Treatment effect identification using alternative parallel assumptions

The core assumption to identify the treatment effect in difference-in-differences estimators is the so-called Parallel Paths assumption, namely that the average change in outcome for the treated in the absence of treatment equals the average change in outcome for the non-treated. We define a family of alternative Parallel assumptions and show for a number of frequently used empirical specifications which parameters of the model identify the treatment effect under the alternative Parallel assumptions. We further propose a fully flexible model which has two desirable features not present in the usual econometric specifications implemented in applied research. First, it allows for flexible dynamics and for testing restrictions on these dynamics. Second, it does not impose equivalence between alternative Parallel assumptions. We illustrate the usefulness of our approach by revising the results of several recent papers in which the difference-in-differences technique has been applied.The core assumption to identify the treatment effect in difference-in-differences estimators is the so-called Parallel Paths assumption, namely that the average change in outcome for the treated in the absence of treatment equals the average change in outcome for the non-treated. We define a family of alternative Parallel assumptions and show for a number of frequently used empirical specifications which parameters of the model identify the treatment effect under the alternative Parallel assumptions. We further propose a fully flexible model which has two desirable features not present in the usual econometric specifications implemented in applied research. First, it allows for flexible dynamics and for testing restrictions on these dynamics. Second, it does not impose equivalence between alternative Parallel assumptions. We illustrate the usefulness of our approach by revising the results of several recent papers in which the difference-in-differences technique has been applied