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 empirical 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, our study expands and unifies the notation for cross-sectional, temporal and cross-temporal reconciliation, thus extending and deepening the probabilistic cross-temporal framework. The results highlight the potential of the proposed cross-temporal forecast reconciliation methods in improving the accuracy of probabilistic forecasting models.

翻译：预测协调是一种事后处理过程，旨在将一组不一致的预测转化为满足多元时间序列给定线性约束的协调预测。本文扩展了当前最先进的横截面概率预测协调方法，将其纳入跨时间框架，同时施加时间约束。我们提出的方法采用参数化高斯方法和非参数自助法，从非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，尤其是在通常的单步残差失效的时间维度上。为解决高维问题，我们提出了协方差矩阵的四种替代方案，利用跨时间结构的双重性质（横截面和时间），并引入了重叠残差的概念。我们通过一项模拟研究（探讨所提出方法的理论和经验特性）以及两项实证预测实验（使用澳大利亚GDP和澳大利亚旅游需求数据集）评估了所提出的跨时间协调方法的有效性。在这两个应用中，最优的跨时间协调方法在连续排名概率得分和能量得分方面显著优于不一致的基础预测。总体而言，我们的研究扩展并统一了横截面、时间和跨时间协调的符号体系，从而深化了概率跨时间框架。研究结果凸显了所提出的跨时间预测协调方法在提升概率预测模型准确性方面的潜力。