Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.

翻译：预测协调是一种后预测过程，旨在根据给定的数据结构，将一组不一致的基础预测修正为一致的预测。大多数基于回归的点预测和概率预测协调结果都建立在所谓的“结构表示”及相关无约束广义最小二乘协调公式之上。然而，结构表示自然地适用于真正的层次/分组时间序列，其中顶层和底层变量被唯一标识。当考虑一般线性约束的多元时间序列时，预测协调根据投影方法自然表达。虽然经典的结构协调公式等价于其投影方法对应形式，但迄今为止尚不完全清楚是否以及如何为一般线性约束多元时间序列推导出类似结构的协调公式。这种表达式将允许以直接方式扩展协调定义、定理和结果。在本文中，我们证明对于一般线性约束多元时间序列，可以根据“类结构”方法表达协调公式，该方法保留自由变量和约束变量（而非底层和上层（聚合）变量），建立概率预测协调框架，并将这些发现应用于获得澳大利亚GDP从收入与支出两侧聚合的完全协调点预测和概率预测，以及按收入、支出和产出三侧及19个国家分解的欧元区GDP的完全协调预测。