REGAIN method learns which forecasts to add for better reconciliation

Traditional forecast reconciliation starts with a fixed measurement system and projects forecasts onto a coherent space. REGAIN flips the question: instead of asking how to reconcile, it asks which additional measurements should be included. The framework learns normalized auxiliary directions, forecasts the induced series using a frozen forecasting oracle, and selects directions based on target-weighted loss reduction after augmented generalized least-squares reconciliation. This approach directly optimizes the downstream impact of auxiliary measurements on final forecasts, rather than relying on variance components or predictability alone.

The paper provides a statistical characterization showing useful auxiliary directions must offer complementary information about unresolved target uncertainty—not just be easy to forecast. It clarifies the covariance-risk reduction mechanism, the role of bias changes in quadratic risk, and ensures stability of estimated gain signals. A stagewise algorithm with held-out gain screening is developed, with an optional joint refinement step. Tests on real-world datasets (Beijing PM2.5, Australian Tourism) demonstrate that gain-selected measurements improve both ordinary multivariate and hierarchical forecasts, particularly when they reveal residual uncertainty not captured by the original measurement system. The method is published as arXiv:2606.04380.