Using compositional and Dirichlet models for market-share regression

When the aim is to model market-shares as a function of explanatory variables, the marketing literature proposes some regression models which can be qualified as attraction models. They are generally derived from an aggregated version of the multinomial logit model widely used in econometrics for discrete choice modeling. But aggregated multinomial logit models (MNL) and the so-called market-share models or generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross-effect parameters. Introducing all possible cross effects is not possible in the MNL and would imply a very large number of parameters in the case of the GMCI. In this paper, we consider alternative models which are the Dirichlet covariate model (DIR) and the compositional model (CODA). DIR allows to introduce brand-specific parameters and CODA allows additionally to consider cross-effect parameters. We show that these last two models can be written in a similar fashion, called attraction form, as the MNL and the GMCI models. As market-share models are usually interpreted in terms of elasticities, we also use this notion to interpret the DIR and CODA models. We compare the main properties of the models in order to explain why CODA and DIR models can outperform traditional market-share models. The benefits of highlighting these relationships is on one hand to propose new models to the marketing literature and on the other hand to improve the interpretation of the CODA and DIR models using the elasticities of the econometrics literature. Finally, an application to the automobile market is presented where we model brands market-shares as a function of media investments, controlling for the brands average price and a scrapping incentive dummy variable. We compare the goodness-of-fit of the various models in terms of quality measures adapted to shares.