Negative Weights Are No Concern in Design-Based Specifications

Recent work shows that popular partially linear regression specifications can put negative weights on some treatment effects, potentially producing incorrectly signed estimands. We show that this is not an issue in design-based specifications, in which low-dimensional controls span the conditional expectation of the treatment. Specifically, the estimands of such specifications are convex averages of causal effects with ex ante weights that average the potentially negative ex post weights across possible treatment realizations. This result extends to design-based instrumental variable estimands under a first-stage monotonicity condition and applies to “formula” treatments and instruments such as shift-share instruments.