Benchmarking Priors: A Fully Bayesian Approach to Small Area Benchmarking

Abstract: We develop a new class of priors for Bayesian small area models under strict external bench- marking constraints. The prior is fully supported on the constrained parameter set defined by the benchmarking constraints and thereby provides a proper constrained posterior that is fully supported on the restricted set. The posterior naturally constrains the small area estimates to conform to the benchmarking values at the aggregate level. In addition, it facilitates valid uncertainty quantification using credible sets that are fully contained within the constrained set. This novel framework has a wide range of applicability, ranging from the standard Gaus- sian Fay-Herriot model to more general Fay-Herriot models, such as robust Fay-Herriot models, where the area-level random effects are distributed as multivariate t, or Fay-Herriot models where there is a complex nonlinear relationship between the area means and the covariates. We illustrate the usefulness of the proposed methodology using simulation experiments as well as by estimating the state-level proportion of children aged 5-17 living under poverty based on the direct estimates obtained from the Small Area Income and Poverty Estimates (SAIPE) program of the U.S. Census Bureau and the national benchmark of the child poverty rate obtained from the Annual Social and Economic Supplement of the Current Population Survey (CPS ASEC) conducted by the U.S. Census Bureau. The numerical results reaffirm the advantages of the proposed benchmarking priors.