Probabilisitic forecasts in hierarchical time series

Forecast reconciliation involves adjusting forecasts to ensure coherence with aggregation constraints. We extend this concept from point forecasts to probabilistic forecasts by redefining forecast reconciliation in terms of linear functions in general, and projections more specifically. New theorems establish that the true predictive distribution can be recovered in the elliptical case by linear reconciliation, and general conditions are derived for when this is a projection. A geometric interpretation is also used to prove two new theoretical results for point forecasting; that reconciliation via projection both preserves unbiasedness and dominates unreconciled forecasts in a mean squared error sense. Strategies for forecast evaluation based on scoring rules are discussed, and it is shown that the popular log score is an improper scoring rule with respect to the class of unreconciled forecasts when the true predictive distribution coheres with aggregation constraints. Finally, evidence from a simulation study shows that reconciliation based on an oblique projection, derived from the MinT method of Wickramasuriya, Athanasopoulos and Hyndman (2018) for point forecasting, outperforms both reconciled and unreconciled alternatives.