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Long Story Short: Omitted Variable Bias in Causal Machine Learning

Victor Chernozhukov, Carlos Cinelli, Whitney Newey, Amit Sharma, Vasilis Syrgkanis

TL;DR

The paper develops a general, nonparametric theory of omitted variable bias for causal parameters by expressing target functionals as linear functionals of the conditional expectation and leveraging Riesz representers. OVB is shown to equal the covariance between the regression error and the representer error, with sharp bounds that depend only on the latent confounders’ maximal explanatory power, reparameterized via $R^2$-like quantities to aid sensitivity analysis. The authors provide debiased machine learning-based inference for the bounds, enabling flexible estimation of the identifiable components and construction of valid confidence intervals. The framework is illustrated through an empirical 401(k) eligibility example and complemented by sensitivity contour plots and benchmarking procedures, demonstrating practical robustness checks against unobserved confounding. The approach unifies linear and nonlinear causal estimands under a single OVB methodology, with potential extensions to instrumental variables and dynamic settings.

Abstract

We develop a general theory of omitted variable bias for a wide range of common causal parameters, including (but not limited to) averages of potential outcomes, average treatment effects, average causal derivatives, and policy effects from covariate shifts. Our theory applies to nonparametric models, while naturally allowing for (semi-)parametric restrictions (such as partial linearity) when such assumptions are made. We show how simple plausibility judgments on the maximum explanatory power of omitted variables are sufficient to bound the magnitude of the bias, thus facilitating sensitivity analysis in otherwise complex, nonlinear models. Finally, we provide flexible and efficient statistical inference methods for the bounds, which can leverage modern machine learning algorithms for estimation. These results allow empirical researchers to perform sensitivity analyses in a flexible class of machine-learned causal models using very simple, and interpretable, tools. We demonstrate the utility of our approach with two empirical examples.

Long Story Short: Omitted Variable Bias in Causal Machine Learning

TL;DR

The paper develops a general, nonparametric theory of omitted variable bias for causal parameters by expressing target functionals as linear functionals of the conditional expectation and leveraging Riesz representers. OVB is shown to equal the covariance between the regression error and the representer error, with sharp bounds that depend only on the latent confounders’ maximal explanatory power, reparameterized via -like quantities to aid sensitivity analysis. The authors provide debiased machine learning-based inference for the bounds, enabling flexible estimation of the identifiable components and construction of valid confidence intervals. The framework is illustrated through an empirical 401(k) eligibility example and complemented by sensitivity contour plots and benchmarking procedures, demonstrating practical robustness checks against unobserved confounding. The approach unifies linear and nonlinear causal estimands under a single OVB methodology, with potential extensions to instrumental variables and dynamic settings.

Abstract

We develop a general theory of omitted variable bias for a wide range of common causal parameters, including (but not limited to) averages of potential outcomes, average treatment effects, average causal derivatives, and policy effects from covariate shifts. Our theory applies to nonparametric models, while naturally allowing for (semi-)parametric restrictions (such as partial linearity) when such assumptions are made. We show how simple plausibility judgments on the maximum explanatory power of omitted variables are sufficient to bound the magnitude of the bias, thus facilitating sensitivity analysis in otherwise complex, nonlinear models. Finally, we provide flexible and efficient statistical inference methods for the bounds, which can leverage modern machine learning algorithms for estimation. These results allow empirical researchers to perform sensitivity analyses in a flexible class of machine-learned causal models using very simple, and interpretable, tools. We demonstrate the utility of our approach with two empirical examples.
Paper Structure (43 sections, 9 theorems, 133 equations, 6 figures, 5 tables)

This paper contains 43 sections, 9 theorems, 133 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Warm-Up: Omitted Variable Bias in Partially Linear Models
  3. Problem set-up
  4. OVB as the covariance of approximation errors
  5. Further characterization of the bias
  6. Main Results: Omitted Variable Bias in Nonparametric Causal Models
  7. Problem set-up
  8. Omitted variable bias for linear functionals of the CEF
  9. Characterization of the OVB bounds
  10. Theoretical details for leading causal estimands
  11. Statistical Inference on the Bounds
  12. Omitted Firm Characteristics in Evaluating the Effects of 401(k) Plan.
  13. Estimates under conditional ignorability.
  14. Sensitivity analysis
  15. Minimal sensitivity reporting.
  16. ...and 28 more sections

Key Result

Theorem 1

Assume that $Y$ and $D$ are square integrable with: Then the OVB for the partially linear model of equations (eq:PLM1) - (eq:PLM2) is given by that is, it is the covariance between the regression error and the RR error. Furthermore, the squared bias can be bounded as where The bound $B^2$ is the product of additional variations that omitted confounders generate in the regression function and i

Figures (6)

  • Figure 1: Examples of different DAGs that imply $Y(d) \perp\!\!\!\!\perp D \mid \{X, A\}$.
  • Figure 2: Two possible causal DAGs for the 401(K) example.
  • Figure 3: Estimate (black), bounds (red), and confidence bounds (blue) for the ATE by income quartiles. Confounding scenario: $\rho^2 = 1$; $C^2_Y \approx 0.04$; $C^2_D\approx 0.03$. Significance level of $5\%$.
  • Figure 4: Sensitivity contour plots 401(k), PLM. Significance level $a=0.05$.
  • Figure 5: Sensitivity contour plots 401(k), NPM. Significance level $a=0.05$.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1: OVB and Sharp Bounds---PLM
  • Corollary 1: Interpreting OVB Bounds in Terms of $R^2$---PLM
  • Remark 1: Regularity Conditions for ATE and ACD
  • Lemma 1: Riesz Representation
  • Theorem 2: OVB and Sharp Bounds
  • Corollary 2: Interpreting OVB Bounds in Terms of $R^2$
  • Remark 2: Interpretation of $1-R^2_{\alpha \sim \alpha_s}$ for the ATE with a Binary Treatment
  • Remark 3: Interpretation of $1-R^2_{\alpha \sim \alpha_s}$ for Average Causal Derivatives
  • Example 1: Weighted Average Potential Outcome
  • Example 2: Weighted Average Treatment Effects
  • ...and 13 more