Controlling for Unmeasured Confounding in the Presence of Time: Instrumental Variable for Trend

Unmeasured confounding is a key threat to reliable causal inference based on observational studies. We propose a new method called instrumental variable for trend that explicitly leverages exogenous randomness in the exposure trend to estimate the average and conditional average treatment effect in the presence of unmeasured confounding. Specifically, we use an instrumental variable for trend, a variable that (i) is associated with trend in exposure; (ii) is independent of the potential exposures, potential trends in outcome and individual treatment effect; and (iii) has no direct effect on the trend in outcome and does not modify the individual treatment effect. We develop the identification assumptions using the potential outcomes framework and we propose two measures of weak identification. In addition, we present a Wald estimator and a class of multiply robust and efficient semiparametric estimators, with provable consistency and asymptotic normality. Furthermore, we propose a two-sample summary-data Wald estimator to facilitate investigations of delayed treatment effect. We demonstrate our results in simulated and real datasets.