  The comparison of subnational areas is ubiquitous but survey samples of these
areas are often biased or prohibitively small. Researchers turn to methods such
as multilevel regression and poststratification (MRP) to improve the efficiency
of estimates by partially pooling data across areas via random effects.
However, the random effect approach can pool observations only through
area-level aggregates. We instead propose a weighting estimator, the synthetic
area estimator, which weights on variables measured only in the survey to
partially pool observations individually. The proposed method consists of
two-step weighting: first to adjust differences across areas and then to adjust
for differences between the sample and population. Unlike MRP, our estimator
can directly use the national weights that are often estimated from pollsters
using proprietary information. Our approach also clarifies the assumptions
needed for valid partial pooling, without imposing an outcome model. We apply
the proposed method to estimate the support for immigration policies at the
congressional district level in Florida. Our empirical results show that small
area estimation models with insufficient covariates can mask opinion
heterogeneities across districts.
