  Under what circumstances is it a threat to the parallel trends assumption
required for Difference in Differences (DiD) studies if treatment decisions are
based on past values of the outcome? We explore via simulation studies whether
parallel trends holds across a grid of data generating processes generally
conducive to parallel trends (random walk, Hidden Markov Model, and constant
direct additive confounding), study designs (never treated, not yet treated, or
later treated control groups), and outcome responsiveness of treatment (yes or
no). We interpret the upshot of our simulation results to be that parallel
trends is typically not a credible assumption when treatments are influenced by
past outcomes. This is due to a combination of regression to the mean and
selection on future treatment values, depending on the control group. Since
timing of treatment initiation is frequently influenced by past outcomes when
the treatment is targeted at the outcome, perhaps DiD is generally better
suited for studying unintended consequences of interventions?
