Hierarchical Forecasting

Overview

Hierarchical forecasting is the process of forecasting related time series arranged in levels and ensuring that predictions are consistent across those levels. In logistics, a hierarchy might connect individual SKU forecasts to product families and total company demand. A coherent forecast makes the sum of the lower-level predictions equal to the corresponding higher-level prediction. Forecasting: Principles and Practice distinguishes strict hierarchies from grouped structures with multiple aggregation paths.

These forecasts support decisions at different operational and strategic horizons: a procurement planner needs SKU-level reorder quantities, while a distribution executive requires aggregate network volumes for warehouse footprint and labor planning. Translating coherent high-level demand projections into actionable warehouse execution often connects upstream planning with execution systems that utilize automated replenishment workflows and SKU-based inventory records. While an inventory tracking system does not typically perform hierarchical matrix reconciliation, accurate transactional baselines remain critical operational inputs.

Etymology and Origins

The term combines hierarchical, from Greek hierarkhia (ranked rule or order), with forecasting. English forecast is Middle English, from fore- plus cast (“to plan or reckon beforehand”); Old Norse forspá (“prophecy”) is a parallel formation, not the direct source of the English verb. In statistical demand management, the hierarchy describes formal mathematical aggregation and linear constraints rather than managerial authority. The literature also uses forecast reconciliation for the adjustment of independently produced base forecasts so that they add consistently across categories, geographies, or time buckets. Hierarchical and grouped time series are treated as related structural cases of the same problem.

Aggregation and disaggregation were used in sales and operations planning long before optimal reconciliation algorithms existed. Earlier econometric work, including Orcutt, Watts, and Edwards (1968) on data aggregation and information loss, examined how aggregation changes statistical conclusions, but it was not a forecasting-reconciliation method. Top-down allocation for product-family forecasting was studied empirically by Gross and Sohl (1990). The modern linear-combination approach is usually dated to Hyndman, Ahmed, Athanasopoulos, and Shang’s 2011 paper on optimal combination forecasts for hierarchical time series.

Historical Development

1970s–1980s (heuristic disaggregation in MRP): Corporate planning commonly used bottom-up summation or top-down historical proportions inside material requirements planning (MRP) and related planning systems. These heuristics could preserve either local item detail or a stable total, but not both. The label “enterprise resource planning” (ERP) became current around 1990; later ERP suites inherited similar aggregation rules rather than originating them in the 1970s.
1990: Charles W. Gross and Jeffrey E. Sohl published a comparative study of disaggregation methods for top-down forecasting in the International Journal of Forecasting, contrasting historical proportions with proportions based on forecasts.
2000s: Cheaper computation made it practical to forecast very large retail assortments item by item. Independent bottom-up forecasts could still disagree with forecasts made directly at distribution-center or company level, which increased interest in joint multi-level estimation.
2011: Rob J. Hyndman, Roman A. Ahmed, George Athanasopoulos, and Han Lin Shang published Optimal Combination Forecasts for Hierarchical Time Series. Their ordinary least squares (and weighted least squares) framework combined base forecasts from every level instead of using only a pure top-down or bottom-up path.
2017: George Athanasopoulos, Rob J. Hyndman, Nikolaos Kourentzes, and Fotios Petropoulos formalized forecasting with temporal hierarchies, reconciling series aggregated over different time buckets such as weeks, months, and quarters.
2019: Shanika L. Wickramasuriya, George Athanasopoulos, and Rob J. Hyndman published minimum trace (MinT) reconciliation. The paper, available online in 2018 and issued in the 2019 Journal of the American Statistical Association, uses a generalized least squares estimator based on the forecast-error covariance to produce minimum-variance unbiased coherent point forecasts when the base forecasts are unbiased.
2020s: Libraries such as Nixtla’s open-source HierarchicalForecast implement MinT-style, probabilistic, and machine-learning reconciliation, including cross-sectional and temporal structures. In warehouse operations, those statistical outputs still have to be turned into executable supply chain planning decisions; the planning system does not itself define the reconciliation mathematics.

Core Concepts

Structure and coherence

A hierarchy defines which series aggregate into which higher-level totals. If two SKUs are forecast independently at 60 and 40 units, a coherent category total is 100. An aggregate model that projects 110 is incoherent with those item forecasts until a reconciliation rule adjusts one or both levels.

A summing matrix $\mathbf{S}$ encodes the aggregation: the vector of all series $\mathbf{y}_t$ is linked to the bottom-level series $\mathbf{b}_t$ by $\mathbf{y}_t = \mathbf{S}\mathbf{b}_t$. Given base forecasts $\hat{\mathbf{y}}_t$ that need not be coherent, a linear mapping $\mathbf{P}$ produces reconciled forecasts $\tilde{\mathbf{y}}_t = \mathbf{S}\mathbf{P}\hat{\mathbf{y}}_t$. The matrix formulation in Forecasting: Principles and Practice is the standard reference for this representation and scales, in principle, to large item hierarchies.

Hierarchical, grouped, and temporal views

Strict hierarchies: Nested trees in which each series has one parent (for example, SKU → subcategory → category → company).
Grouped structures: Crossed attributes without a single tree, such as product family × geography, or channel × warehouse. Grouped time series let different teams see the same reconciled totals sliced on different attributes.
Temporal and cross-temporal hierarchies: Reconcile forecasts across time buckets—daily counts used for short-term labor planning, weekly purchasing cycles, and monthly or quarterly budgets—and, in cross-temporal methods, across both time and product or location. Temporal hierarchies are a distinct aggregation path from the product tree.

Aggregation should use a consistent unit of measure. Summing unlike SKUs by raw piece count can misstate workload or cube. Weight, cartons, pallet equivalents, or revenue are often more relevant, provided the chosen unit is defined consistently at every node. Unit-of-measure conversion in the item master is an operational prerequisite, not a substitute for statistical reconciliation.

Forecasting and Reconciliation Methods

Method	Mechanism	Strengths	Operational vulnerabilities
Bottom-up (BU)	Forecast only the lowest level and sum through $\mathbf{S}$.	Uses item-level promotions and local dynamics; coherent by construction.	Intermittent or noisy SKUs can make the aggregate unstable.
Top-down (TD)	Forecast the top level and split it by historical or forecasted proportions.	Uses a less noisy aggregate; simple to explain.	Misses mix shifts and new-item dynamics. Gross and Sohl documented accuracy differences among proportion rules.
Middle-out (MO)	Forecast an intermediate level, aggregate upward, and allocate downward.	Balances noise and detail when that middle level is well chosen.	Ignores information at levels that are not forecast directly; the choice of level is a modeling decision, not a statistical optimum.
Optimal combination / MinT	Reconcile base forecasts from all levels with least squares, using an estimate of the forecast-error covariance $\mathbf{W}_h$ in the MinT case.	Uses information from every level; MinT minimizes the trace of the reconciled error covariance under stated assumptions.	Covariance estimation is harder in short or intermittent series; unconstrained solutions can be negative.

MinT weights levels according to an estimate of forecast-error dependence. Shrinkage and diagonal approximations are used when the full covariance cannot be estimated reliably. For physical quantities, non-negative reconciliation or a documented post-processing rule is required if negative reconciled values are operationally impossible. Clipping after an unconstrained solution does not, by itself, preserve the optimality proof.

Logistics Applications

Coherent forecasts are used to keep purchasing, inventory positioning, and warehouse execution on the same demand numbers:

Multi-echelon positioning: National or network totals can support supplier commitments, while reconciled node-level forecasts support regional fulfillment. Coherence reduces unexplained gaps between those plans; it does not by itself optimize safety stock or prevent hoarding.
Procurement and replenishment: Bottom-level reconciled SKU forecasts can feed reorder logic and automated purchase-quantity calculations. Alignment with a parent-brand or open-to-buy total is a planning control, not an automatic property of a warehouse management system.
Inbound and outbound scheduling: Aggregate volume, weight, or carton forecasts can inform dock, labor, and carrier planning. Floor execution still follows location-level rules such as stock replenishment.
Shared inventory across channels: Grouped structures can reconcile demand for the same items across direct-to-consumer, wholesale, and store channels. Whether that reduces safety stock depends on lead times, service targets, and whether inventory is actually pooled.

Implementation, Evaluation, and Limitations

The following is a practical sequence used by forecasting teams, not a formal industry standard:

Define the aggregation: Map product, location, channel, and time buckets into an explicit summing matrix. Inconsistent item masters break coherence even if the estimator is correct.
Separate observed sales from demand: Stockouts, cancellations, and backorders censor sales. Physical inventory reconciliation corrects book-to-shelf quantity; it does not reconstruct unconstrained historical demand. That reconstruction is a separate forecasting input.
Fit base models by series behavior: Aggregates often show trend and seasonality suitable for exponential smoothing or ARIMA. Intermittent SKU-location series may need Croston-type or hurdle models. No single model class is required by the reconciliation step.
Reconcile, then enforce operational constraints: Apply a documented method such as bottom-up, top-down, or MinT. If non-negativity or integer quantities are required, state how they are imposed and whether coherence is rechecked afterward.
Evaluate every level that will be used: Use rolling-origin backtests. Point-forecast accuracy guidance cautions against MAPE when actuals are zero or near zero and recommends scaled errors such as MASE; RMSSE was prominent in the M5 competition for intermittent retail series. WAPE (also called WMAPE) avoids a zero denominator by weighting absolute errors by volume, but it is not the same metric as MASE. Inventory cost and service should be reported beside statistical error.

Limitations

Coherence is not accuracy. Coherent forecasts add up. A biased top-level forecast can still pull accurate lower-level forecasts off target.
Hierarchies change. New SKUs, discontinuations, and node realignments change $\mathbf{S}$. Long historical reconciliations are not comparable across incompatible matrices unless the hierarchy is explicitly versioned.
Distributional coherence is harder than point coherence. Linear reconciliation of means does not automatically make quantiles or prediction intervals add up. Coherent probabilistic forecasts generally need additional structure, such as joint sampling or copula-style dependence, and should not be described as a routine extension of MinT.