  Relational arrays represent measures of association between pairs of actors,
often in varied contexts or over time. Trade flows between countries, financial
transactions between individuals, contact frequencies between school children
in classrooms, and dynamic protein-protein interactions are all examples of
relational arrays. Elements of a relational array are often modeled as a linear
function of observable covariates. Uncertainty estimates for regression
coefficient estimators -- and ideally the coefficient estimators themselves --
must account for dependence between elements of the array (e.g. relations
involving the same actor) and existing estimators of standard errors that
recognize such relational dependence rely on estimating extremely complex,
heterogeneous structure across actors. This paper develops a new class of
parsimonious coefficient and standard error estimators for regressions of
relational arrays. We leverage an exchangeability assumption to derive standard
error estimators that pool information across actors and are substantially more
accurate than existing estimators in a variety of settings. This
exchangeability assumption is pervasive in network and array models in the
statistics literature, but not previously considered when adjusting for
dependence in a regression setting with relational data. We demonstrate
improvements in inference theoretically, via a simulation study, and by
analysis of a data set involving international trade.
