Small area estimation under a measurement error bivariate Fay–Herriot model

The bivariate Fay–Herriot model is an area-level linear mixed model that can be used for estimating the domain means of two correlated target variables. Under this model, the dependent variables are direct estimators calculated from survey data and the auxiliary variables are true domain means obtained from external data sources. Administrative registers do not always give good auxiliary variables, so that statisticians sometimes take them from alternative surveys and therefore they are measured with error. We introduce a variant of the bivariate Fay–Herriot model that takes into account the measurement error of the auxiliary variables and we give fitting algorithms to estimate the model parameters. Based on the new model, we introduce empirical best predictors of domain means and we propose a parametric bootstrap procedure for estimating the mean squared error. We finally give an application to estimate poverty proportions and gaps in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.