When Do Treatment Changes Identify Causal Effects?

This paper clarifies the identifying assumptions underlying causal inference based on treatment changes rather than treatment levels, and their relationship to conventional identification strategies. We characterize two distinct structural models, with non-nested identifying assumptions, under which treatment-change identification is valid conditional on observed covariates. We demonstrate that the identifying assumptions relying on 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, conditioning on treatment changes is equivalent to conditioning on treatment levels given lagged treatment. This and other equivalence results motivate overidentification tests by jointly considering methods based on treatment levels and changes. Beyond these tests, the non-nesting results carry a structural double robustness implication: an estimator that differences both the outcome and the treatment over time, such as two-way fixed effects regression, remains consistent if either the treatment-change assumption or the parallel-trends assumption holds, without requiring both simultaneously. We characterize the causal models consistent with each method, investigate finite-sample behavior in a simulation study, and present an empirical application to cigarette demand.