Outstrumental Variable Regression

Estimating the causal effect of a treatment $X$ on an outcome $Y$ from observational data in the presence of unobserved confounding requires additional structure or assumptions. Existing identification strategies such as instrumental variable re- gression exploit exogenous variation in only the treatment . We show that the causal effect can also be identified if the outcome mechanism varies across a finite set of environments, even if the treatment distribution is unchanged. Concretely, we introduce Outstrumental Variable Regression (OVR), which proceeds in three steps: (i) compute the conditional expectation of $Y$ given $X$ in each environ- ment; (ii) form the vector of differences in these conditional expectations between all environments and a reference environment (we call this vector contrast); and (iii) regress Y on X while adjusting for the contrast and environment. We prove that this yields the true causal effect if the following two conditions hold: inclusion restriction, i.e., the contrast determines the conditional confounding bias; and pos- itivity, i.e., the treatment varies sufficiently strongly after adjustment. We prove that a finite-sample version of OVR is consistent under regularity conditions and demonstrate the procedure’s effectiveness on synthetic and real-world data.