MinT Optimal Reconciliation

If you forecast total company revenue, and separately forecast revenue for each of its three divisions, the three division forecasts usually don't sum to the total forecast. MinT ("minimum trace") is a method for reconciling these mismatched forecasts into one consistent set that actually adds up, by nudging each individual forecast just enough — and no more.

The key idea is that not all forecasts deserve equal trust. If the total-revenue forecast has historically been very accurate but one division's forecast has been noisy and volatile, MinT leans on the noisy division forecast to absorb most of the correction, leaving the trustworthy total mostly untouched. It does this by estimating the covariance matrix of forecast errors across the whole hierarchy and choosing the reconciliation that minimizes total forecast error variance (the "trace" of that covariance matrix) subject to the constraint that everything sums correctly.

Worked example. A retailer forecasts total sales at $10 million, and two store forecasts at $4 million and $5.5 million (summing to $9.5 million, a $500,000 gap). If store-level forecasts have historically been noisier than the total, MinT might revise the stores up to $4.2 million and $5.8 million (summing to $10 million) while barely touching the total, rather than spreading the $500,000 gap evenly.

MinT is now the standard approach in forecasting libraries that support hierarchical time series, replacing older ad hoc methods like simply forcing bottom-up sums or top-down splits.