Hierarchical Reconciliation

Hierarchical reconciliation adjusts forecasts so they satisfy aggregation constraints. It is needed when forecasts are produced independently at several levels, such as item, category, region, and total.

Let ${\mathbf{b}}_t$ be bottom-level series at time $t$ and ${\mathbf{y}}_t$ be all series in the hierarchy. A summing matrix $\mathbf{S}$ maps bottom-level values to all levels:

$${\mathbf{y}}_t=\mathbf{S}{\mathbf{b}}_t$$

A reconciled forecast should also satisfy:

$${\tilde{\mathbf{y}}}_{t+h}=\mathbf{S}{\tilde{\mathbf{b}}}_{t+h}$$

where tildes denote reconciled forecasts.

Basic approaches

Bottom-up reconciliation uses bottom-level forecasts and sums them upward. It is simple and coherent, but noisy when bottom-level data is sparse.

Top-down reconciliation forecasts the top level and allocates downward using proportions. It is stable at the total level but may lose local signal.

Middle-out reconciliation starts at an intermediate level, sums upward, and allocates downward. It is useful when that level has the most reliable forecasts.

Optimal-combination methods start from base forecasts at all levels and adjust them jointly. MinT-style methods use an estimate of forecast error covariance to reduce reconciliation error, subject to coherence constraints.

Coherence example

Suppose base forecasts are:

level	forecast
total	110
A	40
B	50

The bottom-level forecasts imply a total of 90, not 110. A bottom-up reconciled total would be 90. A top-down reconciliation could preserve the total 110 and allocate it across A and B using historical shares. An optimal-combination method would adjust all three forecasts based on estimated reliability.

Evaluation

Evaluate base and reconciled forecasts at every planning level. Reconciliation can improve aggregate coherence while worsening bottom-level accuracy. The acceptable tradeoff depends on the decision: inventory placement may prioritize bottom-level accuracy, while financial planning may prioritize aggregate consistency.

Practical guidance

• Reconcile only across aggregation relationships that are operationally meaningful.
• Estimate reconciliation parameters using validation data, not final evaluation data.
• Compare bottom-up, top-down, and optimal-combination baselines.
• Report both coherence and accuracy.
• Treat missing bottom-level series and cold starts explicitly before reconciliation.

Common failure modes

• Reconciliation over an incorrect hierarchy.
• Using noisy historical proportions for top-down allocation.
• Ignoring covariance estimation error in small backtests.
• Reporting only total-level gains.
• Reconciling forecasts produced from inconsistent training cutoffs.

Connections

Reconciliation adjusts base forecasts from hierarchical forecasting so they satisfy aggregation constraints. The same coherence logic appears in temporal reconciliation, and the result still needs backtesting with appropriate forecast error metrics.

References

• Hyndman & Athanasopoulos, FPP3: Forecast reconciliation
• Nixtla HierarchicalForecast documentation

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