results show that the proposed VMD-FMRVR-MOWOA model achieves better performance compared to several other multi-output models in terms of forecasting accuracy and stability. The proposed model can be a potential and effective technique for multi-step-ahead carbon price forecasting in China’s three major regional emission exchanges.