Forecast reconciliation: a geometric view with new insights on bias correction

Anastasios Panagiotelis, George Athanasopoulos, Puwasala Gamakumara, Rob J. Hyndman

Research output: Contribution to journalArticleResearchpeer-review

46 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)343-359
Number of pages17
JournalInternational Journal of Forecasting
Issue number1
Publication statusPublished - Jan 2021


  • Elliptical distributions
  • Forecast reconciliation
  • High-dimensional time series
  • Projections
  • Scoring rules

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