Past articles have provided empirical and mathematical evidence of the "exactness" of various attribution linkink methodologies.  The differences between these methodologies are due principally to each method's definition of a multi-period effect with regard to where inter-period cross products are included.  In single-period Attribution, we track the excess effect created by one type of effect when it is impacted by another type as Interaction and assigns it as appropriate.  By dissecting multi-period effects into pure and inter-period-interaction (IPI) effects, one can offer similar flexibility in multi-period computations, or at the very least better understand the source and materialty of these Inter-Period Interaction effects and where they are assigned in one's chosen methodology.

Pure and Inter-Period Interaction Effects in Multi-Period Attribution.