Linearly constrained multiple time series may be encountered in many practical contexts, such as the National Accounts (e.g., GDP disaggregated by Income, Expenditure and Output), and multilevel frameworks where the variables are organized according to hierarchies or groupings, like the total energy consumption of a country disaggregated by region and energy sources. In these cases, when multiple incoherent base forecasts for each individual variable are available, a forecast combination-and-reconciliation approach, that we call coherent forecast combination, may be used to improve the accuracy of the base forecasts and achieve coherence in the final result. In this paper, we develop an optimization-based technique that combines multiple unbiased base forecasts while assuring the constraints valid for the series. We present closed form expressions for the coherent combined forecast vector and its error covariance matrix in the general case where a different number of forecasts is available for each variable. We also discuss practical issues related to the covariance matrix that is part of the optimal solution. Through simulations and a forecasting experiment on the daily Australian electricity generation hierarchical time series, we show that the proposed methodology, in addition to adhering to sound statistical principles, may yield in significant improvement on base forecasts, single-task combination and single-expert reconciliation approaches as well.

翻译：线性约束多重时间序列可能出现在许多实际情境中，例如国民账户（如按收入、支出和产出分列的国内生产总值），以及变量按层级或分组组织的多级框架，例如按地区和能源来源分列的国家总能源消耗。在这些情况下，当可获得针对每个单独变量的多个非相干基础预测时，可以采用一种预测组合与协调方法——我们称之为相干预测组合——以提高基础预测的准确性，并确保最终结果的协调性。本文提出一种基于优化的技术，该技术能够组合多个无偏基础预测，同时确保约束条件对序列有效。我们针对一般情况（即每个变量可获得不同数量的预测）给出了相干组合预测向量及其误差协方差矩阵的闭式表达式。我们还讨论了与最优解中协方差矩阵相关的实际问题。通过对澳大利亚日度电力生成层级时间序列的模拟和预测实验，我们证明所提出的方法不仅遵循可靠的统计原理，而且相较于基础预测、单任务组合以及单专家协调方法，可能带来显著的性能提升。