A tour of regression models for explaining shares

This paper aims to present and compare statistical modeling methods adapted for shares as dependent variables. Shares are characterized by the following constraints: positivity and sum equal to 1. Four types of models satisfy this requirement: multinomial logit models widely used in discrete choice models of the econometric literature, market-share models from the marketing literature, Dirichlet covariate models and compositional regression models from the statistical literature. We highlight the properties, the similarities and the differences between these models which are coming from the assumptions made on the distribution of the data and from the estimation methods. We prove that all these models can be written in an attraction model form, and that they can be interpreted in terms of direct and cross elasticities. An application to the automobile market is presented where we model brand market-shares as a function of media investments in 6 channels in order to measure their impact, controlling for the brands average price and a scrapping incentive dummy variable. We propose a cross-validation method to choose the best model according to different quality measures.