Hierarchical, Multilevel and Forecast Combination Methods

Introduction

Logistics demand rarely exists at one level of aggregation. A firm may forecast total network demand, regional demand, distribution-centre demand, customer-channel demand, product-family demand and individual stock-keeping-unit (SKU) demand simultaneously. It may also require forecasts for daily, weekly, monthly and quarterly horizons. These forecasts are related, not independent. The sum of regional forecasts should ideally equal the national forecast; channel totals should agree with product totals; and short-horizon forecasts should be compatible with longer-horizon plans.

When independently generated forecasts violate these relationships, the organisation faces contradictory decisions. Procurement may plan for 100,000 units while distribution centres receive forecasts totalling 112,000 units. Sales and operations planning may use a monthly forecast that does not equal the sum of weekly forecasts. Hierarchical and multilevel forecasting methods address this problem by representing demand as a structured system and producing forecasts that are coherent across levels, dimensions and horizons.

Forecast combination complements reconciliation. Rather than selecting one supposedly superior model, a planner can combine statistical models, causal models, machine-learning forecasts and expert judgements. Combination is particularly valuable when demand is uncertain, structural change is present, or no single model captures all relevant information.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Distinguish hierarchical, grouped and multilevel forecasting structures.
2. Explain top-down, bottom-up, middle-out and proportional allocation methods.
3. Formulate forecast reconciliation using a summing matrix.
4. Compare ordinary least squares, weighted least squares and minimum-trace reconciliation.
5. Explain temporal and cross-sectional coherence.
6. Combine forecasts using weighted averages, variance-based weights and forecast-error information.
7. Design an evaluation and governance process for coherent forecast systems.
8. Apply these methods to logistics planning and critically assess their limitations.

1. The structure of a logistics forecasting problem

1.1 Hierarchies and grouped structures

A hierarchical time series has a strict tree structure. For example:

• Total company demand
• Region A
• Warehouse A1
• Warehouse A2
• Region B
• Warehouse B1
• Warehouse B2

Each lower-level series belongs to exactly one parent. A grouped time series has more than one valid aggregation dimension, but not necessarily a tree. Demand may be classified simultaneously by product and location:

• total demand;
• product-family totals;
• regional totals;
• product-by-region combinations.

Product totals and regional totals cross each other. Neither dimension is simply above the other. This structure is often called a grouped or cross-classified hierarchy.

A multilevel forecasting problem extends the idea further. Levels may include product, customer, location, channel, time frequency and planning horizon. A demand observation can therefore be indexed as y_p,l,c,t, where p denotes product, l location, c channel and t time. Multilevel statistical models may also allow parameters to vary by group while sharing information across groups.

1.2 Coherence

Forecasts are coherent when they obey all aggregation constraints. If regional demand sums to national demand, coherence requires:

where ŷ is a forecast and R is the number of regions. Coherence is an accounting property, not a guarantee of accuracy. A coherent set may still be systematically biased or highly uncertain.

Coherence may be required across:

• geographic levels;
• product and SKU levels;
• channels and customer segments;
• weekly, monthly and quarterly time scales;
• operational, tactical and strategic planning horizons.

Temporal coherence requires, for example, that a monthly forecast equal the sum of its constituent weekly forecasts:

This equation assumes that the weekly and monthly periods cover exactly the same calendar demand and that no weeks are omitted or double-counted.

2. The summing-matrix representation

A useful general representation separates the bottom-level demand series from all aggregated series. Let:

• b_t be an m × 1 vector of bottom-level actual demand at time t;
• y_t be an n × 1 vector containing all series, including aggregates;
• S be an n × m summing matrix.

Then:

Every element of S indicates whether, and how, a bottom-level series contributes to an aggregate. In a simple two-region example with bottom series A and B:

If base forecasts for all series are collected in ŷ_t, they need not satisfy the constraints implied by S. Reconciliation constructs forecasts ŷ_t^rec that do:

where ĝ_t is a reconciled forecast vector for the bottom-level series or an equivalent set of latent forecasts. The summing matrix must accurately represent the business structure. Incorrect product-location mappings produce mathematically coherent but operationally meaningless forecasts.

3. Forecasting strategies for hierarchies

3.1 Top-down forecasting

Top-down methods begin with a forecast at an upper level and allocate it to lower levels. If ŷ_T,t is the top-level forecast and p_i is the allocation proportion for lower-level series i, then:

The proportions may be based on:

• historical average shares;
• recent shares;
• proportions of historical totals;
• forecasted shares from separate models.

Top-down forecasting is attractive when lower-level series are noisy or intermittent and the aggregate is more predictable. Its weakness is that lower-level shocks can be hidden. A new customer, stockout, promotion or local disruption may change the composition of demand without changing the top-level forecast.

A proportion based on historical totals is:

This is appropriate only when historical composition is reasonably stable and the numerator and denominator cover the same periods.

3.2 Bottom-up forecasting

Bottom-up forecasting models every bottom-level series and aggregates the forecasts upward:

It preserves local information and is often preferred when SKU-location forecasts are reliable and the bottom-level data are sufficiently dense. However, thousands of low-volume series may produce unstable forecasts. Bottom-up systems can also be computationally expensive and may propagate local errors into aggregate plans.

3.3 Middle-out forecasting

Middle-out methods forecast a strategically useful intermediate level, such as region or product family, and use top-down allocation above it and bottom-up aggregation below it. The middle level should balance statistical reliability with managerial relevance. For example, product-family demand may be stable enough to forecast directly, while individual SKUs are too intermittent and total demand is too broad to support operational decisions.

3.4 Proportional and dynamic allocation

Static proportions are simple but may be inappropriate under changing market conditions. Dynamic allocation estimates proportions that vary with time, price, promotions, customer mix or service restrictions. A simple constrained share model is:

The shares may come from a multinomial model, a machine-learning model followed by normalization, or a separate forecasting process. Dynamic shares require careful validation because errors in both the top-level forecast and shares multiply.

4. Forecast reconciliation

4.1 Why reconciliation is needed

A common workflow produces a base forecast for every series independently. This retains level-specific information but creates incoherence. Reconciliation adjusts the forecasts so that all aggregation identities hold. The adjustment should ideally be small, statistically justified and economically interpretable.

4.2 Ordinary least squares reconciliation

One basic method finds the coherent vector closest to the base forecasts under squared error loss. In matrix form, a common ordinary least squares projection is:

Here, Sᵀ is the transpose of S, and the inverse exists when the columns of S are linearly independent. The method treats all base-forecast errors as equally important and uncorrelated. That assumption is often unrealistic because forecast variances differ substantially between aggregate and SKU series.

4.3 Weighted least squares and MinT

Let W be a positive-definite matrix representing the covariance or a suitable estimate of base-forecast errors. A generalized least squares reconciliation is:

The diagonal version of W uses individual error variances but ignores cross-series correlations. A full covariance matrix accounts for correlated errors, such as common weather or macroeconomic shocks. Minimum-trace (MinT) reconciliation selects the reconciliation mapping that minimises the trace of the covariance matrix of reconciled forecast errors, subject to coherence. In practice, shrinkage covariance estimators are often used because a sample covariance matrix can be unstable when the number of series is large relative to the history.

The weighted formulation assumes that the error covariance estimate is meaningful and that the historical relationship between errors remains sufficiently stable. Poor covariance estimation can make a sophisticated reconciliation method worse than a simpler one.

4.4 Structural and temporal reconciliation

A grouped product-location system can be reconciled across product and location dimensions. If time aggregation is also relevant, forecasts can be reconciled across both cross-sectional and temporal structures. The resulting system must satisfy all identities simultaneously. This is important when a monthly S&OP plan must agree with weekly distribution plans and both must agree with product and location totals.

An important distinction is between cross-sectional coherence, which concerns products, regions or channels, and temporal coherence, which concerns time frequencies. Reconciliation can be sequential, but sequential adjustments may undo earlier adjustments. Joint reconciliation is preferable when computationally feasible and when the full constraint structure is known.

5. Forecast combination

5.1 Rationale

Forecast combination reduces dependence on one model and can improve robustness. Different models may capture different features:

• exponential smoothing captures level, trend and seasonality;
• regression models incorporate prices, promotions and macroeconomic variables;
• intermittent-demand methods address many zero observations;
• machine-learning models capture nonlinearities and interactions;
• expert forecasts incorporate information not yet recorded in the data.

Let ŷ_j,t+h denote forecast j for horizon h. A linear combination is:

The weights commonly satisfy w_j ≥ 0 and ∑_j=1^Jw_j=1. Nonnegative weights summing to one make the combination interpretable as a weighted average, although unconstrained combinations can sometimes perform better in a well-regularised statistical setting.

5.2 Equal-weight combination

The equal-weight forecast is:

It is a strong benchmark because it is transparent, stable and resistant to estimation noise. It is especially useful when the historical sample is short or when model rankings change over time.

5.3 Error-based weighting

Suppose model j has historical mean squared error MSE_j. A commonly used inverse-error weight is:

This requires strictly positive MSE values and comparable evaluation periods. If one model has near-zero error because of leakage or an anomalous validation sample, the weights can become unstable. Weight estimation should therefore use rolling-origin validation, shrinkage, weight caps or a minimum effective weight.

For unbiased forecasts with error variances σ_j² and mutually uncorrelated errors, the minimum-variance combination weights are proportional to inverse variance:

The uncorrelated-error assumption matters. If errors are correlated, the optimal weights depend on the full covariance matrix, not variances alone. In matrix notation, with error covariance matrix Σ, the unconstrained minimum-variance weights satisfying 1ᵀw=1 are:

This solution requires an invertible covariance matrix and can generate negative weights. Constraints or regularisation are commonly applied in operational systems.

5.4 Combining expert and statistical forecasts

Expert judgement should not be treated as automatically superior or inferior to a statistical model. A disciplined process records the expert forecast, rationale, information source, confidence and date. The expert forecast can be combined with statistical forecasts or used as an adjustment subject to approval thresholds.

A useful control is to compare the judgemental adjustment with historical forecast errors. Large deviations require documented evidence, such as a confirmed contract, competitor exit, regulatory change or planned promotion. The system should distinguish genuine information from optimism, anchoring and political pressure.

5.5 Combination and reconciliation together

Combination and reconciliation solve different problems. Combination addresses uncertainty about the generating model; reconciliation addresses structural inconsistency. A practical sequence is:

1. generate base forecasts using several models at relevant levels;
2. combine forecasts within each series or model family;
3. reconcile the combined forecasts across the hierarchy;
4. evaluate both accuracy and coherence;
5. review exceptional adjustments with planners.

Alternatively, forecasts from each model can be reconciled first and then combined. The appropriate order depends on whether the models produce forecasts for the same set of series and whether their errors are comparable. In either case, the final published forecast must be coherent.

6. Worked example: regional allocation and reconciliation

Consider a distributor with total weekly demand forecast of 1,000 units. Historical proportions for three warehouses are 0.50, 0.30 and 0.20. A top-down forecast gives:

Suppose independent bottom-level forecasts are 560, 250 and 230 units. Their aggregate is 1,040 units, so they are incoherent with the total forecast. A simple proportional scaling reconciles them to the total. Let the scaling factor be:

The reconciled forecasts are approximately:

• W1: 560 × 0.961538 = 538.46 units;
• W2: 250 × 0.961538 = 240.38 units;
• W3: 230 × 0.961538 = 221.15 units.

Their sum is 1,000 units, subject to rounding. This adjustment preserves the relative proportions of the independent forecasts but does not account for different forecast-error variances. A MinT or weighted least squares method could produce a different allocation if, for example, W1 has highly reliable data while W3 is intermittent.

Now suppose three forecasts for W1 are 520, 540 and 560 units. An equal-weight combination is:

The example illustrates that combination and reconciliation should be evaluated separately: 540 may be a useful combined W1 forecast, but it must still be reconciled with the total and the other warehouses.

7. Applied case: a multichannel fulfilment network

Background

The following is an illustrative case. OmniFlow, a national distributor, supplies consumer electronics through retail stores, its website and business-to-business contracts. It plans demand by product family, channel, region and fulfilment centre. Its monthly corporate forecast is produced by a central analytics team, while regional planners create local forecasts.

Problem

The forecasts disagree. The sum of regional forecasts exceeds the corporate forecast by 8%. Online demand is growing rapidly, making historical channel shares unreliable. Several new SKUs have intermittent demand, and monthly forecasts do not equal the sum of weekly operational forecasts. Managers respond by manually overwriting numbers, reducing traceability.

Analysis

The analytics team maps the hierarchy and identifies a grouped structure: product family × channel × region. It audits master data, removes duplicate location codes and identifies stockout periods. Forecast accuracy is assessed using rolling-origin back-testing rather than a single holdout period. Weighted interval measures are included because high-volume products dominate aggregate error while service risk is concentrated in volatile SKUs.

The team finds that aggregate demand is easier to forecast than individual SKUs, but channel composition is changing. Historical fixed proportions therefore perform poorly for online demand. A dynamic share model uses recent channel growth, promotions and confirmed contract information, with shares constrained to sum to one.

Approach and solution

The redesigned process has five stages:

1. Statistical base forecasts are generated for total demand, product families, channels, regions and bottom-level series.
2. Forecasts from exponential smoothing, regression and intermittent-demand models are combined using equal weights initially, with weights reviewed through rolling validation.
3. A shrinkage covariance estimate is used in weighted reconciliation to account for scale differences and common errors.
4. Weekly forecasts are reconciled with monthly totals through a joint temporal-cross-sectional constraint system.
5. Expert overrides are recorded as signed interventions with reasons, evidence and expiry dates.

The system publishes point forecasts and prediction intervals. It also reports the size of each reconciliation adjustment. Adjustments above a pre-agreed threshold trigger review rather than silent acceptance.

Outcome

The company obtains one coherent planning dataset. Procurement uses SKU forecasts, fulfilment centres use location forecasts, and executives use product-family and channel totals without contradictory totals. Forecast accuracy improves most at aggregate levels, while operational benefits arise from reduced manual reconciliation, clearer accountability and earlier detection of unusual channel shifts.

Learning points

The case demonstrates that technical reconciliation cannot compensate for poor master data. It also shows that a changing demand mix requires dynamic allocation or models with explanatory variables. Forecast combination should be treated as a controlled learning process, not as an automatic average of every available model. Finally, governance is essential: planners need visibility into adjustments, uncertainty and the assumptions behind the published forecast.

8. Evaluation, implementation and governance

Accuracy and coherence metrics

Accuracy should be assessed at every decision-relevant level. Common measures include mean absolute error:

and root mean squared error:

Here, n is the number of observations, y_t actual demand and ŷ_t forecast demand. MAE is in the same units as demand; RMSE penalises large errors more strongly. Percentage measures can be misleading when actual demand is zero or close to zero. Scaled measures, weighted measures and service-level outcomes are often more appropriate for logistics.

Coherence should be measured directly. For every aggregation identity, calculate the absolute or percentage discrepancy before reconciliation. After reconciliation, discrepancies should be zero apart from numerical rounding. Accuracy evaluation must use the same information set that would have been available when the forecast was issued; otherwise leakage produces over-optimistic results.

Implementation architecture

A production system typically requires:

• a governed hierarchy and versioned summing matrix;
• a forecast store containing model, horizon, timestamp and data version;
• rolling-origin validation and back-testing;
• covariance estimation and reconciliation services;
• override workflows and audit logs;
• monitoring for missing data, structural breaks and unusual adjustments;
• access controls separating model development, approval and publication.

AI and machine learning are useful for nonlinear demand models, dynamic share estimation, anomaly detection and text-based extraction of confirmed customer information. They do not remove the need for structural constraints, explainability and human oversight. An unconstrained machine-learning model may produce excellent SKU forecasts but contradictory totals.

Limitations and risks

Hierarchical methods depend on stable and accurate relationships. Business reorganisations, product substitutions, acquisitions and channel migration can invalidate historical structures. Reconciliation can also redistribute error rather than eliminate it. Forecast combination may dilute a genuinely superior model or preserve a consistently biased model if validation is poorly designed.

Important risks include:

• data leakage, especially when future promotions or realised sales enter training data;
• structural breaks, such as pandemics, strikes or regulatory changes;
• stockout censoring, where observed sales understate unconstrained demand;
• forecast gaming, in which units manipulate forecasts to obtain inventory or targets;
• algorithmic opacity, which prevents managers from challenging implausible outputs;
• privacy and commercial confidentiality, particularly in customer-level forecasts.

Best practice is to maintain a documented hierarchy, use rolling validation, compare against simple benchmarks, publish uncertainty, monitor adjustment magnitudes and retain human accountability for material decisions.

Contemporary developments

Research and practice increasingly focus on probabilistic hierarchical forecasting. Instead of reconciling only point forecasts, distributions or simulated paths can be reconciled so that uncertainty is coherent across levels. This supports inventory optimisation and risk-aware capacity planning.

Other developments include global machine-learning models that learn across many related series, probabilistic forecast combination, reconciliation under intermittent demand, online updating for structural change and joint forecasting of demand and returns. These methods are promising but require careful calibration, computational governance and clear evidence that complexity improves decisions rather than merely improving in-sample fit.

Chapter Summary

Hierarchical and grouped demand systems represent the multiple levels at which logistics decisions are made. Coherent forecasts satisfy all aggregation identities across products, locations, channels and time. Top-down methods allocate an aggregate forecast, bottom-up methods aggregate detailed forecasts, and middle-out methods forecast at a strategically selected intermediate level. Reconciliation uses a summing matrix to adjust base forecasts so that they become coherent. Ordinary least squares is simple, while weighted least squares and MinT account for unequal and correlated forecast errors.

Forecast combination reduces dependence on one model. Equal weights are a robust benchmark; error-based and covariance-based weights can improve performance when estimated with reliable rolling validation. Expert judgement can add valuable information, but it should be documented, evidence-based and auditable. In practice, successful systems combine accurate data structures, appropriate models, reconciliation, uncertainty measurement, governance and managerial review. Coherence is necessary for integrated logistics planning, but it is not itself a guarantee of forecast accuracy.

Key Terms

Hierarchical time series: Time series arranged in a strict parent–child aggregation structure.

Grouped time series: Series organised by multiple crossing aggregation dimensions.

Multilevel forecasting: Forecasting across several related organisational, product, geographic or temporal levels.

Coherence: Compliance with all required aggregation relationships.

Summing matrix: Matrix that maps bottom-level series into aggregate series.

Base forecast: Forecast produced before structural reconciliation.

Top-down forecasting: Allocating an upper-level forecast to lower levels.

Bottom-up forecasting: Forecasting detailed series and aggregating them upward.

Middle-out forecasting: Forecasting at an intermediate level and using both upward and downward procedures.

Dynamic proportions: Time-varying shares used to allocate aggregate demand.

Forecast reconciliation: Adjustment of forecasts to satisfy structural constraints.

Ordinary least squares reconciliation: Reconciliation treating base forecast errors as equally weighted.

Weighted least squares: Reconciliation that uses an error-variance or covariance weighting matrix.

MinT: Minimum-trace reconciliation that seeks low total reconciled forecast-error variance.

Forecast combination: Weighted integration of forecasts from multiple models or experts.

Rolling-origin validation: Repeated historical testing that mimics real forecast production.

Covariance matrix: Matrix describing variances and cross-correlations of forecast errors.

Forecast override: Deliberate human adjustment to a statistical forecast.

Temporal coherence: Agreement among forecasts at different time frequencies.

Prediction interval: Range intended to contain future demand with a stated probability.

Review Questions

Short Answer Questions

1. What is meant by forecast coherence?
2. Distinguish a hierarchy from a grouped time-series structure.
3. State one advantage and one limitation of bottom-up forecasting.
4. What role does the summing matrix play in reconciliation?
5. Why can percentage accuracy measures be problematic for intermittent demand?

Long Answer Questions

1. Compare top-down, bottom-up and middle-out forecasting for a logistics network, including appropriate use conditions.
2. Explain ordinary least squares, weighted least squares and MinT reconciliation, including their assumptions.
3. Discuss how forecast combination can improve robustness and why equal weighting is a useful benchmark.
4. Design a governance process for expert overrides in a hierarchical forecasting system.
5. Explain why temporal coherence matters in weekly operational planning and monthly S&OP.

Analytical / Application Questions

1. A national forecast is 12,000 units, while three independent regional forecasts are 6,500, 3,800 and 2,900 units. Calculate the proportional scaling factor required to reconcile the regional forecasts to the national total, and calculate the reconciled regional forecasts.
2. A planner has three forecasts of 420, 450 and 510 units. Calculate the equal-weight combination and explain one reason why it may be preferred to selecting the historically best model.
3. A company experiences rapidly changing online demand, frequent stockouts and new SKUs. Recommend a forecasting and reconciliation design, specifying how you would address dynamic shares, censored demand, model combination and governance.

Multiple Choice Questions with Answers

Question 1

Which condition defines a coherent hierarchical forecast?

A. Every series uses the same forecasting model

B. All aggregation identities are satisfied

C. All forecasts have equal variances

D. Only bottom-level forecasts are published

Correct Answer: B. All aggregation identities are satisfied

Question 2

In bottom-up forecasting, aggregate demand is obtained by:

A. Allocating the top-level forecast using historical shares

B. Averaging all forecasts across levels

C. Summing forecasts of the bottom-level series

D. Discarding forecasts for intermittent items

Correct Answer: C. Summing forecasts of the bottom-level series

Question 3

What does the summing matrix represent?

A. The covariance of forecast errors

B. The mapping from bottom-level series to aggregate series

C. The weights used in expert judgement

D. The seasonal pattern of demand

Correct Answer: B. The mapping from bottom-level series to aggregate series

Question 4

A top-level forecast is 1,000 units and a lower-level historical share is 0.30. Under a fixed-share top-down method, the lower-level forecast is:

A. 30 units

B. 300 units

C. 700 units

D. 1,300 units

Correct Answer: B. 300 units

Question 5

Which method explicitly uses a forecast-error covariance or weighting matrix?

A. Simple bottom-up aggregation

B. Weighted least squares reconciliation

C. Naive seasonal forecasting only

D. Historical-share allocation only

Correct Answer: B. Weighted least squares reconciliation

Question 6

The equal-weight combination of forecasts 120, 150 and 180 is:

A. 130

B. 140

C. 150

D. 160

Correct Answer: C. 150

Question 7

Why is rolling-origin validation useful?

A. It guarantees zero forecast error

B. It mimics repeated real-time forecast production

C. It eliminates the need for benchmarks

D. It forces all models to have equal weights

Correct Answer: B. It mimics repeated real-time forecast production

Question 8

If forecast errors are correlated, inverse-variance weighting alone may be inadequate because:

A. Correlation affects the variance of the combined forecast

B. All forecasts become identical

C. The hierarchy disappears

D. MAE cannot be calculated

Correct Answer: A. Correlation affects the variance of the combined forecast

Question 9

Which situation most strongly supports dynamic rather than fixed allocation proportions?

A. Stable channel composition over many years

B. No changes in product assortment

C. Rapid migration of demand between channels

D. A hierarchy with only one bottom-level series

Correct Answer: C. Rapid migration of demand between channels

Question 10

A monthly forecast is temporally coherent with weekly forecasts when:

A. Each weekly forecast is identical

B. The monthly forecast equals the sum of its constituent weekly forecasts

C. The monthly forecast is always larger than every weekly forecast

D. Weekly forecasts are generated independently and never adjusted

Correct Answer: B. The monthly forecast equals the sum of its constituent weekly forecasts