Hierarchical Beta regression models for the estimation of poverty and inequality parameters in small areas.

Many parameters that describe poverty, social exclusion and inequality can take values in the (0,1) interval. This class includes headcount ratios such as the at-risk-of-poverty or material deprivation rates, the median poverty gap and the Gini inequality index, just to cite a few. Social scientists and policy planners often need estimates of this type of parameters for small subpopulations for which only small or no samples are available. In this work, we discuss area level models for the estimation of these parameters. Given their nature, Beta regression models are suitable, as the Beta distribution is very flexible over the (0,1) range and it allows for asymmetric sampling distributions. We adopt a Bayesian approach with approximate inference for relevant posterior distributions relying on MCMC algorithms. We focus on specific problems that we think are particularly relevant for small area applied researchers and discuss them with an application to real data. The problems we consider are: i) the estimation of the at-risk-of-poverty rate; ii) the joint estimation of the material deprivation and severe material deprivation rates (i.e. two rates based on increasing thresholds); iii) the joint estimation of two correlated parameters, namely the at-risk-of-poverty rate and the Gini inequality index. The data set that we analyze is a subset of the Italian sample of the EUSILC survey, and the small areas we target are Italian health districts, whose administrations play an important role in the implementation of many social and health expenditure programs related to the contrast of poverty and social exclusion. When estimating the at-risk-of-poverty rate we face the problem of areas with no poor in the sample that leads us to consider zero-mixture Beta regressions, a class of models that will be extended to the multivariate setting. In fact, the joint estimation of parameters in the (0,1) range requires multivariate extensions of the Beta regressions: for material deprivation and severe deprivation rates, based on increasing thresholds, we discuss a multivariate logistic-normal model for the expected values of the Beta distributions, while for the joint estimation of the at-risk-of-poverty rate and the Gini inequality index, we model the correlation between direct estimators using copula functions.