Policy Eligibility as an Instrument

Eligibility Is an Assignment Mechanism, Not the Treatment

A policy may make people eligible for a service without ensuring that they receive it. That distinction is exactly why eligibility can serve as an instrument: eligibility may shift treatment receipt, while receipt itself may be related to health, income, motivation, or other causes of the outcome. The central question is not whether eligible people have different outcomes. It is whether the policy rule supplies a credible source of variation in treatment that can be used to interpret an instrumental-variable contrast.

Let \(Z=1\) indicate that a person is eligible under the policy rule and \(Z=0\) indicate ineligibility. Let \(D=1\) indicate receipt of the treatment or service of interest and \(D=0\) indicate no receipt. Let \(Y\) be the outcome. These variables must refer to a clearly defined population, policy, treatment, and follow-up period. Eligibility is the proposed instrument; it is not interchangeable with treatment receipt.

The four possible compliance types are compliers \((0,1)\), who take treatment only when eligible; never-takers \((0,0)\), who do not take it under either eligibility status; always-takers \((1,1)\), who take it regardless; and defiers \((1,0)\), who take it when ineligible but not when eligible. These types are defined by potential behavior, not by a person’s observed status alone. For any one person, only one of \(D(0)\) and \(D(1)\) is observed.

What Must Be Plausible for Eligibility to Work?

An eligibility rule is a substantive claim about how policy creates variation, not a validity guarantee. The instrumental-variable interpretation needs assumptions about the rule, treatment, outcomes, and population. Consistency and no interference require that observed treatment and outcome agree with the relevant potential values and that one person’s eligibility or treatment does not alter another person’s outcome in a way that matters for the analysis. The treatment must also be defined clearly enough that materially different versions are not being treated as the same intervention.

In a policy setting, independence is often the most demanding assumption. Eligibility might be determined by age, income, disability, location, or prior need, and those characteristics can also predict treatment uptake or outcomes. Conditional independence can be plausible if the relevant causes are measured before eligibility and compared appropriately, but it does not justify an unadjusted comparison of all eligible and ineligible people. Exclusion also deserves a direct substantive argument: the eligibility designation may change outcomes through other services, financial support, information, or administrative contact, even when the person does not receive the treatment being studied.

Monotonicity is a claim about responses to eligibility. It may be reasonable when eligibility only opens access to a service and does not make anyone less likely to receive it. It is less plausible if eligibility changes incentives in opposite directions for different groups, or if ineligible people can obtain a substitute that eligible people cannot. Relevance should be assessed empirically, but a detectable first stage alone cannot establish independence or exclusion.

Wald Contrasts and Conditional Standardization

Under unconditional independence, exclusion, consistency, and monotonicity, the difference in mean treatment receipt between eligibility groups equals the proportion of compliers. The outcome difference equals that proportion multiplied by the average treatment effect among compliers. Their ratio is therefore the complier average causal effect, also called the local average treatment effect (LATE), when the first-stage difference is nonzero.

When independence holds only conditional on pre-eligibility covariates \(X\), these identities apply within covariate strata. Let \(\Delta_D(x)=E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]\) and \(\Delta_Y(x)=E[Y\mid Z=1,X=x]-E[Y\mid Z=0,X=x]\). To form an overall contrast, choose one common target distribution \(q(x)\) over covariate values in which both eligibility statuses are represented. Standardize each conditional contrast over that same distribution.

For continuous \(X\), the sums are replaced by integrals. To see the interpretation, conditional independence allows the observed within-\(x\) contrasts to identify the corresponding contrasts in potential values. With binary treatment and monotonicity, only compliers have \(D(1)\ne D(0)\), and each complier has \(D(1)-D(0)=1\). Thus:

Here \(C\) denotes the complier type. Standardizing gives \(\Delta_D^q=\sum_x q(x)P(C\mid X=x)\), while \(\Delta_Y^q=\sum_x q(x)P(C\mid X=x)E[Y(1)-Y(0)\mid C,X=x]\). If \(\Delta_D^q>0\), their ratio is a weighted average of covariate-specific complier effects, with weights proportional to \(q(x)P(C\mid X=x)\). It is the complier effect for the complier population induced by the chosen target distribution, not automatically the effect for all eligible people or the entire population.

Conditional independence does not generally justify the raw contrasts \(E[D\mid Z=1]-E[D\mid Z=0]\) and \(E[Y\mid Z=1]-E[Y\mid Z=0]\) as the two terms in this derivation. The covariate distributions can differ across eligibility groups. Unadjusted contrasts are justified by unconditional independence; under conditional independence, use conditional contrasts and a common standardization distribution, or an equivalent correctly specified adjustment.

Working Through Compliance Types

A policy’s first stage has a direct interpretation under monotonicity: it is the share of compliers, either overall under unconditional independence or within covariate strata under conditional independence. Never-takers and always-takers may be present, but their treatment status does not change with eligibility. This interpretation depends on the assumptions; the observed first stage by itself does not label any individual’s compliance type.

Substantive Checks and Common Failure Modes

A policy instrument is strongest when its assignment mechanism can be described precisely and the assumptions can be defended using institutional knowledge, not just statistical output. Useful checks include examining how eligibility is determined, confirming that the relevant covariates precede the rule, estimating the first stage in the intended population, and checking whether both eligibility statuses occur across the covariate strata used for adjustment. If some strata contain only eligible or only ineligible people, their within-stratum contrasts are not identified from observed data without additional modeling or extrapolation.

Covariate balance can reveal conspicuous differences, but balance on measured variables does not prove independence from unmeasured potential outcomes or treatment statuses. A strong first stage is also not evidence that exclusion holds. In addition, the treatment and outcome definitions should be audited for hidden pathways: if eligibility provides a separate benefit that affects the outcome, the exclusion restriction fails even if eligibility substantially changes treatment receipt.

The main alternative is to compare people receiving and not receiving treatment after adjustment for measured confounders, as in regression adjustment or propensity score methods. Those approaches require that the relevant treatment-outcome confounders be measured and controlled. An instrumental-variable analysis instead relies on the eligibility rule satisfying independence and exclusion and generally identifies an effect for compliers. A randomized encouragement design has the same conceptual distinction between assignment and receipt, but policy eligibility is not randomized merely because it is written as a rule.

A common mistake is to say, “Eligibility is independent of potential outcomes within each risk group, so I can compare all eligible people with all ineligible people.” That reasoning is incorrect because the two groups can contain different proportions of risk groups, and those groups can have different baseline treatment rates and outcomes. The correction is to use within-group contrasts and standardize both the outcome contrast and the treatment contrast over the same explicitly chosen distribution of pre-eligibility covariates. Another mistake is to call the resulting ratio “the policy’s effect on everyone.” It is a complier effect only under the assumptions, and its target depends on the standardization population.