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.

翻译：当我们将线性回归解释为由准实验处理变异所证实的因果效应估计时，我们究竟意指什么？本文形式化了准实验解释的一个最小准则，并刻画了其必然蕴含。一个最基本的要求是：在真实处理分配过程下，回归必须始终估计潜在结果的某种对比。这一要求意味着对真实处理分布的线性约束。若要将回归作准实验性解释，这些约束便暗示了真实处理分布的候选形式，我们称之为隐含设计。回归估计量在数值上等价于使用隐含设计的增强逆倾向加权（AIPW）估计量。隐含设计作为一个统一框架，能够融合并拓展现有关于回归因果解释的理论成果——涵盖差异显著的多种情境（包括多重处理、面板数据及工具变量）。该框架为广泛使用但理解尚浅的设定提供了新的理论见解。