Hierarchical Time Series Forecasting with Robust Reconciliation
Shuhei Aikawa, Aru Suzuki, Kei Yoshitake, Kanata Teshigawara, Akira Iwabuchi, Ken Kobayashi, Kazuhide Nakata
TL;DR
This work addresses hierarchical time series forecasting where coherence requires $\mathbf{y}^{(t)}=\mathbf{S}\mathbf{b}^{(t)}$. It introduces a robust reconciliation framework that accounts for uncertainty in the forecast-error covariance by solving a min-max problem $\min_{\mathbf{P}} \max_{\mathbf{W}\in\mathcal{W}} \sum_{t=1}^{T} (\mathbf{y}^{(t)} - \mathbf{S}\mathbf{P}\hat{\mathbf{y}}^{(t)})^T \mathbf{W} (\mathbf{y}^{(t)} - \mathbf{S}\mathbf{P}\hat{\mathbf{y}}^{(t)})$, which is reformulated as a semidefinite program. Empirical results on four real datasets show that the robust approach often yields superior forecast accuracy compared to BU, TD, GLS, and MinT, especially when the estimated covariance deviates from the truth, though benefits can diminish when the gap is small and computational costs rise. The method provides a principled way to incorporate covariance uncertainty into reconciliation, with practical guidance on constructing the uncertainty set via bootstrap and validation-based tuning of the width parameter $\alpha$. Overall, robust reconciliation offers meaningful improvements for hierarchical forecasting in settings with substantial covariance estimation risk, while highlighting trade-offs in scalability and dataset-dependent effectiveness.
Abstract
This paper focuses on forecasting hierarchical time-series data, where each higher-level observation equals the sum of its corresponding lower-level time series. In such contexts, the forecast values should be coherent, meaning that the forecast value of each parent series exactly matches the sum of the forecast values of its child series. Existing hierarchical forecasting methods typically generate base forecasts independently for each series and then apply a reconciliation procedure to adjust them so that the resulting forecast values are coherent across the hierarchy. These methods generally derive an optimal reconciliation, using a covariance matrix of the forecast error. In practice, however, the true covariance matrix is unknown and has to be estimated from finite samples in advance. This gap between the true and estimated covariance matrix may degrade forecast performance. To address this issue, we propose a robust optimization framework for hierarchical reconciliation that accounts for uncertainty in the estimated covariance matrix. We first introduce an uncertainty set for the estimated covariance matrix and formulate a reconciliation problem that minimizes the worst-case expected squared error over this uncertainty set. We show that our problem can be cast as a semidefinite optimization problem. Numerical experiments demonstrate that the proposed robust reconciliation method achieved better forecast performance than existing hierarchical forecasting methods, which indicates the effectiveness of integrating uncertainty into the reconciliation process.