Generalizing Difference-in-Differences to Non-Canonical Settings: Identifying an Array of Estimands

TL;DR

This paper extends the Difference-in-Differences methodology to a variety of data structures and assumptions, identifying multiple causal estimands beyond the canonical setting.

Contribution

It characterizes the conditions under which the generalized DiD (gDiD) formula identifies meaningful causal effects across diverse data and assumption configurations.

Findings

Identifies multiple causal estimands under different data structures.

Provides conditions for the validity of gDiD in non-canonical settings.

Discusses empirical and structural validation of parallel trends assumptions.

Abstract

Consider a general setting in which data on an outcome is collected in two `groups' at two time periods, with certain group-periods deemed `treated' and others `untreated'. A special case is the canonical Difference-in-Differences (DiD) setting in which one group is treated only in the second period while the other is treated in neither period. Then it is well known that under a parallel trends assumption across the two groups the classic DiD formula (subtracting the average change in outcome across periods in the treated group by the average change in the outcome across periods in the untreated group) identifies the average treatment effect on the treated in the second period. But other relations between group, period, and treatment are possible. For example, the groups might be demographic (or other baseline covariate) categories with all units in both groups treated in the second period and none treated in the first, i.e. a pre-post design. Or one group might be treated in both periods while the other is treated in neither. Furthermore, other parallel trends assumptions under other treatment regimes are possible. For example, we could assume the two groups' potential outcomes would evolve in parallel under a regime of `do not switch treatment in the second period'. In fact, there is a literal array of data structures and parallel trends assumptions. The difference between the changes in outcomes of the two groups, which we dub the `group DiD' (gDiD) formula, identifies different causal estimands depending on the data structure and parallel trends assumption adopted. Here, we determine under which combinations of data structure and assumptions the gDiD formula identifies meaningful causal estimands. We also explore when parallel trends assumptions are amenable to empirical check or structural justification via Single World Intervention Graphs.