Forecast reconciliation is a post-forecasting process that involves transforming a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper we extend the current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent cross-temporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the two-fold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We assess the effectiveness of the proposed cross-temporal reconciliation approaches through a simulation study that investigates their theoretical and empirical properties and two forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the Continuous Ranked Probability Score and the Energy Score. Overall, the results highlight the potential of the proposed methods to improve the accuracy of probabilistic forecasts and to address the challenge of integrating disparate scenarios while coherently taking into account short-term operational, medium-term tactical, and long-term strategic planning.

翻译：预测协调是一种预测后处理过程，旨在将一组不一致的预测转换为满足多变量时间序列特定线性约束条件的协调预测。本文扩展了当前最先进的横截面概率预测协调方法，将其应用于涵盖时间约束的跨时间框架。我们提出的方法论同时采用参数化高斯方法和非参数化自助法，从不一致的跨时间分布中抽取样本。为改进预测误差协方差矩阵的估计，我们提出使用多步残差，尤其是在通常单步残差失效的时间维度上。针对高维度问题，我们提出四种协方差矩阵替代方案，充分利用跨时间结构的两重性（横截面与时间维度），并引入重叠残差的概念。通过模拟研究（探究其理论与经验性质）以及两项预测实验（使用澳大利亚GDP与澳大利亚旅游需求数据集），我们评估了所提出的跨时间协调方法的有效性。在两项应用中，最优跨时间协调方法在连续排序概率评分与能量评分指标上均显著优于不一致的基础预测。总体而言，研究结果凸显了所提方法在提升概率预测准确性方面的潜力，并解决了整合不同情景的挑战，同时协调地考虑了短期运营、中期战术与长期战略规划。