Forecasting hierarchical and grouped time series through trace minimization

Shanika L Wickramasuriya, George Athanasopoulos, Rob J Hyndman
(2015)

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Large collections of time series often have aggregation constraints due to product or geographical groupings. The forecasts for the most disaggregated series are usually required to add-up exactly to the forecasts of the aggregated series, a constraint we refer to as ``coherent forecasts”. The combination forecasts proposed by Hyndman et al. (2011) are based on a generalized least squares (GLS) estimator and require an estimate of the covariance matrix of the reconciliation errors (i.e., the errors that arise due to incoherent forecasts). We show that this is impossible to estimate in practice due to identifiability conditions.

We propose a new reconciliation forecasting approach that incorporates the information from a full covariance matrix of forecast errors in obtaining a set of coherent forecasts. Our approach minimizes the mean squared error of the coherent forecasts across the entire collection of time series under the assumption of unbiasedness. The minimization problem has a closed form solution. We make this solution scalable by providing a computationally less demanding alternative representation.

We evaluate the performance of the proposed method compared to alternative methods using a series of simulation designs which take into account various features of the collected time series. This is followed by an empirical application using Australian domestic tourism data. The results indicate that the proposed method works well with artificial and real data.