Structural Nested Mean Models Under Parallel Trends Assumptions

In this paper, we generalize methods in the Difference in Differences (DiD) literature by showing that both additive and multiplicative standard and coarse Structural Nested Mean Models (Robins, 1994, 1997, 1998, 2000, 2004; Lok and Degruttola, 2012; Vansteelandt and Joffe, 2014) are identified under parallel trends assumptions. Our methodology enables adjustment for time-varying covariates, identification of effect heterogeneity as a function of time-varying covariates, and estimation of treatment effects under a general class of treatment patterns (e.g. we do not restrict to the `staggered adoption' setting). We stress that these extensions come essentially for free, as our parallel trends assumption is not stronger than other parallel trends assumptions in the DiD literature. However, in contrast to much of the DiD literature, we only consider panel data, not repeated cross sectional data.