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. We also explain how to estimate optimal treatment regimes via optimal regime Structural Nested Mean Models under parallel trends assumptions plus an assumption that there is no effect modification by unobserved confounders. Finally, we illustrate our methods with real data applications estimating effects of bank deregulation on housing prices and effects of floods on flood insurance take-up.
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