Discrete forecast reconciliation

This paper presents a formal framework and proposes algorithms to extend forecast reconciliation to discrete-valued data to extend forecast reconciliation to discrete-valued data, including low counts. A novel method is introduced based on recasting the optimisation of scoring rules as an assignment problem, which is solved using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical time series. Two discrete reconciliation algorithms are also proposed and compared against generalisations of the top-down and bottom-up approaches for count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are forecasting criminal offences in Washington D.C. and product unit sales in the M5 dataset. Compared to benchmarks, the proposed framework shows superior performance in both simulations and empirical studies.