Make aggregate forecasts add up

For a hierarchy with two bottom series A and B, the ordered vector is total, A, B. A summing matrix maps bottom values to that vector by adding them in its first row and copying each in the remaining rows. More complicated grouped structures can contain several overlapping aggregation dimensions. Specify a stable row and column ordering and preserve it in the artifact. A correct matrix applied to forecasts in a different order can generate coherent-looking but semantically wrong outputs.

Use additive quantities on a common interval and unit. Total requests can be summed across disjoint sites; average utilization percentages cannot generally be summed without denominators. A hierarchy change such as moving a site between regions changes the summing matrix and the interpretation of historical residuals. Version membership at the forecast’s intended target period, with a policy for planned future changes. Avoid mixing base forecasts built under different membership snapshots and hoping the reconciliation step will infer the intended structure.

Ordinary least squares finds the coherent vector closest to the base vector in unweighted squared distance. Covariance-weighted reconciliation instead uses a positive-definite forecast-error covariance matrix for the specified horizon. The inverse covariance encodes which combinations of base errors are relatively uncertain. Estimate that matrix from appropriate historical out-of-sample errors, with regularization when the number of series is large relative to the available residual sample. Using training residuals or the wrong horizon can misrepresent the adjustment geometry.

The MinT result concerns a class of linear unbiased coherent estimators under its error and covariance assumptions. It does not promise lower realized error for every individual series or every future period. If the covariance is proportional to the identity, the displayed formula reduces to ordinary least squares. Solve the corresponding linear systems rather than forming explicit inverses in implementation, and detect singular or ill-conditioned estimates. A covariance matrix estimated from only a few highly correlated residuals may need shrinkage or another documented fallback.

The unconstrained projection can produce negative bottom forecasts even when the underlying quantity is a nonnegative count. Enforcing nonnegativity requires a separate constrained optimization and changes the estimator. Rounding each component independently can also break exact additivity; integer allocation requires a declared rounding or optimization rule. Check coherence numerically after all postprocessing, not only immediately after the matrix multiplication. Keep raw base values so reviewers can inspect which series absorbed the largest adjustments.

Marginal quantiles cannot generally be added as though they were means. Distributional reconciliation needs a justified joint construction; reconciling joint samples path by path is one route when the sample dependence model is appropriate. Assess calibration and interval behavior afterward rather than assuming coherence preserves them automatically. In production, promote base models, hierarchy matrices, covariance estimates, and postprocessing rules as a compatible bundle so a forecast cannot accidentally combine one model generation with another hierarchy generation.