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 counter by showing that negative weights are no problem 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.