Potential weights and implicit causal designs in linear regression

2024-07-30

AI Summary

When linear regression is used to estimate treatment effects in quasi-experiments, its causal interpretation rests on implicit assumptions—specifically, under what conditions does the regression coefficient represent a comparable contrast of individual potential outcomes? Method: We formally introduce the concept of “latent weights” to characterize regression’s implicit weighting of unobserved counterfactuals; derive necessary linear constraints on treatment assignment for causal interpretability; and define the “implicit causal design set,” unifying and extending existing theoretical frameworks. Our approach integrates design-based inference, counterfactual modeling, and linear constraint analysis. Contribution: We establish a necessary conditions framework for causal interpretation of regression, provide operational transparency diagnostics, and deliver novel theoretical justification for widely used—but previously under-justified—regression specifications, including covariate-adjusted regression.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

When we interpret linear regression estimates as causal effects justified by quasi-experiments, what do we mean? This paper characterizes the necessary implications when researchers ascribe a design-based interpretation to a given regression. To do so, we define a notion of potential weights, which encode counterfactual decisions a given regression makes to unobserved potential outcomes. A plausible design-based interpretation for a regression estimand implies linear restrictions on the true distribution of treatment; the coefficients in these linear equations are exactly potential weights. Solving these linear restrictions leads to a set of implicit designs that necessarily include the true design if the regression were to admit a causal interpretation. These necessary implications lead to practical diagnostics that add transparency and robustness when design-based interpretation is invoked for a regression. They also lead to new theoretical insights: They serve as a framework that unifies and extends existing results, and they lead to new results for widely used but less understood specifications.

Problem

• Characterize implications of causal linear regression interpretation
• Identify implicit designs for true causal treatment assignment
• Unify theoretical results across diverse regression settings

Innovation

• Implicit designs unify diverse causal settings
• Linear restrictions solve treatment distribution constraints
• Potential outcomes contrast defines causal interpretation