Forecast reconciliation usually starts from a fixed measurement system and asks how forecasts should be projected onto a coherent space. We ask a different question: which additional linear measurements should be forecast and included in the reconciliation system? We propose REGAIN, a reconciliation-gain framework that learns normalized auxiliary directions, forecasts the induced series with a frozen forecasting oracle, and selects directions by their target-weighted loss reduction after augmented generalized least-squares reconciliation. Unlike variance-based components or predictability-based auxiliary selection, REGAIN optimizes the downstream effect of an auxiliary measurement on the final reconciled forecasts. We provide a statistical characterization showing that useful auxiliary directions must provide complementary information about unresolved target uncertainty, rather than merely being easy to forecast. The analysis also clarifies the covariance-risk reduction mechanism, the role of bias changes in realized quadratic risk, and the stability of estimated gain signals. A stagewise learning algorithm with held-out gain screening is developed, together with an optional joint refinement step. Experiments on Beijing PM2.5 and Australian Tourism data show that gain-selected measurements can improve both ordinary multivariate and hierarchical forecasts, especially when they reveal residual uncertainty not captured by the original measurement system.

翻译：预测协调通常始于一个固定的测量系统，并询问如何将预测投影到一致空间中。我们提出一个不同的问题：哪些额外的线性测量应被预测并纳入协调系统？我们提出REGAIN——一种协调增益框架，该框架学习归一化辅助方向，使用冻结的预测预言机预测诱导序列，并通过增广广义最小二乘协调后的目标加权损失减少来选择方向。与基于方差的主成分或基于可预测性的辅助选择不同，REGAIN优化辅助测量对最终协调预测的下游影响。我们提供统计表征，表明有用的辅助方向必须提供关于未解决目标不确定性的互补信息，而不仅仅是易于预测。分析还阐明了协方差-风险降低机制、实现二次风险中偏差变化的作用，以及估计增益信号的稳定性。我们开发了一种包含保留增益筛选的分阶段学习算法，并辅以可选的联合优化步骤。在北京PM2.5和澳大利亚旅游数据上的实验表明，增益选择的测量能改进普通多元预测和层次预测，尤其是当它们揭示原始测量系统未捕捉到的残余不确定性时。