Hierarchical Forecasting

Hierarchical Forecasting — also called Forecast Reconciliation — is the family of methods for producing forecasts at multiple levels of an aggregation hierarchy that are mutually consistent. In workforce management, the hierarchy is typically skill → site → network or queue → line of business → enterprise; forecasts are needed at every level, and they must add up.

Most WFM organizations face this problem and most do not solve it well. Common patterns: forecast at the bottom and sum (loses signal at the aggregate level); forecast at the top and disaggregate (loses signal at the bottom); forecast independently at each level (the levels do not reconcile, and reporting becomes a continuous argument about which number is "right"). Modern reconciliation methods address all of these failure modes.

The hierarchy problem

A typical WFM hierarchy:

```

               Enterprise total
                      |
          +-----------+-----------+
          |                       |
       Site A                  Site B
          |                       |
     +----+----+              +---+---+
     |    |    |              |       |
  Skill1 Skill2 Skill3      Skill4  Skill5

```

The constraint: forecasts at any level must sum (or aggregate appropriately) to forecasts at higher levels. If the skill-level forecasts say 100, 150, 200 calls for skills 1-3 at Site A, then the Site A forecast must be 450; Site A + Site B must equal the Enterprise total.

Without reconciliation, three approaches all fail in different ways:

Bottom-up

Forecast each skill independently; sum to site; sum to enterprise.

Strengths: uses skill-level granularity; the sum is consistent by construction.

Weaknesses: skill-level series are often noisy or intermittent. Errors at the skill level accumulate when summed. Site- and enterprise-level forecasts are less accurate than what could be obtained by forecasting those levels directly.

Top-down

Forecast the enterprise total; disaggregate to sites and skills using historical proportions.

Strengths: the top level forecast is typically more accurate than any single component (more signal, less noise). Aggregate-level decisions (annual budget) are well-served.

Weaknesses: bottom-level forecasts are basically a re-arrangement of the top forecast; they don't reflect skill-specific dynamics. A skill that's growing while the rest are flat will be under-forecasted because it gets the historical share of the average growth rate.

Independent forecasting

Forecast at each level separately using whatever method works best for that level.

Strengths: each level gets the method that fits its data shape.

Weaknesses: the forecasts don't add up. Site A independent forecast might predict 460 while the sum of its skill forecasts is 445. Reconciling after the fact (manually adjusting one or the other) is ad-hoc and political.

Optimal reconciliation (MinT)

The modern approach: forecast at every level independently, then reconcile by minimizing the variance of the reconciled forecasts subject to the aggregation constraint. The method, due to Wickramasuriya, Athanasopoulos, & Hyndman (2019), is called Minimum Trace (MinT) reconciliation.

The math, briefly: let $\widehat{\mathbf{𝐲}}$ be the vector of all base forecasts (every level). Let $\mathbf{𝐒}$ be the summing matrix that encodes which lower-level series sum to which higher-level series. The reconciled forecasts $\widetilde{\mathbf{𝐲}}$ are:

$\widetilde{\mathbf{𝐲}}=\mathbf{𝐒}({\mathbf{𝐒}}^{\prime }{\mathbf{𝐖}}^{-1}\mathbf{𝐒}{)}^{-1}{\mathbf{𝐒}}^{\prime }{\mathbf{𝐖}}^{-1}\widehat{\mathbf{𝐲}}$

where $\mathbf{𝐖}$ is the covariance matrix of forecast errors (estimated from historical data).

In plain language: the reconciliation projects the independent forecasts onto the constraint that they must aggregate consistently, weighting by how reliable each level's forecast is. Levels with smaller forecast errors carry more weight in the reconciliation.

The result outperforms bottom-up, top-down, and unreconciled-independent approaches in essentially every WFM context where it's been tested.

When hierarchical forecasting matters in WFM

The case for reconciliation is strongest when:

• The organization makes decisions at multiple levels. Skill-level scheduling, site-level capacity planning, enterprise budget — all need consistent numbers.
• Skill-level data is noisy or intermittent. Bottom-up forecasts of noisy skills produce noisy aggregates. Top-down or reconciled approaches share signal across the hierarchy.
• Different levels have different growth dynamics. One skill is growing rapidly while others are flat. Top-down disaggregation will under-forecast the growing skill.
• Reporting requires reconciled numbers. Finance, operations, and WFM cannot use forecasts that don't add up; the alternative is a continuous reconciliation argument.

The case is weakest when:

• The hierarchy has only one level that matters. If only the enterprise total drives decisions, forecast the enterprise total directly and stop.
• The hierarchy is shallow and stable. A flat hierarchy with consistent ratios across components may be served well enough by top-down disaggregation.

Grouped time series

A generalization of strict hierarchies. The same series can be aggregated in multiple non-nested ways — for example, by skill AND by site AND by language AND by customer segment. Each aggregation produces a different hierarchy.

WFM example: contacts can be aggregated by skill (sales/support/billing), by site (US/EU/APAC), and by channel (voice/chat/email) simultaneously. None of these aggregations nest cleanly into the others. The grouped time series framework handles this: reconciliation across multiple non-nested aggregation paths.

The math extends naturally; the practical complication is that the constraint matrix $\mathbf{𝐒}$ becomes much larger and the covariance matrix $\mathbf{𝐖}$ is harder to estimate reliably.

Common WFM pitfalls

• Forecast first, then "reconcile" by Excel adjustment. Manual reconciliation introduces bias and inconsistency. Use a defined method; document the approach.
• Reconciling forecasts produced by different methods. Mixing seasonal naive at the bottom with Holt-Winters at the top, then reconciling, can hide method-selection problems. Verify each level's forecast independently before reconciling.
• Using bottom-up because "we have skill-level data." Having the data does not mean bottom-up is the right approach. Test against MinT and top-down on the same data.
• Ignoring the hierarchy in capacity planning. Capacity decisions made on unreconciled forecasts produce inconsistencies between budget (enterprise level) and operational plans (site/skill level).
• Failing to update $\mathbf{𝐖}$. The covariance matrix should be re-estimated periodically as forecast accuracy at each level changes.

Diagnostic check

After producing reconciled forecasts, verify:

1. Aggregation consistency. The sum of any lower-level reconciled forecasts equals the higher-level reconciled forecast. (This holds by construction in MinT; check anyway.)
2. Per-level accuracy. Reconciled forecasts at each level should be at least as accurate as the corresponding independent forecast on out-of-sample data. If reconciliation is making accuracy worse at a level, the covariance estimation is suspect.
3. Behavior under structural change. When one component grows or shrinks unusually, the reconciliation should reflect that — not bury it under an aggregate trend.

Software support

Modern statistical packages support hierarchical forecasting:

• R: the `hts` package (older) and `fable`/`fabletools` (current) support full reconciliation including MinT
• Python: `statsforecast` and `hierarchicalforecast` from Nixtla support MinT and related methods

WFM software platforms vary widely. Most enterprise WFM platforms support bottom-up aggregation (sum the skill forecasts to the site forecast) but not formal reconciliation. The Pillar 3 (Advanced Capacity Planning) tooling in the WFM Ecosystem Architecture is where reconciliation typically lives in modern practice.

Connection to the WFM operating model

Hierarchical forecasting is a Level 3+ capability in the WFM Labs Maturity Model™. Level 2 organizations typically operate with separate forecasts at each level and reconcile manually (or simply don't reconcile). Level 3 introduces formal reconciliation. Level 4 extends it across grouped hierarchies (skill × site × channel × segment).

For practitioners working on this capability: start by documenting the hierarchy, then implement MinT on a single dimension (e.g., skill → site → enterprise), validate the gain over current practice, then extend to additional dimensions if the value justifies the complexity.

References

• Hyndman, R. J., & Athanasopoulos, G. "Forecasting hierarchical and grouped time series." Forecasting: Principles and Practice (Python edition). otexts.com/fpppy.
• Wickramasuriya, S. L., Athanasopoulos, G., & Hyndman, R. J. "Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization." Journal of the American Statistical Association 114(526), 2019. The MinT paper.

See Also

• Forecasting Methods — overview; reconciliation is applied on top of any combination of methods
• Naive and Seasonal Naive Forecasting — base method at any level
• Exponential Smoothing — common base method at higher aggregation levels
• ARIMA Models — base method when explicit autocorrelation is present
• Regression for Forecasting — base method when explanatory variables drive the forecast
• Forecast Accuracy Metrics — how to compare reconciled vs unreconciled forecast quality
• Multi-Objective Optimization in Contact Center — related multi-level problem; capacity allocation across the same hierarchy
• WFM Ecosystem Architecture — Pillar 3 (Advanced Capacity Planning)
• WFM Labs Maturity Model™ — Level 3+ capability