When Does Coarse-to-Fine Forecasting Help? An Exact Error Decomposition and a Checkable Criterion

Coarse-grained future structure is often easier to predict than fine-grained dynamics, motivating a broad class of coarse-to-fine and multiscale forecasting methods. Yet predictability does not necessarily imply predictive value: under the same forecasting protocol, conditioning on a coarse-grained future can substantially improve a matched backbone, have little effect, or even degrade the final prediction. This raises a more fundamental question than whether a particular hierarchy works: when can predictable coarse-grained information actually be converted into better fine-grained forecasts? We study this question by separating three quantities that are often conflated by standard forecasting metrics: the predictability of future coarse-grained structure, the value of the resulting coarse anchor relative to a matched unconditional forecaster, and the additional error that can be recovered through fine-grained refinement. This decomposition yields a simple necessary condition that can be evaluated before training the refinement model. We partition the final error into a component determined by coarse prediction and a component that refinement can potentially recover. If the former already exceeds the error of the target direct forecaster, then no refinement can make the hierarchical predictor outperform that baseline. Across 28 dataset–horizon settings from seven multivariate forecasting benchmarks and four prediction horizons, this condition identifies eight such cases, and the hierarchical method loses in all eight. The remaining cases reveal qualitatively different success and failure regimes. In some settings, the coarse anchor itself accounts for nearly all of the improvement; in others, substantial refinement is required. More importantly, a harmful anchor does not necessarily imply failure, nor does improving an anchor guarantee a net gain. On Electricity, refinement fully compensates for an initially harmful anchor and ultimately surpasses the baseline. On long-horizon Traffic, refinement improves the anchor but remains insufficient to outperform the direct forecaster. Targeted interventions further separate information availability from model realization. On ETTm1, coarse future structure remains predictable at long horizons even when the learned hierarchy fails to exploit it. Improving only coarse extraction or only refinement provides partial recovery, whereas improving both produces a strongly coupled rescue effect. On Traffic, increasing coarse resolution reduces unresolved within-block structure and anchor error, turning H = 336 from a net loss into a net gain and substantially narrowing the deficit at H = 720. These results show that coarse-grained predictability and realized predictive value are distinct properties. The success of coarse-to-fine forecasting depends not only on whether coarse future structure exists and can be predicted, but also on whether it can be converted into a useful anchor relative to the target baseline, and whether refinement can recover the information that remains unresolved after anchoring.