Abstract
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 that nests 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 Minimum Trace (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 that the initial base forecasts are also unbiased. Approaches for dealing with biased base forecasts are proposed. An extensive empirical study of 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.
Highlights
The past decade has seen rapid development in methodologies for forecasting time series that follow a hierarchical aggregation structure
All reconciliation methods that we are aware of consider a linear mapping for ψ, which involves pre-multiplying base forecasts by an n × n matrix that has s as its image
We have provided evidence that bias correction before reconciliation improves forecast accuracy compared to approaches that do not bias correct and/or do not use reconciliation
Summary
The past decade has seen rapid development in methodologies for forecasting time series that follow a hierarchical aggregation structure. Multivariate time series following an aggregation structure arise in many sectors such as retail, energy, insurance, health and welfare and economics (see for example Karmy & Maldonado 2019, Ben Taieb et al 2017, Nystrup et al 2019, Almeida et al 2016, Jeon et al 2019, Mahkya et al 2017, Li & Tang 2019, Shang & Hyndman 2017, Athanasopoulos et al 2019) Forecasts of these series should adhere to aggregation constraints to ensure aligned decision making. We provide a new proof that reconciled forecasts dominate unreconciled forecasts which makes explicit the link between a reconciliation method and a loss function We believe that this link is lacking in previous work that attempts to establish similar results, in particular Van Erven & Cugliari (2015) and Wickramasuriya et al (2019).
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