Aggregated curves are common structures in economics and finance, and the most prominent examples are supply and demand curves. In this study, we exploit the fact that all aggregated curves have an intrinsic hierarchical structure, and thus hierarchical reconciliation methods can be used to improve the forecast accuracy. We provide an in-depth theory on how aggregated curves can be constructed or deconstructed, and conclude that these methods are equivalent under weak assumptions. We consider multiple reconciliation methods for aggregated curves, including previously established bottom-up, top-down, and linear optimal reconciliation approaches. We also present a new benchmark reconciliation method called 'aggregated-down' with similar complexity to bottom-up and top-down approaches, but it tends to provide better accuracy in this setup. We conducted an empirical forecasting study on the German day-ahead power auction market by predicting the demand and supply curves, where their equilibrium determines the electricity price for the next day. Our results demonstrate that hierarchical reconciliation methods can be used to improve the forecasting accuracy of aggregated curves.

翻译：聚合曲线是经济学和金融学中常见的结构，最典型的例子是供给曲线和需求曲线。在本研究中，我们利用所有聚合曲线均具有内在层次结构这一事实，从而可以采用层次化协调方法来提高预测精度。我们深入阐述了聚合曲线的构建与解构理论，并得出结论：在弱假设条件下这些方法是等价的。我们考虑了多种适用于聚合曲线的协调方法，包括已有的自下而上、自上而下及线性最优协调方法。此外，我们提出了一种名为“聚合向下”（aggregated-down）的新型基准协调方法，该方法与自下而上及自上而下方法的复杂度相当，但在本设定中往往能提供更高的预测精度。我们通过对德国日前电力拍卖市场的需求曲线和供给曲线进行实证预测研究——其均衡点决定了次日的电力价格——结果表明，层次化协调方法可用于提升聚合曲线的预测精度。