Multi-Scale Entropy-Gated Deep Quantile Learning via EntropyFusion-VaR: Tail-Risk Forecasting for U.S. Energy-Sector...

Forecasting extreme-tail risk in energy equities involves temporal-memory mismatch, quantile crossing, and scarce extreme observations. This study proposes a multi-scale entropy-gated deep quantile learning model, EntropyFusion-VaR. Weighted permutation entropy (WPE) characterizes local ordinal disorder, Rényi entropy describes concentration in the full binned return distribution, and transfer entropy (TE) describes directional predictive information. These state features drive a gate that coordinates a causal dilated TCN and a Patch Transformer. An incremental cumulative output layer ensures ordered joint quantiles at the 1%, 2.5%, and 5% levels. An exponential aggregation of pinball losses gives greater optimization weight to large forecast errors, while a differentiable coverage penalty regularizes exception frequencies. Experiments on EOG Resources (EOG), Marathon Petroleum (MPC), and Schlumberger (SLB) show lower reported average tail loss and zero quantile crossing. The model attains the best mean rank at each studied horizon and ranks first in eight of nine stock-horizon combinations. The evidence supports state-dependent tail-risk forecasting for these three U.S. energy-sector equities, with coverage assessed separately from loss improvement.