A Forecast Combination Framework for Hierarchical and Grouped Time Series Reconciliation

Abstract

Forecast combining and forecast reconciliation for hierarchical and grouped time series have largely developed as separate research areas. This paper connects the two by developing a forecast combination framework for forecast reconciliation. For each bottom-level series, we construct a maximal linearly independent set of structured candidate forecasts from aggregation constraints, and show that combining these candidates and aggregating the resulting bottom-level forecasts is equivalent to standard unbiased linear reconciliation. Within this representation, we prove that mean-squared-error optimal combination weights exactly recover the widely used Minimum Trace (MinT) reconciliation. We further show that the optimal weight problem is separable across different bottom-level series, each yielding a Bates--Granger optimal forecast combination. This reveals MinT as a collection of optimal combinations over hierarchy-induced candidate forecasts. For finite-sample estimation, we propose a modular penalized framework that nests existing MinT variants and supports rich extensions including covariance shrinkage, weight penalization, and scalable series-wise separate estimation. Empirical results show that the framework is practically implementable, competitive with existing methods, and can improve accuracy while preserving coherence. Overall, the forecast combination perspective offers new interpretations of existing reconciliation approaches and provides a flexible basis for designing new methods.