  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.
