Top-Down Reconciliation

The inverse approach to top-down reconciliation. Forecasts are generated independently at the most granular level (e.g., SKU-by-store) using high-frequency signals. Aggregate forecasts are then derived by summing up the granular predictions.

• Key Advantage: Captures fine-grained patterns and local demand shifts that are lost in aggregation.
• Key Challenge: Noisy data at the lowest level can propagate errors upward, and it requires computationally expensive models at scale.
• Contrast: Unlike top-down, which assumes historical proportions are stable, bottom-up is purely additive.

A hybrid strategy that generates forecasts at an intermediate level of the hierarchy (e.g., product category by region). The forecast is then both disaggregated downward to granular levels and aggregated upward to corporate totals.

• Use Case: Balances the stability of aggregate forecasts with the granularity of lower-level signals.
• Benefit: Mitigates the noise amplification of bottom-up and the rigidity of top-down by anchoring on a mid-tier where data is relatively clean and voluminous.

A modern approach that treats hierarchical coherence as a constrained optimization problem. Instead of a pure top-down or bottom-up approach, independent forecasts are generated at every level and then statistically adjusted to satisfy aggregation constraints.

• Mechanism: Uses the Generalized Least Squares (GLS) estimator to minimize forecast error variance across the entire hierarchy.
• Key Insight: The optimal combination matrix is derived from the covariance of base forecast errors, giving more weight to levels with historically higher accuracy.
• Result: Produces the most accurate coherent forecasts possible, outperforming simple proportional disaggregation.

The foundational data structure that necessitates reconciliation. A hierarchical time series organizes data into a tree-like aggregation structure where nodes at higher levels are the sum of their children.

• Example Structure: Total Sales > Region > Store > Department > SKU.
• Coherence Requirement: The forecast for 'Total Sales' must equal the sum of all regional forecasts, which must equal the sum of all store forecasts.
• Business Reality: Different stakeholders (CFO, Regional Manager, Store Planner) require forecasts at different aggregation levels, all of which must be mathematically consistent.

The historical ratios used in top-down reconciliation to disaggregate an aggregate forecast. These proportions define how the top-level prediction is allocated to lower-level nodes.

• Average Historical Proportions: Calculate each child node's average share of the parent over a lookback window.
• Proportions of Historical Averages: Compute the average value for each child, then calculate their relative share.
• Rolling Proportions: Use only the most recent periods to capture shifting mix dynamics.
• Risk: If a product's mix is rapidly changing (e.g., a declining SKU), static historical proportions will systematically misallocate the forecast.

The systematic tendency of a model to consistently over-predict or under-predict actual demand. In hierarchical contexts, bias at one level can cascade through the reconciliation process.

• Measurement: Calculated as the Mean Forecast Error (MFE) over a historical holdout period.
• Top-Down Risk: If the aggregate forecast has a positive bias, that error is mechanically distributed to every child node via the disaggregation proportions.
• Mitigation: Track bias at every hierarchy level independently. A top-down approach with an unbiased aggregate forecast can still produce biased lower-level forecasts if proportions are stale.