Optimal Forecast Reconciliation for Quantiles

Forecasting multivariate data that adhere to known linear constraints, so-called hierarchical data, benefits from a post-processing step known as reconciliation. While traditional reconciliation methods focus on mean forecasts, in a decision-making setting the optimal action is a functional of a belief distribution, for example a quantile. Building on a general framework, this paper develops a new methodology for forecast reconciliation where the objective is to obtain accurate forecasts for a given quantile level. This is achieved by minimising expected pinball loss, a challenging problem which we propose to overcome in two ways. First, expectations are approximated by drawing samples from the base forecasts, making the approach applicable for any distributional form. Second, the pinball loss is approximated with a smooth function, enabling optimisation with first-order methods. Theoretical results are developed proving that the minimiser of the objective function employing these two approximations converges, in the limit, to the minimiser of expected pinball loss. Applications to both simulated and real-world data demonstrate that the proposed methodology delivers statistically significant improvements in forecast accuracy over the widely used MinT benchmark.