Joint Invariance in Multiperiod Arithmetic Attribution

Executive overview

A multiperiod report can be coherent in two directions. Merging sectors before or after linking should agree, and collapsing a block of periods and continuing should agree with linking the whole horizon. Exact reconciliation guarantees neither.

This paper proves that, inside the common-coefficient form, temporal associativity is exactly a block-factorization law for the coefficients. It then builds a continuous one-parameter family of linkers that satisfy every structural requirement, with reverse Frongello, Cariño and forward Frongello as three points on the same line.

Findings, in order

1. Cross-sectional coherence. A self-contained proof that aggregation consistency on a rich signed domain implies common period coefficients.
2. Associativity is factorization. Temporal associativity holds exactly when each full-history coefficient equals the block coefficient times the within-block coefficient.
3. A continuous family. An infinite one-parameter family of total, exactly reconciling, arbitrarily factorizing linkers on the positive gross-return group.
4. History sits on one line. Reverse Frongello, Cariño and forward Frongello appear at parameters −1, 0 and 1, so reconciliation and factorization alone do not pick a historical formula.
5. Symmetry selects Cariño. Within the family, both reversal symmetry and full order invariance single out Cariño.
6. Order symmetry is not associativity. Published order-symmetric constructions by Reztsov are non-examples of block factorization.

Novelty