Instrumented difference-in-differences

Abstract

Unmeasured confounding is a key threat to reliable causal inference based on observational studies. Motivated from two powerful natural experiment devices, the instrumental variables and difference-in-differences, we propose a new method called instrumented difference-in-differences that explicitly leverages exogenous randomness in an exposure trend to estimate the average and conditional average treatment effect in the presence of unmeasured confounding. We develop the identification assumptions using the potential outcomes framework. We propose a Wald estimator and a class of multiply robust and efficient semiparametric estimators, with provable consistency and asymptotic normality. In addition, we extend the instrumented difference-in-differences to a two-sample design to facilitate investigations of delayed treatment effect and provide a measure of weak identification. We demonstrate our results in simulated and real datasets.

Keywords: causal inference; effect modification; exclusion restriction; instrumental variables; multiple robustness.

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.

FIGURE 2

Changes in prevalence of cigarette smoking for men and women aged 20–29, lung cancer mortality rates for men and women aged 55–64 years among four successive 10-year birth cohorts: 1911–1920, 1921–1930, 1931–1940, 1941–1950