  Recently, there has been a surge in methodological development for the
difference-in-differences (DiD) approach to evaluate causal effects. Standard
methods in the literature rely on the parallel trends assumption to identify
the average treatment effect on the treated. However, the parallel trends
assumption may be violated in the presence of unmeasured confounding, and the
average treatment effect on the treated may not be useful in learning a
treatment assignment policy for the entire population. In this article, we
propose a general instrumented DiD approach for learning the optimal treatment
policy. Specifically, we establish identification results using a binary
instrumental variable (IV) when the parallel trends assumption fails to hold.
Additionally, we construct a Wald estimator, novel inverse probability
weighting (IPW) estimators, and a class of semiparametric efficient and
multiply robust estimators, with theoretical guarantees on consistency and
asymptotic normality, even when relying on flexible machine learning algorithms
for nuisance parameters estimation. Furthermore, we extend the instrumented DiD
to the panel data setting. We evaluate our methods in extensive simulations and
a real data application.
