Abstract
Recent work shows that popular partially-linear regression specifications can put negative weights on some treatment effects, potentially producing incorrectly-signed estimands. We show 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.
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