FIGURE 1.

Directed acyclic graph (DAG) for instrumented difference-in-differences (DID). Suppose the existence of an unmeasured confounder $U_t$ such that $\left(D_0,D_1\right)\perp \left(Y_0,Y_1\right)|U_0,U_1,\mathit{X}$. Assumption 2(a) states that $Z$ must be associated with the change in exposure $D_1-D_0$, Assumption 2(b) states that $Z$ is independent of any unmeasured confounders $U_0,U_1$ and cannot have any direct effect on the change in outcome $Y_1-Y_0$ and does not modify the treatment effect.