Contamination Bias in Linear Regressions

We study regressions with multiple treatments and a set of controls that is flexible enough to purge omitted variable bias. We show that these regressions generally fail to estimate convex averages of heterogeneous treatment effects—instead, estimates of each treatment’s effect are contaminated by non-convex averages of the effects of other treatments. We discuss three estimation approaches that avoid such contamination bias, including the targeting of easiest-to-estimate weighted average effects. A re-analysis of nine empirical applications finds economically and statistically meaningful contamination bias in observational studies; contamination bias in experimental studies is more limited due to smaller variability in propensity scores.