Three-fold Fay-Herriot model with unequal variance random effects

Abstract

This study extends the classical Fay–Herriot model to a threefold hierarchical structure incorporating heteroscedastic random effects. Three submodels are also introduced within this framework. Small area best linear unbiased predictors are derived for linear indicators, and their mean squared errors (MSEs) are estimated using both analytical and parametric bootstrap methods. Model performance and reliability are evaluated through diagnostic tools and influence measures, specifically designed for small area estimation. Simulation experiments are conducted to analyze the empirical properties of the predictors and MSE estimators. The proposed approach is applied to data from the 2019–2021 Spanish Living Conditions Survey to estimate the proportion of women and men below the poverty line, disaggregated by province, age group, and year.