When Do Treatment Changes Identify Causal Effects?

Abstract

This paper clarifies the identifying assumptions underlying causal inference based on treatment changes rather than levels, and their relationship to conventional identification strategies. We characterize two structural models, with non-nested assumptions, under which treatment-change identification is valid conditional on observed covariates by differencing out time-constant confounders that are additive in the treatment equation. The assumptions underlying treatment changes are generally not nested with those of methods relying on treatment levels, such as selection-on-observables strategies that control for past outcomes, treatments, and covariates, or difference-in-differences approaches that difference outcomes rather than treatments over time. We show, however, that under a random-walk restriction on the treatment process, exploiting treatment changes for identification is equivalent to using treatment levels given lagged treatment. This and other equivalence results motivate overidentification tests based on methods considering treatment levels and changes. Under an alternative model that does not assume a random walk but instead rules out dynamic treatment effects (among other conditions), treatment changes can still be used as an instrument to identify a treatment effect that is constant given covariates. However, without random walk, different identification strategies are generally not nested. In partially linear models, the non-nesting results carry a double robustness implication for two-way fixed effects regression that differences both the outcome and the treatment over time, which under certain conditions remains consistent if either the treatment-change assumption or the parallel-trends assumption holds. We characterize the causal models consistent with each method, run simulations for illustration, and present an empirical application to cigarette demand.