Variable Selection in the Presence of Factors: A Model Selection Perspective

In the context of a Gaussian multiple regression model, we address the problem of variable selection when in the list of potential predictors there are factors, that is, categorical variables. We adopt a model selection perspective, that is, we approach the problem by constructing a class of models, each corresponding to a particular selection of active variables. The methodology is Bayesian and proceeds by computing the posterior probability of each of these models. We highlight the fact that the set of competing models depends on the dummy variable representation of the factors, an issue already documented by Fernández et al. in a particular example but that has not received any attention since then. We construct methodology that circumvents this problem and that presents very competitive frequentist behavior when compared with recently proposed techniques. Additionally, it is fully automatic, in that it does not require the specification of any tuning parameters.