Doubly Robust Difference-in-Differences with General Treatment Patterns

We develop a difference-in-differences method in a general setting in which the treatment variable of interest may be non-binary and its value may change in each time period. It is generally difficult to estimate treatment parameters defined with the potential outcome given the entire path of treatment adoption, as each treatment path may be experienced by only a small number of observations. We propose an empirically tractable alternative using the concept of effective treatment, which summarizes the treatment path into a low-dimensional variable. Under a parallel trends assumption conditional on observed covariates, we show that doubly robust difference-in-differences estimands can identify certain average treatment effects for movers, even when the chosen effective treatment is misspecified. We consider doubly robust estimation and multiplier bootstrap inference, which are asymptotically justifiable if either an outcome regression function for stayers or a generalized propensity score is correctly parametrically specified. We illustrate the usefulness of our method by estimating the instantaneous and dynamic effects of union membership on wages.