Forecast Reconciliation: A geometric View with New Insights on Bias Correction

A geometric interpretation is developed for so-called reconciliation methodologies used to forecast time series that adhere to known linear constraints. In particular, a general framework is established nesting many existing popular reconciliation methods within the class of projections. This interpretation facilitates the derivation of novel theoretical results. First, reconciliation via projection is guaranteed to improve forecast accuracy with respect to a class of loss functions based on a generalised distance metric. Second, the MinT method minimises expected loss for this same class of loss functions. Third, the geometric interpretation provides a new proof that forecast reconciliation using projections results in unbiased forecasts provided the initial base forecasts are also unbiased. Approaches for dealing with biased base forecasts are proposed. An extensive empirical study on Australian tourism flows demonstrates the theoretical results of the paper and shows that bias correction prior to reconciliation outperforms alternatives that only bias-correct or only reconcile forecasts.