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Automatic debiased machine learning and sensitivity analysis for sample selection models

Jakob Bjelac, Victor Chernozhukov, Phil-Adrian Klotz, Jannis Kueck, Theresa M. A. Schmitz

TL;DR

The paper addresses causal inference when outcomes are missing non-randomly and treatment assignment is non-random by extending the Riesz representation to sample-selection models, enabling stable estimation and a transparent OVB decomposition. It develops ForestRiesz to learn the Riesz representer jointly with the outcome regression, avoiding unstable inverse-probability weighting and providing a Neyman-orthogonal, cross-fitted estimator. A key contribution is the decomposition of omitted-variable bias into a data-driven scale factor and two confounding strengths, with a quasi-Gaussian calibration to map latent selection confounding to interpretable bounds. Empirically, ForestRiesz yields larger ATEs for the gender wage gap than standard DML approaches, and sensitivity analysis shows the results are robust to plausible unobserved confounding, highlighting the method's practical value for reliable causal inference under sample selection. The framework thus offers a unified, robust approach to causal effects when outcome observability is selective and helps practitioners quantify bias risk in a transparent way.

Abstract

In this paper, we extend the Riesz representation framework to causal inference under sample selection, where both treatment assignment and outcome observability are non-random. Formulating the problem in terms of a Riesz representer enables stable estimation and a transparent decomposition of omitted variable bias into three interpretable components: a data-identified scale factor, outcome confounding strength, and selection confounding strength. For estimation, we employ the ForestRiesz estimator, which accounts for selective outcome observability while avoiding the instability associated with direct propensity score inversion. We assess finite-sample performance through a simulation study and show that conventional double machine learning approaches can be highly sensitive to tuning parameters due to their reliance on inverse probability weighting, whereas the ForestRiesz estimator delivers more stable performance by leveraging automatic debiased machine learning. In an empirical application to the gender wage gap in the U.S., we find that our ForestRiesz approach yields larger treatment effect estimates than a standard double machine learning approach, suggesting that ignoring sample selection leads to an underestimation of the gender wage gap. Sensitivity analysis indicates that implausibly strong unobserved confounding would be required to overturn our results. Overall, our approach provides a unified, robust, and computationally attractive framework for causal inference under sample selection.

Automatic debiased machine learning and sensitivity analysis for sample selection models

TL;DR

The paper addresses causal inference when outcomes are missing non-randomly and treatment assignment is non-random by extending the Riesz representation to sample-selection models, enabling stable estimation and a transparent OVB decomposition. It develops ForestRiesz to learn the Riesz representer jointly with the outcome regression, avoiding unstable inverse-probability weighting and providing a Neyman-orthogonal, cross-fitted estimator. A key contribution is the decomposition of omitted-variable bias into a data-driven scale factor and two confounding strengths, with a quasi-Gaussian calibration to map latent selection confounding to interpretable bounds. Empirically, ForestRiesz yields larger ATEs for the gender wage gap than standard DML approaches, and sensitivity analysis shows the results are robust to plausible unobserved confounding, highlighting the method's practical value for reliable causal inference under sample selection. The framework thus offers a unified, robust approach to causal effects when outcome observability is selective and helps practitioners quantify bias risk in a transparent way.

Abstract

In this paper, we extend the Riesz representation framework to causal inference under sample selection, where both treatment assignment and outcome observability are non-random. Formulating the problem in terms of a Riesz representer enables stable estimation and a transparent decomposition of omitted variable bias into three interpretable components: a data-identified scale factor, outcome confounding strength, and selection confounding strength. For estimation, we employ the ForestRiesz estimator, which accounts for selective outcome observability while avoiding the instability associated with direct propensity score inversion. We assess finite-sample performance through a simulation study and show that conventional double machine learning approaches can be highly sensitive to tuning parameters due to their reliance on inverse probability weighting, whereas the ForestRiesz estimator delivers more stable performance by leveraging automatic debiased machine learning. In an empirical application to the gender wage gap in the U.S., we find that our ForestRiesz approach yields larger treatment effect estimates than a standard double machine learning approach, suggesting that ignoring sample selection leads to an underestimation of the gender wage gap. Sensitivity analysis indicates that implausibly strong unobserved confounding would be required to overturn our results. Overall, our approach provides a unified, robust, and computationally attractive framework for causal inference under sample selection.
Paper Structure (24 sections, 1 theorem, 61 equations, 5 figures, 7 tables)

This paper contains 24 sections, 1 theorem, 61 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Identification under Confounding and Sample Selection
  3. Defining Causal Effects: The Potential Outcomes Framework
  4. The Sample Selection Problem
  5. Riesz Representers and Automatic Debiased Machine Learning
  6. Neyman-Orthogonal Scores and the Role of the Riesz Representer
  7. Riesz Representation Approach under Sample Selection
  8. Sensitivity Analysis
  9. Estimation
  10. Simulation Study
  11. Application
  12. Conclusion
  13. Proof of Theorem \ref{['Rieszrepresenter']}
  14. Omitted Variable Bias in Sample Selection Models
  15. Quasi-Gaussian Latent-Index Model for Selection
  16. ...and 9 more sections

Key Result

Theorem 1

Under the Assumptions A1, A4, confoundinD and A3, the Riesz representers of the long parameter $\theta_0$ and the short parameter $\theta_s$ are given by and where $p_d(X) := \mathbb{P}(D = d| X)$ is the propensity score for $d \in \{0,1\}$, $\pi_0(d, X, A) = \mathbb{P}(S = 1| D = d, X, A)$ accounts for selection in the long parameter, and $\pi_s(d, X) = \mathbb{P}(S = 1| D = d, X)$ accounts for

Figures (5)

  • Figure 1: Quasi-Gaussian calibration curve in a synthetic example. The horizontal axis shows values of the classical interpretable parameter, and the vertical axis shows values of the implied technical parameter.
  • Figure 2: This figure displays the distribution of ATE estimates based on $\theta_0 = 1$ and $200$ Monte Carlo iterations. It illustrates that as the sample size grows, the IRM suffers from an increasing downward bias, while both the SSM and FR converge to the simulated ATE.
  • Figure 3: This figure displays the distribution of SSM ATE estimates based on $\theta_0 =1$ and $200$ Monte Carlo iterations. Given the linearity of the DGP, it illustrates that the Lasso/Logistic specification of Bia2024 converges faster to the true ATE than the random forest specification with $max\_depth = 20$.
  • Figure 4: Sensitivity of the estimated gender wage gap (log wages) to potential omitted confounding. The plot compares the Riesz Representer ATE estimate from the full model with the counterfactual confidence interval that would arise if an unobserved confounder were as influential as marital status. While confidence intervals widen under this hypothetical confounder, the estimated ATE remains negative, indicating that the gender wage gap is robust to confounding of realistic magnitude.
  • Figure 5: Contour plot of bias bounds as a function of outcome sensitivity $\eta^2_{Y \sim A | D,X ,S=1}$ and selection sensitivity $\eta^2_{S^*\sim~ A |D, X}$. The figure shows how large an omitted confounder must be in terms of explanatory power for both wages and selection into observed wages to overturn the observed gender wage gap. Only confounders with combined sensitivity above the robustness threshold ($RV = 0.063$) could eliminate the estimated effect, implying strong robustness to selection and outcome confounding.

Theorems & Definitions (1)

  • Theorem 1