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 idiosyncratic effect heterogeneity.

翻译：我们研究了包含多个处理变量且控制变量集足够灵活以消除遗漏变量偏差的回归模型。研究表明，这些回归通常无法估计异质性处理效应的凸性均值——相反，每个处理效应的估计值会受到其他处理效应非凸性均值的污染。我们讨论三种能够避免此类污染偏差的估计方法，包括针对最易估计的加权平均效应。对九项实证应用案例的重新分析发现，观察性研究中存在经济且统计意义上显著的污染偏差；由于异质性效应具有特异性，实验研究中的污染偏差更为有限。