  Suppose it is of interest to characterize effect heterogeneity of an
intervention across levels of a baseline covariate using only pre- and post-
intervention outcome measurements from those who received the intervention,
i.e. with no control group. For example, a researcher concerned with equity may
wish to ascertain whether a minority group benefited less from an intervention
than the majority group. We introduce the `subgroup parallel trends' assumption
that the counterfactual untreated outcomes in each subgroup of interest follow
parallel trends pre- and post- intervention. Under the subgroup parallel trends
assumption, it is straightforward to show that a simple `subgroup difference in
differences' (SDiD) expression (i.e., the average pre/post outcome difference
in one subgroup subtracted by the average pre/post outcome difference in the
other subgroup) identifies the difference between the intervention's effects in
the two subgroups. This difference in effects across subgroups is identified
even though the conditional effects in each subgroup are not. The subgroup
parallel trends assumption is not stronger than the standard parallel trends
assumption across treatment groups when a control group is available, and there
are circumstances where it is more plausible. Thus, when effect modification by
a baseline covariate is of interest, researchers might consider SDiD whether or
not a control group is available.
