Variance reduction and interval efficiency in forecast reconciliation

Abstract Forecast reconciliation is commonly motivated by variance reduction, whereas finite-sample interval widths depend on order statistics. We construct a nonelliptical residual family for two- and three leaf hierarchies in which the full covariance, oracle minimum-trace projector, calibration size and actual marginal coverage remain fixed, yet the expected ordinary-width ordering reverses at least three times. Exact rational sign witnesses at calibration size 39 and a uniform Gaussian smoothing bound establish the result for continuous laws with exact 90% or 95% single-node coverage. The construction identifies a boundary of variance-based efficiency arguments outside elliptical residual families. We also compare existing covariance estimators on two related Australian retail partitions and a tourism hierarchy. In 72-month sequential evaluations with annual expanding-window refits, past-only covariance fitting, rolling calibration and fixed normalization scales, centered diagonal target shrinkage reduces mean normalized interval scores by 2.0%, 3.3% and 0.8% relative to raw forecasts at nominal 90%. Realized coverage remains below nominal, and centered shrinkage is not uniformly best. Historical comparisons additionally show why cross-design scores require common normalization. The study introduces no estimator and makes no temporal-dependence coverage guarantee.