Generalized Difference-In-Differences Models: Robust Bounds

Abstract: The difference-in-differences (DID) method identifies the average treatment effects on the treated (ATT) under mainly the so-called parallel trends (PT) assumption. The most common and widely used approach to justify the PT assumption is the pretreatment  period examination. If a null hypothesis of the same trend in the outcome mean for both treatment and control groups in the pre-treatment periods is rejected, researchers believe less in PT and the DID results. This paper fills this gap by developing a generalized DID framework that utilizes all the information available not only from the pre-treatment periods but also from multiple data sources. Our approach interprets PT in a different way using a notion of selection bias, which enables us to generalize the standard DID estimand by defining an information set that may contain multiple pre-treatment periods or other baseline covariates. Our main assumption states that the selection bias in the post-treatment period lies within the convex hull of all selection biases in the pre-treatment periods. We provide a sufficient condition for this assumption to hold. Based on the baseline information set we construct, we first provide an identified set for the ATT that always contains the true ATT under our identifying assumption, and also the standard DID estimand. Secondly, we propose a class of criteria on the selection biases from the perspective of policymakers that can achieve a point identification of the ATT. Finally, we illustrate our methodology through some numerical and empirical examples.