Optimal combination forecasts for hierarchical time series

Abstract

In many applications, there are multiple time series that are hierarchically organized and can be aggregated at several different levels in groups based on products, geography or some other features. We call these "hierarchical time series". They are commonly forecast using either a "bottom-up" or a "top-down" method. In this paper we propose a new approach to hierarchical forecasting which provides optimal forecasts that are better than forecasts produced by either a top-down or a bottom-up approach. Our method is based on independently forecasting all series at all levels of the hierarchy and then using a regression model to optimally combine and reconcile these forecasts. The resulting revised forecasts add up appropriately across the hierarchy, are unbiased and have minimum variance amongst all combination forecasts under some simple assumptions. We show in a simulation study that our method performs well compared to the top-down approach and the bottom-up method. We demonstrate method by forecasting Australian tourism demand where the data are disaggregated by purpose of visit and geographical region.

Forecasting with temporal hierarchies

“…We refer to (5) as the temporal reconciliation regression model. It is analogous to the cross-sectional hierarchical reconciliation regression model proposed by Hyndman et al (2011) and also applied in Athanasopoulos et al (2009) for reconciling forecasts of structures of tourism demand. A similar idea has been used for imposing aggregation constraints on time series produced by national statistical agencies (Quenneville and Fortier, 2012).…”

“…is referred to as the "summing" matrix (drawing from the work of Hyndman et al 2011) and y [1] i is the vector of observations of the time series observed at the highest available frequency. It is not always possible to represent the aggregated series in a single tree such as Figure 1.…”

“…Hyndman et al (2011) and Athanasopoulos et al (2009) avoid estimating Σ by using ordinary least squares (OLS), replacing Σ by σ 2 I in (6). This is only optimal under some special conditions, such as when the reconciliation errors are equivariant and uncorrelated, or when the base forecast errors satisfy the same aggregation constraints as the original data (Hyndman et al, 2011), neither of which is likely to be true. For the temporal hierarchy case the errors are not equivariant by definition.…”

“…Hyndman et al (2011) consider the analogous cross-sectional problem and derive the weights from the structure of the hierarchy. In this paper, by constructing temporal hierarchical structures we are able to apply a similar approach to temporal hierarchies.…”

“…To counter this significant limitation Hyndman et al (2011) introduced a regression based approach that instead optimally combines forecasts from all the series of the hierarchy and results in reconciled forecasts. This method is used by Athanasopoulos et al (2009) with good empirical results and further developed by Wickramasuriya et al (2015).…”