Hierarchical Forecasting and Reconciliation

A retailer forecasts total company revenue directly, and separately forecasts revenue for each of its three regions. If the three regional forecasts are made independently — each its own model, trained on its own regional history — there's no mathematical reason they'll sum to the same number as the top-level company forecast. One might say the regions total $410 million while the company-level model says $395 million. Both numbers came from reasonable models trained on real data, and yet they contradict each other, which is a problem the moment anyone tries to use both forecasts in the same planning process. Hierarchical forecasting is the discipline of forecasting quantities that have this parent-child structure — country made of regions, regions made of stores, a portfolio made of sectors made of individual names — and reconciliation is the step that forces the forecasts back into agreement.

Bottom-up, top-down, and why both have a flaw

The simplest fix is bottom-up: forecast only the most granular level (each store, say), and get every higher level's forecast for free by summing up — region forecast = sum of its stores' forecasts, company forecast = sum of all regions. This guarantees consistency by construction, but it throws away information: the most granular series are usually the noisiest, and any pattern that's easier to see in the aggregate (a company-wide seasonal promotion effect, say) has to be independently rediscovered by every single bottom-level model, or it's lost.

Top-down does the reverse: forecast only the aggregate, then split it down to lower levels using historical proportions — if a region has historically earned 28% of company revenue, its forecast is just 28% of the top-level number. This uses the most reliable, least noisy series (the aggregate) as the anchor, but it assumes the historical split proportions stay stable, which fails exactly when it matters most — a region growing faster than the rest of the company will keep being under-forecast by a fixed historical share.

Worked example: reconciling three regional forecasts to a company total

Independent models produce these forecasts for next quarter: Region A $150M, Region B $120M, Region C $140M — summing to $410M — while a separate, independently trained company-level model forecasts $395M. A common reconciliation approach distributes the $15M discrepancy across the regions in proportion to their own forecast size, rather than picking one number arbitrarily. Region A's share of the $410M total is 150/410 ≈ 36.6%, so its adjusted forecast becomes 395 × 0.366 ≈ $144.5M. Region B's share is 120/410 ≈ 29.3%, giving 395 × 0.293 ≈ $115.7M. Region C's share is 140/410 ≈ 34.1%, giving 395 × 0.341 ≈ $134.8M. Check: 144.5 + 115.7 + 134.8 = 395.0, matching the top-level forecast exactly, and each region's adjustment is proportional to how large its own original forecast was, which is a simple, defensible way to blend the bottom-up and top-down information rather than trusting either level alone.

What this means in practice

The proportional method shown above is a simplification of a broader family of optimal reconciliation approaches, of which MinT (Minimum Trace) is the best known: rather than a fixed rule like "top-down" or "bottom-up," MinT weights each level's original forecast by an estimate of how reliable it is (its forecast error variance), so a genuinely accurate bottom-level model gets to keep more of its own forecast, while a noisy one gets pulled harder toward the more reliable aggregate. In a quant context, this structure shows up constantly: a portfolio-level return forecast, sector-level forecasts, and single-stock forecasts should sum coherently through the portfolio's actual weights, and treating each level as an unrelated forecasting problem — the default if every model is built independently — routinely produces inconsistencies that only surface at the point of use, when a risk report can't explain why the parts don't add up to the whole.

Practice

1. A hierarchy has two children summing to a parent. Child forecasts are 80 and 60 (sum 140); the parent's independent forecast is 130. Using the proportional method above, what are the reconciled child forecasts?
2. Explain, in one or two sentences, why a bottom-up approach can miss a pattern that top-down would have captured, using a company-wide promotional campaign as the example.