Difference-in-Differences with Multiple Time Periods and an Application on the Minimum Wage and Employment

Difference-in-Differences (DID) is one of the most important and popular designs for evaluating causal effects of policy changes. In its standard format, there are two time periods and two groups: in the first period no one is treated, and in the second period a "treatment group" becomes treated, whereas a "control group" remains untreated. However, many em- pirical applications of the DID design have more than two periods and variation in treatment timing. In this article, we consider identification and estimation of treatment effect parameters using DID with (i) multiple time periods, (ii) variation in treatment timing, and (iii) when the "parallel trends assumption" holds potentially only after conditioning on observed covariates. We propose a simple two-step estimation strategy, establish the asymptotic prop- erties of the proposed estimators, and prove the validity of a computationally convenient bootstrap procedure. Furthermore we propose a semiparametric data-driven testing procedure to assess the credibility of the DID design in our context. Finally, we analyze the effect of the minimum wage on teen employment from 2001-2007. By using our proposed methods we confront the challenges related to variation in the timing of the state-level minimum wage policy changes. Open-source software is available for implementing the proposed methods.