Quantifying Omitted Variable Bias in Nonlinear Instrumental Variable Estimators

1. Problem Statement

Instrumental Variable (IV) estimators are standard tools for addressing endogeneity in causal inference. However, their validity relies on the assumption that all relevant confounders are observed. When relevant variables are omitted (Omitted Variable Bias, or OVB), standard IV estimates become biased.

While sensitivity analysis for OVB is well-established in linear IV models (e.g., the work of Cinelli and Hazlett, 2022), there is a lack of rigorous frameworks for nonlinear IV estimators. This paper addresses this gap by developing a framework to quantify OVB in:

• Local Average Treatment Effect (LATE): The effect on compliers.
• LATE for the Treated (LATT): The effect on compliers who actually received treatment.
• Partially Linear IV Models (PLIVM): Models with partial linearity in covariates.

The core challenge is that these estimators are often ratios of expectations (e.g., $\theta = \lambda / \gamma$). OVB affects both the numerator (reduced form, $\lambda$) and the denominator (first stage, $\gamma$) in complex, non-linear ways, making direct bias decomposition difficult.

2. Methodology

The paper proposes a unified framework that extends the results of Chernozhukov et al. (2024a) to nonlinear settings. The methodology consists of four main components:

A. Bias Decomposition and Partial Identification

The authors define the true parameter $\theta$ and the short-form estimator $\theta_s$ (based only on observed covariates $W_s = (Z, X)$) as ratios:
$$\theta = \frac{\lambda}{\gamma}, \quad \theta_s = \frac{\lambda_s}{\gamma_s}$$
where $\lambda = E[\alpha(W)g_Y(W)]$ and $\gamma = E[\alpha(W)g_D(W)]$. Here, $\alpha$ is a weighting function (Riesz representer), and $g$ represents conditional expectations.

The bias is characterized by decomposing the difference between the long and short forms:
$$\lambda - \lambda_s = E[(\alpha - \alpha_s)(g_Y - g_{Ys})]$$
$$\gamma - \gamma_s = E[(\alpha - \alpha_s)(g_D - g_{Ds})]$$

Using the Cauchy-Schwarz inequality, the authors derive bounds for the bias magnitude:
$$|\lambda - \lambda_s| = |\rho_Y| C_Y C_\alpha S_Y$$
$$|\gamma - \gamma_s| = |\rho_D| C_D C_\alpha S_D$$
where:

• $\rho$: Correlation between the omitted variable's effect on the weight and the outcome/treatment.
• $S$: Standard deviations of residuals (estimable from data).
• $C$: Sensitivity parameters representing the "strength" of the omitted variable.

This leads to partial identification bounds for $\lambda$ and $\gamma$ (denoted $[\lambda^-, \lambda^+]$ and $[\gamma^-, \gamma^+]$).

B. Constructing OVB Bounds for $\theta$

The paper derives Theorem 1, which maps the bounds of $\lambda$ and $\gamma$ to bounds for the ratio $\theta_0$. The resulting identification set depends on the signs of the bounds for $\gamma$:

• If $\gamma$ is strictly positive or strictly negative, the bounds for $\theta$ are closed intervals derived from the ratios of the extreme values of $\lambda$ and $\gamma$.
• If the bounds for $\gamma$ straddle zero (i.e., $\gamma^- < 0 < \gamma^+$), the identification set for $\theta$ may be the entire real line or a union of disjoint intervals, rendering the estimate uninformative.

C. Sensitivity Parameters and Benchmarking

The sensitivity parameters ($C_\alpha, C_Y, C_D$) are unit-free measures of the omitted variable's predictive power.

• $C_\alpha^2$: Measures the gain in prediction precision for the instrument $Z$ when adding omitted variables.
• $C_Y^2$ and $C_D^2$: Measure the reduction in Mean Squared Error (MSE) for predicting $Y$ and $D$ when adding omitted variables.

To avoid arbitrary specification, the paper adopts a Benchmarking Analysis (following Cinelli and Hazlett, 2022). Researchers compare the omitted variable's strength to that of an observed covariate $X_j$. By setting a relative strength indicator $k=1$ (the omitted variable is as important as the strongest observed covariate), the authors calibrate the maximum plausible values for the sensitivity parameters.

D. Estimation and Inference (Double Machine Learning)

To handle high-dimensional covariates and complex functional forms, the paper employs Double Machine Learning (DML):

1. Nuisance Parameter Estimation: Uses $K$-fold cross-fitting and machine learning algorithms (e.g., Random Forests) to estimate conditional expectations ($E[Y|X]$, $E[D|X]$, propensity scores).
2. Orthogonality: Utilizes Neyman-orthogonal moment conditions (influence functions) to ensure the estimator is robust to errors in nuisance parameter estimation.
3. Confidence Intervals: Constructs OVB-adjusted Confidence Intervals using the Stoye (2009) shrinkage method. This method ensures uniform validity even when the identification region is narrow (i.e., when OVB is small), avoiding the "superefficiency" issues of standard inversion methods.

3. Key Contributions

1. Generalization to Nonlinear IV: The paper provides the first systematic framework for OVB analysis in nonlinear IV estimators (LATE, LATT, PLIVM), moving beyond the linear restrictions of previous literature.
2. Bias Decomposition: It derives explicit bias decompositions for the numerator and denominator of ratio estimators, linking them to sensitivity parameters that have clear interpretations (predictive power).
3. Robust Inference Tools: It develops a rigorous statistical inference procedure that produces OVB-adjusted confidence intervals, allowing researchers to test if causal conclusions hold under plausible levels of unobserved confounding.
4. Practical Benchmarking: It integrates the benchmarking approach to calibrate sensitivity parameters, making the analysis more accessible and grounded in empirical reality rather than arbitrary assumptions.

4. Empirical Results (JTPA Data)

The framework is applied to the U.S. Job Training Partnership Act (JTPA) experiment to estimate the impact of job training on earnings.

• Data: 11,204 adult workers (males and females).
• Instrument: Random assignment of job training offers.
• Findings:
  • First-Stage Robustness: The estimate of the compliance probability ($\gamma$, or $P(T=C)$) is highly robust to omitted variables for both genders. The bounds remain tight and positive even under strong sensitivity assumptions.
  • Treatment Effect Sensitivity:
    • Females: The LATE and LATT estimates remain statistically significant and robust even after accounting for OVB. The adjusted confidence intervals exclude zero.
    • Males: The LATE and LATT estimates are fragile. While point estimates are positive, the OVB-adjusted confidence intervals are wide and include zero, suggesting that unobserved confounding could explain the observed effects.
  • Gender Heterogeneity: The analysis highlights that program impacts are robust for females but fragile for males, a nuance that might be missed by standard point-estimate analysis without sensitivity checks.

5. Significance

This paper significantly advances the field of causal inference by providing a "stress test" for nonlinear IV estimates.

• For Applied Researchers: It offers a practical toolkit (DML + Benchmarking + Shrinkage CIs) to assess the credibility of causal claims in the presence of unobserved confounders.
• For Policy Analysis: The JTPA application demonstrates that ignoring OVB in nonlinear settings can lead to overconfident conclusions, particularly for subgroups where effects are weaker (e.g., males in this study).
• Theoretical Impact: By formalizing the relationship between omitted variables and Riesz representers in nonlinear settings, it bridges the gap between sensitivity analysis and modern machine learning-based econometrics.