  In many social science applications, researchers use the
difference-in-differences (DID) estimator to establish causal relationships,
exploiting cross-sectional variation in a baseline factor and temporal
variation in exposure to an event that presumably may affect all units. This
approach, often referred to as generalized DID (GDID), differs from canonical
DID in that it lacks a "clean control group" unexposed to the event after the
event occurs. In this paper, we clarify GDID as a research design in terms of
its data structure, feasible estimands, and identifying assumptions that allow
the DID estimator to recover these estimands. We frame GDID as a factorial
design with two factors: the baseline factor, denoted by $G$, and the exposure
level to the event, denoted by $Z$, and define effect modification and causal
interaction as the associative and causal effects of $G$ on the effect of $Z$,
respectively. We show that under the canonical no anticipation and parallel
trends assumptions, the DID estimator identifies only the effect modification
of $G$ in GDID, and propose an additional generalized parallel trends
assumption to identify causal interaction. Moreover, we show that the canonical
DID research design can be framed as a special case of the GDID research design
with an additional exclusion restriction assumption, thereby reconciling the
two approaches. We illustrate these findings with empirical examples from
economics and political science, and provide recommendations for improving
practice and interpretation under GDID.
