Rules for consistent aggregation: With application to cost efficiency decompositions

This study offers a complete discussion of existing and novel rules for consistent aggregation, here interpreted as rules ensuring that measurement scores for units can somehow be added or averaged so that their sum or mean has the same structure and interpretation as the unit-level scores. Examples are the denominator rule and the numerator rule, but there are many more, dependent on the choice of aggregation function, either the arithmetic, harmonic or geometric mean, and associated weights, either symmetric or asymmetric. Additionally, when the aggregates are decomposed, this can be done based on equal or unequal weights for the different components. Several of these general rules have appeared dispersedly in the literature to evaluate applied research in fields such as index numbers, efficiency measures, productivity analysis, cross-efficiency measures, and composite indicators. To address this fragmentation, we present a unified framework for assessing and comparing these rules, highlighting their individual characteristics. In addition, we conduct an empirical application that brings all the decompositions together and compares them using a common dataset of financial institutions. From a methodological standpoint, the overall conclusion is that for consistent decomposition a geometric mean, based on symmetric weights — so that there is no need to distinguish between denominator and numerator rules — is advisable. Consistent factorial decompositions appear then also straightforward.