Multivariate Beta Regression with Application in Small Area Estimation

Multivariate beta regression models for jointly modelling two or more variables whose values belong in the (0,1) interval, such as indexes, rates or proportions, are proposed for making small area predictions. The multivariate model can help the estimation process by borrowing strength between units and obtaining more precise estimates, especially for small samples. Each response variable is assumed to have a beta distribution so the models could accommodate multivariate asymmetric data. Copula functions are used to construct the joint distribution of the dependent variables; all the marginal distributions are fixed as beta. A hierarchical beta regression model is additionally proposed with correlated random effects. We present an illustration of the proposed approach by estimating two indexes of educational attainment at school level in a Brazilian state. Our predictions are compared with separate univariate beta regressions. The inference process was conducted using a full Bayesian approach.