The link between multiplicative competitive interaction models and compositional data regression with a total

Résumé

This article sheds light on the relationship between compositional data (CoDa) regression models and multiplicative competitive interaction (MCI) models, which are two approaches for modeling shares. We demonstrate that MCI models are special cases of CoDa models and that a reparameterization links both. Recognizing this relation offers mutual benefits for the CoDa and MCI literature, each with its own rich tradition. The CoDa tradition, with its rigorous mathematical foundation, provides additional theoretical guarantees and mathematical tools that we apply to improve the estimation of MCI models. Simultaneously, the MCI model emerged from almost a century-long tradition in marketing research that may enrich the CoDa literature. One aspect is the grounding of the MCI specification in intuitive assumptions on the behavior of individuals. From this basis, the MCI tradition also provides credible justifications for heteroskedastic error structures -- an idea we develop further and that is relevant to many CoDa models beyond the marketing context. Additionally, MCI models have always been interpreted in terms of elasticities, a method only recently revealed in CoDa. Regarding this interpretation, the change from the MCI to the CoDa perspective leads to a decomposition of the influence of the explanatory variables into contributions from relative and absolute information. This decomposition also opens the door for testing hypothesis about the importance of each information type.