Multivariate reconciliation for hierarchical time series

2026-05-18

AI Summary

This study addresses the limitations of existing hierarchical time series forecasting methods, which are predominantly univariate and struggle to simultaneously satisfy aggregation constraints and exploit inter-variable correlations. To overcome this, we propose a multivariate joint reconciliation framework that explicitly incorporates the correlation structure among variables into the reconciliation process—marking the first such approach to move beyond traditional univariate, independent reconciliation. Built upon a multivariate regression framework, our method integrates base forecasts with covariance information and achieves coherent predictions across both variables and hierarchy levels by minimizing a multivariate loss function. Empirical evaluations on both simulated data and real-world Brazilian employment statistics demonstrate that the proposed approach significantly outperforms state-of-the-art methods, yielding markedly improved forecast accuracy.

Abstract

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.

Problem

• hierarchical time series
• forecast reconciliation
• multivariate time series
• coherence
• correlation

Innovation

• multivariate reconciliation
• hierarchical time series
• forecast coherence
• cross-variable correlation
• aggregation constraints