Front-Door Difference-in-Differences Estimators

We develop front-door difference-in-differences estimators as an extension of front-door estimators. Under one-sided noncompliance, an exclusion restriction, and assumptions analogous to parallel trends assumptions, this extension allows identification when the front-door criterion does not hold. Even if the assumptions are relaxed, we show that the front-door and front-door difference-in-differences estimators may be combined to form bounds. Finally, we show that under one-sided noncompliance, these techniques do not require the use of control units. We illustrate these points with an application to a job training study and with an application to Florida's early in-person voting program. For the job training study, we show that these techniques can recover an experimental benchmark. For the Florida program, we find some evidence that early in-person voting had small positive effects on turnout in 2008. This provides a counterpoint to recent claims that early voting had a negative effect on turnout in 2008.