  Ecological inference (EI) is the process of learning about individual
behavior from aggregate data. We study a partially identified linear contextual
effects model for EI and describe how to estimate the district level parameter
averaging over many precincts in the presence of the non-identified parameter
of the contextual effect. This may be regarded as a first attempt in this
venerable literature to limit the scope of the key form of non-identifiability
in EI. To study the operating characteristics of our model, we have amassed the
largest collection of data with known ground truth ever applied to evaluate
solutions to the EI problem. We collect and study 459 datasets from a variety
of fields including public health, political science, and sociology. The
datasets contain a total of 2,370,854 geographic units (e.g., precincts), with
an average of 5,165 geographic units per dataset. Our replication data are
publicly available via the Harvard Dataverse (Jiang et al. 2018) and may serve
as a useful resource for future researchers. For all real data sets in our
collection that fit our proposed rules, our approach reduces the width of the
Duncan and Davis (1953) deterministic bound, on average, by about 45\%, while
still capturing the true district level parameter in excess of 97\% of the
time. .
