Contamination Bias in Linear Regressions

We study the interpretation of regressions with multiple treatments and flexible controls. Such are often used to analyze stratified randomized control trials intervention arms, estimate value-added (for, e.g., teachers) observational data, leverage quasi-random assignment decision-makers (e.g. bail judges). show that these generally fail convex averages heterogeneous treatment effects, even when conditionally randomly assigned controls sufficiently avoid omitted variables bias. Instead, estimates each treatment's effects contaminated by a non-convex average other treatments. Thus, recent concerns about heterogeneity-induced bias in leveraging potential outcome restrictions parallel trends assumptions) also arise "design-based" identification strategies. discuss solutions contamination propose new class efficient estimators weighted In re-analysis Project STAR trial, we find minimal because effect heterogeneity is largely idiosyncratic. But sizeable arises becomes correlated propensity scores.