CODA methods and the multivariate Student distribution: an application to political economy

In a multiparty election, the vote shares form a composition vector (mathematically, a vector belonging to a simplex). Political economists are interested by the impact of the characteristics of the geographical units on the outcome of the elections. Because vote shares data often exhibit heavy tail behavior, we decide to use a Student error distribution. We describe how to adapt the CODA regression model to the multivariate Student error distribution. For a Gaussian errors vector, the assumption of independent coordinates is equivalent to the assumption of correlated coordinates. However, this equivalence is no longer true when considering a multivariate Student distribution. In this paper, we recall these two types of multivariate Student distribution for the error term, and concentrate on building a CODA regression model using the multivariate independent Student error vectors. We compare this model to a model which uses the multivariate Gaussian distribution. The models are fitted on French electoral data of the 2015 departmental elections. We illustrate on this data set a method for selecting between the Gaussian and the Student models based on the Mahalanobis distance.