Understanding forecast reconciliation

A series of recent papers introduce the concept of Forecast Reconciliation, a process by which independently generated forecasts of a collection of linearly related time series are reconciled via the introduction of accounting aggregations that naturally apply to the data. Aside from its clear presentational and operational virtues, the reconciliation approach generally improves the accuracy of the combined forecasts. In this paper, we examine the mechanisms by which this improvement is generated by re-formulating the reconciliation problem as a combination of direct forecasts of each time series with additional indirect forecasts derived from the linear constraints. Our work establishes a direct link between the nascent Forecast Reconciliation literature and the extensive work on Forecast Combination. In the original hierarchical setting, our approach clarifies for the first time how unbiased forecasts for the entire collection can be generated from base forecasts made at any level of the hierarchy, and we illustrate more generally how simple robust combined forecasts can be generated in any multivariate setting subject to linear constraints. In an empirical example, we show that simple combinations of such forecasts generate significant improvements in forecast accuracy where it matters most: where noise levels are highest and the forecasting task is at its most challenging.