When an Information Constraint Rules Out Hidden Confounding

The problem behind causal comparisons

To evaluate a training program, researchers would like to compare each person’s earnings with training and without it. These are potential outcomes, but only the outcome under the person’s actual treatment is observed. Differences between participants and non-participants can therefore reflect both the program and selection into it.

A standard assumption, called unconfoundedness, says that after conditioning on observed characteristics, treatment assignment provides no further information about the relevant potential outcome. In his November 2026 preprint, Unconfoundedness as an Information Constraint, University of Bonn economist Christoph Breunig develops a theoretical characterization of when an information restriction is equivalent to this assumption.

Bounding information about treatment

The paper uses mutual information, a measure of statistical dependence. Conditional mutual information measures dependence remaining after accounting for other variables.

Breunig’s restriction compares two quantities: the information about treatment contained in a potential outcome after conditioning on observed covariates, and the information about treatment contained in latent outcome heterogeneity without conditioning on those covariates. The first must not exceed the second.

For potential wages, latent heterogeneity can be represented by a person’s rank in the wage distribution conditional on observed characteristics. The restriction allows potential wages to remain informative about program participation, provided that this information does not exceed what the latent rank reveals about participation.

The comparison depends on the chosen latent representation. The baseline model assumes that latent heterogeneity is independent of the covariates and adds no information about assignment once the potential outcome and covariates are known. The latter condition holds automatically when those variables determine the latent heterogeneity.

An inequality with exact consequences

Under these maintained conditions, Breunig establishes that the information restriction is equivalent to a particular assignment structure: each treatment probability factorizes into a function of latent heterogeneity multiplied by a separate function of observed covariates.

Equivalently, latent heterogeneity and covariates must remain independent within every treatment group. Although written as an inequality, the information restriction holds with equality whenever it is satisfied in this model.

This structure does not universally eliminate hidden selection. One example involves non-participation and two training options: latent earnings heterogeneity determines participation, while observed characteristics influence which option participants receive. The information restriction can hold even though treatment remains confounded.

When variation eliminates hidden selection

The central result concerns variation in treatment probabilities across covariate values. For binary treatment, Breunig shows that any nonconstant propensity score—the probability of treatment given the covariates—makes the information restriction equivalent to unconfoundedness for the potential outcome under consideration.

The intuition is that the two assignment probabilities must sum to one. Within the multiplicative structure, variation in the propensity score forces the normalized latent factors to equal one, removing assignment’s dependence on latent heterogeneity conditional on covariates.

For multiple treatment states, the sufficient condition is full rank of the second-moment matrix of the propensity-score vector. This requires sufficiently rich variation across treatment probabilities, not merely some variation in one probability.

Importantly, relevance is distinct from overlap, which concerns whether assignment probabilities approach zero or one. Neither condition implies the other. Observable propensity-score variation alone does not establish unconfoundedness: the information restriction and maintained model conditions remain essential.

Restrictions for individual treatment states

The preprint also considers information restrictions concerning only one treatment state. These bound residual selection rather than necessarily eliminating it.

If that state’s assignment probability can approach one on sets of positive probability, the corresponding restriction becomes equivalent to conditional independence between the potential outcome and the indicator for that state. For binary treatment, this yields unconfoundedness for that potential outcome.

Breunig applies this result to the average treatment effect among participants. A restriction concerning non-participation and untreated potential earnings, combined with control probabilities approaching one and appropriate overlap, identifies that effect.

Scope and practical cautions

Extensions address additional conditioning variables, continuous treatments and instrument exogeneity. Continuous treatments require a completeness condition for full unconfoundedness; weaker pointwise relevance, together with stated regularity conditions, can identify an average potential outcome at a particular treatment value.

The paper includes simulations, not an empirical program evaluation. They illustrate that weighting can remain biased even with true propensity scores when residual confounding persists. They also show that the relevance test can overreject in smaller samples in the discrete-covariate design.

The contribution is therefore a conditional theoretical characterization, not an observable-data certificate that hidden confounding is absent.