Potential weights and implicit causal designs in linear regression

Applied researchers routinely use linear regression to estimate causal effects, justified by quasi-experimental treatment variation, while leaving assumptions on treatment assignment implicit. We formalize a minimal criterion for quasi-experimental interpretation -- that the regression estimates some contrast of potential outcomes under the true assignment process, regardless of potential outcomes -- and characterize its implications for arbitrary regressions. This criterion implies linear restrictions on the true treatment distribution, whose solutions we call implicit designs. A regression is exactly quasi-experimental if and only if the true design is an implicit design, and approximately so when it is close to one, in a sense we formalize. Our framework unifies existing results and uncovers new ones across many settings. Qualitatively, an AI-assisted census of 1,051 recent papers finds quasi-experimental regression pervasive and often vulnerable to our negative results. Quantitatively, we assess exact and approximate quasi-experimental interpretation in nine studies by computing their implicit designs and estimands.