Some time series can be hierarchically organized into levels based on certain characteristics, such as geography or other attributes of interest. These series are referred to as hierarchical time series. Typically, forecasts are generated at all levels to ensure coherence, meaning that the forecasts should satisfy the same aggregation constraints as the observed data. Various approaches have been proposed to guarantee this coherence by using a set of base forecasts. The process through which these forecasts are adjusted to become coherent is known as forecast reconciliation. Similar to the univariate case, multivariate time series can also be structured hierarchically. However, all existing approaches are limited to a single variable. As a result, ensuring coherent forecasts requires reconciling each variable separately. However, this process does not account for correlations among multiple variables. To address this limitation, this paper proposes a multivariate reconciliation methodology that ensures coherent forecasts and incorporates relationships among variables. The proposed methodology was tested through numerical simulations, considering distinct scenarios within the series hierarchy and across multiple variables. Additionally, some base forecasting models were evaluated. The methodology was also applied to real employment data of admissions and dismissals in Brazil. The results demonstrated that multivariate reconciliation yielded more accurate outcomes than the other methods considered, both in simulated data and in practical applications.

翻译：某些时间序列可依据地理或其他感兴趣属性等特征，按层级结构组织成不同层次。这类序列被称为层级时间序列。通常，为确保一致性，需要在所有层级生成预测，即这些预测应满足与观测数据相同的聚合约束。已有多种方法通过使用一组基础预测来保证这种一致性，而调整这些预测以实现一致的过程被称为预测协调。与单变量情形类似，多元时间序列也可构建层级结构。然而，现有方法均局限于单一变量。因此，确保一致预测需要分别协调每个变量，但该过程未考虑多个变量间的相关性。为解决这一局限，本文提出一种多元协调方法，既能保证预测的一致性，又能整合变量间的关联关系。通过数值模拟对所提方法进行检验，考虑了序列层级内部及跨多变量的不同场景，并评估了若干基础预测模型。该方法还应用于巴西就业数据（包含入职与离职）的实际案例分析。结果表明，无论是在模拟数据还是实际应用中，多元协调方法均比所考虑的其他方法取得更优的预测效果。