Can the assumptions behind a causal claim survive the data?

How can researchers test a claim about outcomes they can never observe together? Someone assigned to a smoking-cessation program cannot simultaneously reveal what their smoking history would have been without it. Yet assumptions connecting those alternative histories can still impose restrictions on observable data.

In their preprint, Testing Exclusion and Shape Restrictions in Potential Outcomes Models, Hiroaki Kaido and Kirill Ponomarev develop a graph-based framework for deriving those restrictions. The contribution is a systematic way to check modeling assumptions—not a method that automatically establishes a causal explanation.

Making hidden assumptions testable

The paper studies potential responses: the outcomes or choices a person would exhibit under different inputs. An input might be a randomly assigned treatment or an instrument that influences treatment uptake. The framework assumes that the input is statistically independent of potential responses, possibly after conditioning on observed characteristics.

Researchers then restrict which response patterns are allowed. An exclusion restriction says that an input cannot affect an outcome through a particular route. For example, an instrument might influence an outcome only by changing treatment received. A monotonicity restriction rules out responses in the opposite direction: an encouragement might increase treatment uptake for some people but must not decrease it for anyone.

These assumptions concern individuals, not just averages. The authors express them as restrictions on the set of possible response types, where each type describes what would happen across alternative circumstances.

Their graph represents observable response–input combinations as vertices. Two vertices are connected when at least one permitted response type could generate both combinations under their respective inputs.

Collections of mutually unconnected vertices therefore represent incompatible possibilities. Under the independence assumption, their conditional probabilities must sum to no more than one. Enumerating these collections produces inequalities that researchers can test statistically.

For a simple example, suppose encouragement cannot reduce anyone’s treatment uptake. The probability of taking treatment without encouragement cannot exceed the probability of taking it with encouragement. More elaborate assumptions generate more elaborate checks.

When do the checks capture everything?

The graph inequalities are always necessary under the maintained assumptions. But they do not always capture every observable restriction of a model.

The authors establish completeness, which economists call sharpness, under two regularity conditions. First, combinations that are compatible pair by pair must also be jointly compatible. Second, the graph must belong to the mathematical class called perfect graphs.

When those conditions hold, satisfying all the inequalities means that some distribution of permitted response types can reproduce the observed distribution. It does not establish that those are the actual response types, or that the assumptions are uniquely correct.

The conditions cover many monotonicity models and, in the finite-response setting, all restrictions with a binary input. They can fail in important cases, including instrumental-variable models with three or more instrument values and exclusion alone. Such cases can require additional inequalities or direct tests using the permitted response types. The paper also develops extensions for continuous outcomes and observed characteristics.

A smoking-cessation illustration

The empirical application revisits the US Lung Health Study, which randomized 5,887 participants with early chronic obstructive pulmonary disease to usual care or intensive smoking-cessation therapy combined with either an inhaled bronchodilator or a placebo.

For the spouse analysis, the authors use 3,874 married units with complete observations. At the 5% significance level, their tests reject the hypothesis that treatment has no spillover effect on spouses’ smoking. They do not reject the weaker restriction that spillovers never increase smoking. The results are consistent with beneficial spillovers, but do not establish that every spouse benefits or identify the mechanism.

They also test an exposure classification that treats having a spouse in either intervention arm as the same indirect exposure. Neither that classification nor its combination with a restriction that greater exposure cannot increase smoking is rejected.

A separate analysis of 5,738 participants rejects no treatment effect on the initial smoking-free spell across five annual surveys. It does not reject the ordering that cessation lasts at least as long under placebo-backed therapy as under control, and at least as long with the bronchodilator as with placebo. Non-rejection is not proof of that ordering.

Practical promise, bounded conclusions

The authors report computing the application’s collections of incompatible events within three seconds per hypothesis. That is not a universal speed guarantee: the number of inequalities can grow exponentially. Simulations also show that some violations remain difficult to detect, although conditioning on observed characteristics or using richer instruments can improve detection.

This account uses the supplied full-paper text extraction of an arXiv preprint, not an independently verified replication. The extraction omits figures and may damage equations, tables and layout; no OCR was used. The tests rely on maintained independence assumptions and statistical procedures with large-sample guarantees. Their value is to expose observable contradictions in causal models—not to turn a model that survives testing into a proven description of reality.