Potential weights and implicit causal designs in linear regression

When we interpret linear regression as estimating causal effects justified by quasi-experimental treatment variation, what do we mean? This paper formalizes a minimal criterion for quasi-experimental interpretation and characterizes its necessary implications. A minimal requirement is that the regression always estimates some contrast of potential outcomes under the true treatment assignment process. This requirement implies linear restrictions on the true distribution of treatment. If the regression were to be interpreted quasi-experimentally, these restrictions imply candidates for the true distribution of treatment, which we call implicit designs. Regression estimators are numerically equivalent to augmented inverse propensity weighting (AIPW) estimators using an implicit design. Implicit designs serve as a framework that unifies and extends existing theoretical results on causal interpretation of regression across starkly distinct settings (including multiple treatment, panel, and instrumental variables). They lead to new theoretical insights for widely used but less understood specifications.