  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 pre-treatment period examination. If a null hypothesis of the
same trend in the outcome means for both treatment and control groups in the
pre-treatment periods is rejected, researchers believe less in PT and the DID
results. This paper develops a robust generalized DID method 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 provide an identified set for the ATT that always contains the
true ATT under our identifying assumption, and also the standard DID estimand.
We extend our proposed approach to multiple treatment periods DID settings. We
propose a flexible and easy way to implement the method. Finally, we illustrate
our methodology through some numerical and empirical examples.
