Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Potential weights and implicit causal designs in linear regression

Jiafeng Chen

TL;DR

The paper develops a design-based framework to interpret linear regressions with finite-valued treatments as quasi-experiments by introducing potential weights and implicit designs. It shows that a regression’s estimand is a design-based contrast of potential outcomes and that, when a valid implicit design exists, OLS equals an AIPW estimator, yielding a doubly robust interpretation. The authors provide a constructive procedure to compute implicit designs, diagnose their plausibility, and refine the estimand via reweighting, with practical diagnostics and empirical illustrations. They further extend the framework to two-stage least squares and highlight the fragility of quasi-experimental interpretations for interacted regressions and TWFE in panel data, offering guidance for transparent causal analysis.

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.

Potential weights and implicit causal designs in linear regression

TL;DR

The paper develops a design-based framework to interpret linear regressions with finite-valued treatments as quasi-experiments by introducing potential weights and implicit designs. It shows that a regression’s estimand is a design-based contrast of potential outcomes and that, when a valid implicit design exists, OLS equals an AIPW estimator, yielding a doubly robust interpretation. The authors provide a constructive procedure to compute implicit designs, diagnose their plausibility, and refine the estimand via reweighting, with practical diagnostics and empirical illustrations. They further extend the framework to two-stage least squares and highlight the fragility of quasi-experimental interpretations for interacted regressions and TWFE in panel data, offering guidance for transparent causal analysis.

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.
Paper Structure (34 sections, 10 theorems, 176 equations, 6 figures, 1 table)

This paper contains 34 sections, 10 theorems, 176 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Potential weights and implicit designs
  3. Core intuition
  4. General setup
  5. Takeaways for practitioners
  6. Theoretical applications and examples
  7. A unified analysis of quasi-experimental interpretation in regression
  8. Interactions and impossibility of regression estimation of ATE
  9. The fragility of quasi-experimental TWFE
  10. Extension: Two-stage least-squares
  11. Empirical illustration of diagnostics
  12. Diagnostics for the implicit design
  13. Known design
  14. Unknown design
  15. Refining the implicit design and implicit estimand
  16. ...and 19 more sections

Key Result

theorem 1

$\tau$ is minimally quasi-experimental if and only if When this happens, the estimand $\tau$ is equal to the implicit estimand under $\bm{\pi}$.

Figures (6)

  • Figure 1: Estimated implicit design by $x_i$ across two samples in cervellati2024random
  • Figure 2: Permutation tests for \ref{['item:M2']}
  • Figure 3: The distribution of the estimated implicit design for specification \ref{['eq:blakeslee_interact']}. 55 out of 786 observations (or 7% of the observations) are outside $[0,1]$.
  • Figure 4: Calibration of the implicit design. This is a binned scatterplot of $W_i$ on $\hat{\pi}_i$, with associated pointwise confidence intervals and uniform confidence bands cattaneo2024binscatter.
  • Figure 5: Alternative coefficient estimates for $\tau_1$ in \ref{['eq:blakeslee_interact']}
  • ...and 1 more figures

Theorems & Definitions (43)

  • theorem 1
  • theorem 2: Double robustness of $\Lambda \hat\beta$ under \ref{['item:M1']}
  • theorem 3
  • proof : Notes
  • proof : Notes
  • proof : Notes
  • proof : Notes
  • proof
  • proof : Proof of \ref{['thm:main']}
  • proof : Proof of \ref{['cor:binary_main']}
  • ...and 33 more