Properties of the reconciled distributions for Gaussian and count forecasts

Reconciliation enforces coherence between hierarchical forecasts, in order to satisfy a set of linear constraints. While most works focus on the reconciliation of point forecasts, we consider probabilistic reconciliation and we analyze the properties of distributions reconciled via conditioning. We provide a formal analysis of the variance of the reconciled distribution, treating the case of Gaussian and count forecasts separately. We also study the reconciled upper mean in the case of one-level hierarchies, again treating Gaussian and count forecasts separately. We then show experiments on the reconciliation of intermittent time series related to the count of extreme market events. The experiments confirm our theoretical results and show that reconciliation largely improves the performance of probabilistic forecasting.