分层与分组时间序列协调的预测组合框架
A Forecast Combination Framework for Hierarchical and Grouped Time Series Reconciliation
AI总结：
该研究提出用于分层与分组时间序列协调的预测组合框架，将预测组合与协调关联，证明MinT协调等价于最优预测组合，提出模块化惩罚框架，该框架实用且可提升预测准确性。
AI中文摘要：
分层与分组时间序列的预测组合和预测协调在很大程度上已发展为独立的研究领域。本文通过开发一种用于预测协调的预测组合框架将两者联系起来。对于每个底层序列，我们从聚合约束中构建一个结构化候选预测的最大线性无关集，证明将这些候选预测进行组合并聚合得到的底层预测等价于标准无偏线性协调。在该表示中，我们证明均方误差最优组合权重恰好恢复广泛使用的最小迹（MinT）协调。我们进一步证明最优权重问题可在不同底层序列间分离，每个序列产生一个贝茨-格兰杰（Bates-Granger）最优预测组合。这揭示MinT是基于层级诱导候选预测的最优组合集合。对于有限样本估计，我们提出一个模块化惩罚框架，该框架嵌套现有MinT变体并支持丰富扩展，包括协方差收缩、权重惩罚以及可扩展的序列级单独估计。实证结果表明，该框架实际可实现，与现有方法相比具有竞争力，可在保持一致性的同时提高准确性。总体而言，预测组合视角为现有协调方法提供了新的解释，并为设计新方法提供了灵活基础。
英文摘要：
Forecast combining and forecast reconciliation for hierarchical and grouped time series have largely developed as separate research areas. This paper connects the two by developing a forecast combination framework for forecast reconciliation. For each bottom-level series, we construct a maximal linearly independent set of structured candidate forecasts from aggregation constraints, and show that combining these candidates and aggregating the resulting bottom-level forecasts is equivalent to standard unbiased linear reconciliation. Within this representation, we prove that mean-squared-error optimal combination weights exactly recover the widely used Minimum Trace (MinT) reconciliation. We further show that the optimal weight problem is separable across different bottom-level series, each yielding a Bates--Granger optimal forecast combination. This reveals MinT as a collection of optimal combinations over hierarchy-induced candidate forecasts. For finite-sample estimation, we propose a modular penalized framework that nests existing MinT variants and supports rich extensions including covariance shrinkage, weight penalization, and scalable series-wise separate estimation. Empirical results show that the framework is practically implementable, competitive with existing methods, and can improve accuracy while preserving coherence. Overall, the forecast combination perspective offers new interpretations of existing reconciliation approaches and provides a flexible basis for designing new methods.
