Manski Worst-Case Bounds for Average Treatment Effects

Why Bound the Average Treatment Effect?

When treatment is not randomly assigned, the treated and untreated groups may differ in ways that also affect their outcomes. Their observed mean difference therefore need not equal the causal effect of treatment. If we do not impose assumptions that resolve this selection problem, we can still ask what values the population average treatment effect could take, given the observed data and a known range of possible outcomes.

Manski worst-case bounds answer that question using only the treatment share, the observed outcome means in each treatment group, and the outcome support. They make no assumption that treatment assignment is independent of potential outcomes. The result can be a wide interval, but its width honestly reflects how much the missing counterfactual outcomes leave unknown.

Consistency means that a treated individual’s observed outcome is their \(Y(1)\), while an untreated individual’s observed outcome is their \(Y(0)\). Thus the data reveal \(Y(1)\) for the treated and \(Y(0)\) for the untreated. They do not, in general, reveal \(Y(0)\) for the treated or \(Y(1)\) for the untreated. Manski bounds use the known support \([a,b]\) to limit these missing counterfactuals.

The support assumption must apply to both potential outcomes, including the unobserved ones. It is not enough that the outcomes actually observed in the data happen to fall between \(a\) and \(b\). The interval must be justified as a bound on the outcomes that could occur under either treatment status.

Deriving the Worst-Case Bounds

Split each population potential-outcome mean according to treatment status. The observed part is known, and the missing part is restricted only by the outcome support:

For the treated, the observed treated mean identifies \(E[Y(1)\mid D=1]=m_1\). For the untreated, \(E[Y(0)\mid D=0]=m_0\). By contrast, the conditional means of the unobserved counterfactuals are unknown. Since each lies in \([a,b]\), the smallest possible \(E[Y(1)]\) uses \(a\) for the untreated group’s missing \(Y(1)\), and the largest uses \(b\). For \(E[Y(0)]\), the smallest value uses \(a\) for the treated group’s missing \(Y(0)\), and the largest uses \(b\).

To minimize the difference \(E[Y(1)]-E[Y(0)]\), make the first mean as small as possible and the second as large as possible. To maximize it, do the reverse. This gives the Manski worst-case bounds:

The bounds can also be understood by first bounding the two population means separately. The lower bound subtracts the largest compatible control potential-outcome mean from the smallest compatible treatment potential-outcome mean. The upper bound subtracts the smallest compatible control mean from the largest compatible treatment mean. This separate optimization is valid because the assumptions place no restriction linking the two missing potential outcomes for an individual.

Worked Examples

Why the Bounds Are Sharp

A bound is sharp when every value in the stated identified set is compatible with the assumptions and observed information. The endpoint constructions in the third example show why the lower and upper limits cannot be tightened using only the stated restrictions: the unobserved potential outcomes can take support-endpoint values while leaving all observed outcomes unchanged.

The same reasoning applies generally. For the lower endpoint, set the unobserved \(Y(1)\) among untreated individuals to \(a\), and the unobserved \(Y(0)\) among treated individuals to \(b\). For the upper endpoint, set them to \(b\) and \(a\), respectively. These assignments attain the formulas’ endpoints without altering the observed treatment-specific means. Intermediate ATE values can also be generated by choosing intermediate values for the missing potential outcomes. Thus, absent further restrictions, the interval describes exactly the set of compatible ATE values—not merely a conservative outer range.

Common Mistake and Practical Interpretation

A frequent mistake is to report \(m_1-m_0\) as the ATE simply because it is the difference between the observed group means. Consistency identifies \(m_1\) as the mean of \(Y(1)\) among the treated and \(m_0\) as the mean of \(Y(0)\) among the untreated. But the ATE also depends on \(Y(1)\) among the untreated and \(Y(0)\) among the treated. Without an assumption addressing the relationship between treatment assignment and potential outcomes, the observed difference does not fill in those missing means.

The correction is to state what the data and support assumption imply: calculate the worst-case interval, then interpret its endpoints in the outcome’s units. If the interval includes zero, the assumptions do not determine even the sign of the ATE. If it lies entirely above or below zero, the sign is determined under these assumptions, though the exact effect may remain unknown. A narrower interval may require additional, defensible restrictions; it cannot be obtained just by treating the observed difference as causal.

In an application, the treatment share and group means are estimated from a sample. Substituting their sample counterparts gives estimated bound endpoints; sampling uncertainty in those estimates is a separate issue from the identification bounds themselves. The bounds describe uncertainty about which population ATE is compatible with the assumptions, not a confidence interval for an ATE.