Forecasting hierarchical and grouped time series through trace minimization

Large collections of time series often have aggregation constraints due to product or geographical hierarchies. The forecasts for the disaggregated series are usually required to add up exactly to the forecasts of the aggregated series, a constraint known as “aggregate consistency”. 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 aggregate inconsistency). We show that this is impossible to estimate in practice due to identifiability conditions.