Hierarchical Forecasting

Time series are organized into a strict tree or matrix hierarchy. A retail example:

• Level 0 (Top): Total national demand
• Level 1: Regional demand (East, West)
• Level 2: Product category (Electronics, Apparel)
• Level 3 (Bottom): Individual SKU

Each level imposes a linear aggregation constraint. The sum of bottom-level forecasts must equal the forecast of their parent node, ensuring a single, unified view of demand across the enterprise.

The process begins by generating base forecasts independently at each node using any time series model (ARIMA, ETS, DeepAR). These base forecasts are statistically optimal at their specific level but are almost always mathematically incoherent—the sum of regional forecasts will not equal the national forecast.

A reconciliation step then adjusts these base forecasts to produce coherent forecasts that satisfy all aggregation constraints while minimizing information loss.

Several mathematical approaches enforce coherence:

• Bottom-Up: Forecast only at the most granular level and sum upwards. Ignores aggregate-level signal.
• Top-Down: Forecast at the aggregate level and disaggregate using historical proportions. Loses granular detail.
• Middle-Out: Forecast at an intermediate level, sum upwards, and disaggregate downwards.
• Optimal Reconciliation (MinT): Uses a generalized least squares estimator to combine forecasts from all levels, weighting them by the covariance of their errors. This is the state-of-the-art approach, minimizing total forecast variance.

The hierarchical structure is formally encoded in a summing matrix (S). This sparse matrix defines how bottom-level series aggregate into higher-level nodes.

For a hierarchy with n total nodes and m bottom-level nodes, S is an n × m matrix where each row specifies which bottom-level series contribute to that node. This matrix is the fundamental algebraic object used in all optimal reconciliation formulas, including the MinT estimator.

A critical challenge in hierarchical forecasting is that errors compound across levels. An over-forecast at the SKU level propagates upward, inflating category and regional forecasts. Conversely, a top-down error cascades downward, distorting granular allocation.

Optimal reconciliation explicitly models the covariance matrix of base forecast errors to minimize this propagation. By understanding which nodes have correlated errors, the reconciliation algorithm can intelligently distribute adjustments rather than applying naive proportional scaling.

Hierarchical forecasting enables information pooling across related series. A slow-moving SKU with sparse history can borrow statistical strength from its category-level trend or from sibling SKUs with similar seasonality.

Modern approaches like hierarchical global models (e.g., Hierarchical DeepAR) train a single neural network across all bottom-level series simultaneously, learning shared patterns while producing coherent probabilistic forecasts. This is especially powerful for handling intermittent demand and cold-start items in long-tail retail catalogs.

A coherence strategy that generates a forecast at the highest aggregate level first, then disaggregates it to lower levels using historical proportions.

• Approach: Compute total company forecast, then allocate to regions based on historical share
• Key advantage: Aggregate forecast is often more accurate and stable
• Key drawback: Ignores granular signals; a regional spike is diluted by the global average
• Best for: Stable hierarchies where the mix of lower-level contributions is consistent over time

A coherence strategy that generates forecasts at the most granular level first, then sums them up to derive higher-level predictions.

• Approach: Forecast every SKU-store combination independently, then sum to category and national levels
• Key advantage: Captures fine-grained patterns, promotions, and local demand signals
• Key drawback: Granular forecasts are noisier; errors accumulate when summed
• Best for: Hierarchies where bottom-level dynamics are heterogeneous and rich with data

A hybrid approach that generates forecasts at an intermediate level of the hierarchy, then reconciles both upward and downward.

• Approach: Forecast at the product category level, sum up to division, and disaggregate down to SKU
• Key advantage: Balances the stability of aggregation with the granularity of lower levels
• Key drawback: Requires careful selection of the middle level; performance depends on this choice
• Best for: Deep hierarchies where neither the top nor the bottom level provides optimal signal-to-noise ratio

A statistical reconciliation method that produces coherent forecasts by optimally combining all base forecasts using the hierarchy's covariance structure.

• Mechanism: Minimizes the trace of the forecast error covariance matrix, hence MinT (Minimum Trace)
• Inputs: Base forecasts at every node plus the historical forecast error covariance matrix
• Key advantage: Outperforms simple top-down or bottom-up by weighting each node's forecast by its reliability
• Variants: WLS (weighted least squares) and OLS (ordinary least squares) reconciliation are special cases

The mathematical requirement that forecasts at different levels of a hierarchy must sum correctly to maintain internal consistency.

• Example: The sum of regional sales forecasts must equal the national forecast; the sum of SKU forecasts within a category must equal the category forecast
• Enforcement: Achieved via a summing matrix that maps bottom-level series to all aggregate series
• Business impact: Incoherent forecasts cause conflicting inventory orders, misaligned financial planning, and eroded trust in the forecasting system
• Formal notation: For a hierarchy with n bottom-level series and m total series, a summing matrix S of size m × n enforces y_total = S * y_bottom

The structural grouping of time series across non-temporal dimensions such as geography, product taxonomy, or customer segments.

• Common hierarchies: Store → Region → Country; SKU → Subcategory → Category → Division
• Challenge: Each aggregation level has different signal-to-noise characteristics and data sparsity
• Advanced pattern: Cross-sectional hierarchies can be combined with temporal hierarchies (daily → weekly → monthly) to form cross-temporal hierarchies
• Relevance: The choice of aggregation structure directly impacts forecast accuracy and the optimal reconciliation strategy