Potential weights and implicit causal designs in linear regression

Jiafeng Chen

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Abstract: When we interpret linear regression as estimating causal effects justified by quasi-experimental treatment variation, what do we mean? This paper characterizes the necessary implications when linear regressions are interpreted causally. A minimal requirement for causal interpretation is that the regression estimates some contrast of individual potential outcomes under the true treatment assignment process. This requirement implies linear restrictions on the true distribution of treatment. Solving these linear restrictions leads to a set of implicit designs. Implicit designs are plausible candidates for the true design if the regression were to be causal. The implicit designs serve as a framework that unifies and extends existing theoretical results across starkly distinct settings (including multiple treatment, panel, and instrumental variables). They lead to new theoretical insights for widely used but less understood specifications.

Date: 2024-07, Revised 2025-07
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