  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.
