Potential weights and implicit causal designs in linear regression

When do linear regressions estimate causal effects in quasi-experiments? This paper provides a generic diagnostic that assesses whether a given linear regression specification on a given dataset admits a design-based interpretation. To do so, we define a notion of potential weights, which encode counterfactual decisions a given regression makes to unobserved potential outcomes. If the specification does admit such an interpretation, this diagnostic can find a vector of unit-level treatment assignment probabilities -- which we call an implicit design -- under which the regression estimates a causal effect. This diagnostic also finds the implicit causal effect estimand. Knowing the implicit design and estimand adds transparency, leads to further sanity checks, and opens the door to design-based statistical inference. When applied to regression specifications studied in the causal inference literature, our framework recovers and extends existing theoretical results. When applied to widely-used specifications not covered by existing causal inference literature, our framework generates new theoretical insights.