Benchmarking techniques for reconciling Bayesian small area models at distinct geographic levels

In sample surveys, there is often insufficient sample size to obtain reliable direct estimates for parameters of interest for certain domains. Precision can be increased by introducing small area models which ‘borrow strength’ by connecting different areas through use of explicit linking models, area-specific random effects, and auxiliary covariate information. One consequence of the use of small area models is that small area estimates at a lower (for example, county) geographic level typically will not aggregate to the estimate at the corresponding higher (for example, state) geographic level. Benchmarking is the statistical procedure for reconciling these differences. This paper provides new perspectives for the benchmarking problem, especially for complex Bayesian small area models which require Markov Chain Monte Carlo estimation. Two new approaches to Bayesian benchmarking are introduced: one procedure based on minimum discrimination information, and another procedure for fully Bayesian self-consistent conditional benchmarking. Notably the proposed procedures construct adjusted posterior distributions whose first and higher order moments are consistent with the benchmarking constraints. It is shown that certain existing benchmarked estimators are special cases of the proposed methodology under normality, giving a distributional justification for the use of benchmarked estimates. Additionally, a ‘flexible’ benchmarking constraint is introduced, where the higher geographic level estimate is not considered fixed, and is simultaneously adjusted, along with lower level estimates.