Interpreting the impact of explanatory variables in compositional models

Regression models have been developed for the case where the dependent variable is a vector of shares. Some of them, fromthe marketing literature, are easy to interpret but they are quite simple and can only be complexified at the expense of a large number of parameters. Other models, compositional regression models, are based on the simplicial geometry and use a log-ratio transformation of shares. They are flexible in terms of explanatory variables, but their interpretation is not straightforward, due to the link between shares. This paper combines both literatures in order to obtain a performing market-share model allowing to get relevant interpretations, which can be used for decision making in practical cases. For example, we are interested in modeling the impact of media investments on automobile manufacturers sales. In order to take into account the competition, we model the brands market-shares as a function of brands media investments. We furthermore focus on compositional models where some explanatory variables are compositional. Two specifications are possible: in Model A, a unique coefficient is associated to each compositional explanatory variable, whereas in Model B a compositional explanatory variable is associated to component-specific and cross-effect coefficients. Model A and Model B are estimated for our application in the B segment of the French automobile market, from 2003 to 2015. In order to enhance the interpretability of these models, we present different impact measures (marginal effects, elasticities, odds ratios) and we show that elasticities are particularly useful to isolate the impact of an explanatory variable on a particular share. We prove that elasticities can be equivalently computed from the transformed model and from the initial model. Direct and cross effects of media investments are computed for both models. Model B shows interesting non-symmetric synergies between brands