  A popular method for estimating a causal treatment effect with observational
data is the difference-in-differences (DiD) model. In this work, we consider an
extension of the classical DiD setting to the hierarchical context in which
data cannot be matched at the most granular level (e.g., individual-level
differences are unobservable). We propose a Bayesian hierarchical
difference-in-differences (HDiD) model which estimates the treatment effect by
regressing the treatment on a latent variable representing the mean change in
group-level outcome. We present theoretical and empirical results showing that
an HDiD model that fails to adjust for a particular class of confounding
variables, or confounding with the baseline (pre-treatment) outcomes, biases
the treatment effect estimate. We propose and implement various approaches to
perform variable selection using a structured Bayesian spike-and-slab model in
the HDiD context. Our proposed methods leverage the temporal structure within
the DiD context to select those covariates that lead to unbiased and efficient
estimation of the causal treatment effect. We evaluate the methods' properties
through theoretical results and simulation, and we use them to assess the
impact of primary care redesign of clinics in Minnesota on the management of
diabetes outcomes from 2008 to 2017.
