We propose to estimate the weight matrix used for forecast reconciliation as parameters in a general linear model in order to quantify its uncertainty. This implies that forecast reconciliation can be formulated as an orthogonal projection from the space of base-forecast errors into a coherent linear subspace. We use variance decomposition together with the Wishart distribution to derive the central estimator for the forecast-error covariance matrix. In addition, we prove that distance-reducing properties apply to the reconciled forecasts at all levels of the hierarchy as well as to the forecast-error covariance. A covariance matrix for the reconciliation weight matrix is derived, which leads to improved estimates of the forecast-error covariance matrix. We show how shrinkage can be introduced in the formulated model by imposing specific priors on the weight matrix and the forecast-error covariance matrix. The method is illustrated in a simulation study that shows consistent improvements in the log-score. Finally, standard errors for the weight matrix and the variance-separation formula are illustrated using a case study of forecasting electricity load in Sweden.

翻译：我们提出将用于预测协调的权重矩阵估计为广义线性模型中的参数，以量化其不确定性。这意味着预测协调可被表述为从基预测误差空间到一致线性子空间的正交投影。我们利用方差分解与Wishart分布推导出预测误差协方差矩阵的中心估计量。此外，我们证明了距离缩减特性不仅适用于层次结构各层级协调后的预测值，也适用于预测误差协方差。我们推导了协调权重矩阵的协方差矩阵，从而改进了对预测误差协方差矩阵的估计。通过向权重矩阵和预测误差协方差矩阵施加特定的先验分布，我们展示了如何在所构建的模型中引入收缩方法。仿真研究验证了该方法在对数得分上的一致性改进。最后，通过瑞典电力负荷预测的案例研究，展示了权重矩阵的标准误差及方差分离公式的应用。