Potential weights and implicit causal designs in linear regression

TL;DR

This paper formalizes the conditions under which linear regression can be interpreted as a causal effect estimator, introducing the concept of implicit designs that unify various causal inference frameworks.

Contribution

It introduces the concept of implicit designs in linear regression, providing a unified theoretical framework for causal interpretation across diverse settings.

Findings

Regression estimates correspond to contrasts of potential outcomes under implicit designs.

Implicit designs impose linear restrictions on the true treatment distribution.

The framework extends existing causal inference results to complex settings like multiple treatments and instrumental variables.

Abstract

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.