Hierarchical Forecasting and Reconciliation
Why independently forecasting every level of a hierarchy produces numbers that do not add up, what reconciliation does about it, and why the optimal method also improves accuracy.
Sales forecast by SKU, by category, by store, by region and nationally. Each level is forecast by whatever model works best for it, and then the SKU forecasts do not sum to the category forecast, the stores do not sum to the region, and the business has several inconsistent numbers describing the same future. Reconciliation is the machinery that fixes this, and the interesting result is that fixing it also makes the forecasts more accurate.
The structure
A hierarchy is described by a summing matrix \(S\) mapping bottom-level series to every aggregate. Forecasts are coherent if they satisfy that structure, meaning the aggregates equal the sums of their components.
The traditional approaches enforce coherence by discarding information.
Bottom-up forecasts the lowest level and sums. It uses all the detail and forecasts the noisiest series, and aggregation of noisy forecasts can be worse than forecasting the aggregate directly.
Top-down forecasts the total and disaggregates by historical proportions. The top series is smoothest and easiest to forecast, and the proportions are assumed stable, which they are not when the mix is changing, which is usually the interesting case.
Middle-out picks a level and does both, which combines the objections.
Optimal reconciliation
The modern approach forecasts every level independently, then projects the whole vector of base forecasts onto the coherent subspace, choosing the projection that minimises expected error (Wickramasuriya, Athanasopoulos and Hyndman, 2019, JASA).
with \(W\) the covariance of the base forecast errors. The projection uses information from every level, so a strong signal at the aggregate level improves the bottom-level forecasts and vice versa.
The important property is that this is not only a consistency fix. Because the projection is minimum-variance within the coherent subspace, reconciled forecasts are typically more accurate than the base forecasts at every level, including the level that was forecast best originally. Enforcing a constraint the truth satisfies adds information rather than costing accuracy.
When it breaks
Estimating the error covariance is the hard part. \(W\) is a full covariance over all series, which for thousands of series cannot be estimated reliably. Practical implementations use structural approximations, diagonal, shrinkage toward diagonal, or scaled by series variance, and the choice matters enough to be reported.
Grouped hierarchies are not trees. Products crossed with regions crossed with channels form a lattice with multiple aggregation paths, and the machinery generalises while the covariance estimation becomes harder still.
Coherence in the point forecast does not give coherent intervals. Prediction intervals require the joint distribution, and reconciling means and reporting independently computed intervals produces uncertainty statements that do not add up even though the means do.
Business hierarchies change. Reorganisations, new categories and store openings alter \(S\), and a reconciliation system that assumes a fixed structure needs an explicit process for evolving it.
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