Cross-temporal forecast reconciliation aims to ensure consistency across forecasts made at different temporal and cross-sectional levels. We explore the relationships between sequential, iterative, and optimal combination approaches, and discuss the conditions under which a sequential reconciliation approach (either first-cross-sectional-then-temporal, or first-temporal-then-cross-sectional) is equivalent to a fully (i.e., cross-temporally) coherent iterative heuristic. Furthermore, we show that for specific patterns of the error covariance matrix in the regression model on which the optimal combination approach grounds, iterative reconciliation naturally converges to the optimal combination solution, regardless the order of application of the uni-dimensional cross-sectional and temporal reconciliation approaches. Theoretical and empirical properties of the proposed approaches are investigated through a forecasting experiment using a dataset of hourly photovoltaic power generation. The study presents a comprehensive framework for understanding and enhancing cross-temporal forecast reconciliation, considering both forecast accuracy and the often overlooked computational aspects, showing that significant improvement can be achieved in terms of memory space and computation time, two particularly important aspects in the high-dimensional contexts that usually arise in cross-temporal forecast reconciliation.

翻译：跨时间预测协调旨在确保不同时间层面和横截面层面的预测结果之间的一致性。本文探讨了序列法、迭代法与最优组合法之间的关系，并讨论了在何种条件下序列协调方法（无论是先横截面后时间，还是先时间后横截面）等价于完全（即跨时间）一致的迭代启发式方法。此外，我们证明当回归模型中的误差协方差矩阵呈现特定模式时（最优组合法正是基于该回归模型），无论采用何种顺序应用单维横截面与时间协调方法，迭代协调都会自然收敛到最优组合解。通过使用小时级光伏发电数据集进行预测实验，研究了所提出方法的理论与实证特性。本研究提出了一个全面理解与改进跨时间预测协调的框架，同时考虑了预测精度与常被忽视的计算因素，表明在内存空间和计算时间方面均可实现显著改进——这两个维度在跨时间预测协调通常面临的高维场景中尤为重要。