  Researchers increasingly leverage movement across multiple treatments to
estimate causal effects. While these "mover regressions" are often motivated by
a linear constant-effects model, it is not clear what they capture under weaker
quasi-experimental assumptions. I show that binary treatment mover regressions
recover a convex average of four difference-in-difference comparisons and are
thus causally interpretable under a standard parallel trends assumption.
Estimates from multiple-treatment models, however, need not be causal without
stronger restrictions on the heterogeneity of treatment effects and
time-varying shocks. I propose a class of two-step estimators to isolate and
combine the large set of difference-in-difference quasi-experiments generated
by a mover design, identifying mover average treatment effects under
conditional-on-covariate parallel trends and effect homogeneity restrictions. I
characterize the efficient estimators in this class and derive specification
tests based on the model's overidentifying restrictions. Future drafts will
apply the theory to the Finkelstein et al. (2016) movers design, analyzing the
causal effects of geography on healthcare utilization.
