Game-Theoretically Optimal Reconciliation of Contemporaneous Hierarchical Time Series Forecasts

  • Tim van ErvenEmail author
  • Jairo Cugliari
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 217)


In hierarchical time series (HTS) forecasting, the hierarchical relation between multiple time series is exploited to make better forecasts. This hierarchical relation implies one or more aggregate consistency constraints that the series are known to satisfy. Many existing approaches, like for example bottom-up or top-down forecasting, therefore attempt to achieve this goal in a way that guarantees that the forecasts will also be aggregate consistent. We propose to split the problem of HTS into two independent steps: first one comes up with the best possible forecasts for the time series without worrying about aggregate consistency; and then a reconciliation procedure is used to make the forecasts aggregate consistent. We introduce a Game-Theoretically OPtimal (GTOP) reconciliation method, which is guaranteed to only improve any given set of forecasts. This opens up new possibilities for constructing the forecasts. For example, it is not necessary to assume that bottom-level forecasts are unbiased, and aggregate forecasts may be constructed by regressing both on bottom-level forecasts and on other covariates that may only be available at the aggregate level. We illustrate the benefits of our approach both on simulated data and on real electricity consumption data.


Prediction Interval Bregman Divergence Regional Forecast Initial Forecast Bregman Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to thank Mesrob Ohannessian for useful discussions, which led to the closed-form solution for the GTOP predictions in Example 1. We also thank two anonymous referees for useful suggestions to improve the presentation. This work was supported in part by NWO Rubicon grant 680-50-1112.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Université Paris-SudOrsay CedexFrance
  2. 2.INRIA Saclay – Ile de France, Select teamUniversité Paris-SudOrsay CedexFrance
  3. 3.Laboratoire ERICUniversité Lumière Lyon2Bron CedexFrance

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