When Many AI-Generated Controls Can Help Regression—and When They Cannot

The problem: an important control is missing

Researchers estimating a treatment effect may need to account for something they cannot directly observe, such as a patient’s underlying health. If that characteristic influences both treatment assignment and outcomes, a regression that ignores it can produce a misleading coefficient.

In Regression with Many Generated Controls, Timothy Christensen and Daniel Vebman examine whether many imperfect measurements can substitute for this missing control. Their paper, posted as arXiv preprint version 2611.06855v1 on November 4, 2026, develops mathematical results and illustrates them with simulations. Its conclusions are conditional on a statistical model, not evidence that arbitrary collections of AI outputs eliminate confounding.

The motivating possibilities include ratings generated by different language-model prompts, personas or random seeds, alongside human annotations and conventional machine-learning predictions. The target is the coefficient on an observed explanatory variable—not a coefficient on the generated measurements themselves.

More measurements do not automatically remove bias

The authors model each proxy as a function of an observation-specific latent variable, a proxy-specific component and independent noise. Different proxies can therefore encode the latent information in different, potentially nonlinear ways.

A central assumption is that the latent variable would adequately control for confounding if it were observed. The framework also imposes independence conditions, moment requirements and enough remaining variation in the explanatory variables after adjustment. Generating more measurements does not establish these assumptions.

Within this framework, the authors compare ordinary least squares using every proxy with regression using principal components extracted from the proxy matrix. Principal component analysis, or PCA, compresses the measurements into factors representing shared variation.

Under the paper’s respective regularity and growth conditions, these two approaches converge to the same population target. PCA can reduce noise and the number of regressors, but it does not automatically correct a mismatch between the available factors and the adjustment required. A tighter estimate can still center on the wrong coefficient.

Why flexible adjustment matters

The distinction turns on two conditions the authors call “strong richness” and “weak richness.”

Strong richness requires the latent-variable-dependent mean of the regression error to be approximable by linear combinations of the factors revealed by the proxies. It is sufficient for consistency of both linear approaches, alongside their other assumptions.

Weak richness asks instead whether the factors contain enough information to recover that mean through a potentially nonlinear function. This can hold even when strong richness fails.

For example, suppose every proxy measures the same latent variable with a different slope and some noise. Together, the measurements can reveal that variable. But if the confounding component depends on its square, controlling only linearly for the recovered factor does not represent the necessary adjustment. A flexible function of the factor can.

The authors therefore recommend a partially linear approach: extract principal components, flexibly predict both the outcome and the explanatory variables from those components, and regress the resulting outcome residuals on the explanatory-variable residuals. They prove consistency under weak richness and additional conditions ensuring that factor estimation and the flexible prediction steps work adequately as the samples grow.

This is not a universal ranking of bias. Outside the richness conditions, the paper notes that the bias magnitudes of the linear and partially linear approaches are not generally ordered.

Instrumental variables require different caution

The paper also examines using one proxy as an instrumental variable for another. The intuition is to isolate their shared signal while removing measurement noise.

Its detailed two-proxy results show that the limiting estimate generally depends on which proxies are selected. For consistency across almost every proxy choice in an endogenous setting, the shared proxy signal must have a restrictive one-factor structure, together with a condition eliminating residual bias. Diverse relationships with the latent variable, which can help regression-based adjustment, therefore create difficulties for this IV strategy. The result concerns the studied setup, not every instrumental-variables method.

What the simulations establish

The simulations use 500 repetitions for each of four designs ranging from 250 observations and 50 proxies to 5,000 observations and 1,500 proxies.

In the squared-confounding example, the partially linear estimator approaches zero bias while the linear all-proxy and factor regressions retain their predicted asymptotic bias. In a binary-confounder example, all three have zero asymptotic bias, but the partially linear estimates are better centered in the reported smaller samples. The two-proxy IV estimates show substantially greater dispersion and remain biased.

These are illustrations, not real-world evaluations of generated controls. The authors also acknowledge that their simulation factor counts exceed the underlying signal rank and thus fall outside their stated factor-selection conditions. Their practical message remains conditional: sufficiently informative proxies, used with an appropriate method, can permit consistent estimation without validation data—but quantity alone is not enough.