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 evaluate the proposed methods through a simulation study that investigates their theoretical and empirical properties. We further assess the effectiveness of the proposed cross-temporal reconciliation approach by applying it to two empirical forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, we show that 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：预测调和是一种预测后处理过程，通过该过程可将一组不一致的预测转化为满足多变量时间序列给定线性约束条件的一致预测。本文在现有最先进的截面概率预测调和框架基础上，将其拓展至跨时域框架，并同时引入时间约束。我们提出的方法采用参数化高斯方法与非参数化自助法两种途径，从不一致的跨时域分布中抽取样本。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，特别是在常规单步残差失效的时间维度上。针对高维问题，我们提出四种协方差矩阵替代方案，充分利用跨时域结构的双重特性（截面性与时域性），并引入重叠残差概念。通过仿真研究评估了所提出方法的理论与实证性质。进一步，我们利用澳大利亚GDP与澳大利亚旅游需求数据集开展两项实证预测实验，验证了所提跨时域调和方法的有效性。两项应用均表明，在连续排名概率得分与能量得分指标上，最优跨时域调和方法显著优于不一致的基础预测。总体而言，本研究拓展并统一了截面、时域及跨时域调和的符号体系，深化了概率性跨时域框架的理论基础。研究结果凸显了所提跨时域预测调和方法在提升概率预测模型精度方面的潜力。