What Exclusion Rules Out
An instrument is useful when its assignment changes treatment receipt without independently changing the outcome. The exclusion restriction formalizes the second part of that idea: holding treatment receipt fixed, changing the instrument must not change the outcome. It does not say that the instrument and outcome must be unrelated in the observed data. If the instrument changes treatment receipt, the resulting outcome can change through treatment.
Let \(Z\) be a binary instrument, with values 0 and 1, and let \(D\) be a binary treatment. Write \(D(z)\) for the treatment a person would receive if assigned instrument value \(z\). Write \(Y(z,d)\) for the outcome that person would have under instrument value \(z\) and treatment receipt \(d\). These are potential outcomes: typically, only one instrument value and one treatment receipt are observed for each person.
For example, if \(Z\) is an offer of a service, \(D\) is receipt of that service, and \(Y\) is a health outcome, exclusion requires the offer to have no effect on health for a person whose receipt of the service is fixed. An offer that also changes health through reassurance, extra clinical attention, or another pathway would not satisfy this assumption unless that pathway is part of the defined treatment.
From Potential Outcomes to the Observed Outcome
Under consistency, a person assigned \(Z\) receives \(D(Z)\), and their observed outcome is the potential outcome corresponding to that assignment and receipt:
Exclusion lets us replace the instrument-indexed outcome with the common treatment-indexed outcome. This substitution gives a precise account of the pathway restriction.
In particular, comparing a person’s potential outcome under the two instrument assignments gives
The equality follows because exclusion applies separately at the treatment level received under each assignment: \(Y(1,D(1))=Y(D(1))\) and \(Y(0,D(0))=Y(D(0))\). The two outcomes can differ if the assignments lead to different treatment receipt. Exclusion says that any such difference operates through treatment receipt, not that the instrument has no total effect on the outcome.
Worked Example: Exclusion Allows an Outcome Difference
Suppose a hypothetical individual has the following potential outcomes, measured in a health score. The person would not receive treatment under \(Z=0\), but would receive it under \(Z=1\): \(D(0)=0\) and \(D(1)=1\).
| Instrument value \(z\) | Outcome if \(d=0\), \(Y(z,0)\) | Outcome if \(d=1\), \(Y(z,1)\) |
|---|---|---|
| 0 | 150 | 142 |
| 1 | 150 | 142 |
At each fixed treatment level, the outcomes are equal across instrument values: \(Y(1,0)=Y(0,0)=150\), and \(Y(1,1)=Y(0,1)=142\). Exclusion holds for this individual. Under \(Z=0\), the observed outcome is \(Y(0,D(0))=Y(0,0)=150\). Under \(Z=1\), it is \(Y(1,D(1))=Y(1,1)=142\). The difference is \(142-150=-8\) health-score points. It arises because treatment receipt differs, not because the instrument changes the outcome while treatment is fixed.
Exclusion, Independence, and Relevance Are Different
Instrumental-variable reasoning involves distinct assumptions that answer distinct questions. Exclusion concerns the potential outcome at fixed treatment receipt. Independence concerns how the instrument is assigned in relation to potential outcomes and treatment receipt. Relevance concerns whether treatment receipt differs with the instrument.
For a simple randomized instrument, independence is expressed by saying \(Z\) is independent of the collection of potential treatment receipts and outcomes. Relevance is commonly described by a nonzero first-stage contrast, \(E[D\mid Z=1]-E[D\mid Z=0]\ne 0\). In a randomized assignment setting, that observed contrast reflects a difference in average potential treatment receipt; without independence, it may instead reflect differences between the groups assigned to each instrument value.
| Assumption | Question it answers | What it does not establish |
|---|---|---|
| Exclusion | At a fixed treatment level, would changing \(Z\) change \(Y\)? | That \(Z\) is independent of potential outcomes or that \(Z\) changes \(D\) |
| Independence | Is assignment of \(Z\) unrelated to the relevant potential outcomes and treatment receipts? | That \(Z\) changes treatment receipt or has no direct effect on \(Y\) |
| Relevance | Does treatment receipt differ across instrument values? | That the difference is unconfounded or that exclusion holds |
These assumptions do not substitute for one another. For instance, a randomly assigned instrument could satisfy independence and exclusion but fail relevance if nobody’s treatment receipt changes with assignment. Conversely, treatment receipt could be associated with the instrument (relevance in the observed first stage) even though assignment is related to prognosis or the instrument also affects the outcome directly. A strong first stage alone does not establish exclusion.
Worked Example: Checking Exclusion at Fixed Treatment Levels
Consider a hypothetical intervention offer \(Z\), treatment receipt \(D\), and outcome \(Y\) measured in symptom-score points. A researcher proposes the following potential-outcome model for one person:
If \(d=0\), then \(Y(0,0)=150\) and \(Y(1,0)=150-3=147\). If \(d=1\), then \(Y(0,1)=150-8=142\) and \(Y(1,1)=150-8-3=139\). Therefore, at either fixed treatment level, changing \(z\) lowers the score by 3 points. Since \(Y(1,d)\ne Y(0,d)\), exclusion fails.
The \(-3\) term represents a direct pathway from the offer to the outcome in this hypothetical model—for example, reassurance caused by receiving the offer, whether or not treatment is received. If this person also has \(D(0)=0\) and \(D(1)=1\), the total potential-outcome contrast is \(139-150=-11\) points. The model separates this into an 8-point change associated with treatment receipt and a 3-point direct change associated with the instrument. That decomposition is a feature of the stipulated model; observing a total difference alone would not establish it.
A Worked Check of Three Separate Conditions
Worked Example: A Randomized Instrument with Exclusion and Relevance
Suppose a hypothetical finite population has four people. The table gives each person’s treatment receipts under both instrument values and their outcome under each treatment value. Assignment is randomized independently of these potential outcomes.
| Person | \(D(0)\) | \(D(1)\) | \(Y(0)\) | \(Y(1)\) |
|---|---|---|---|---|
| A | 0 | 0 | 12 | 10 |
| B | 0 | 1 | 8 | 6 |
| C | 1 | 1 | 15 | 13 |
| D | 1 | 1 | 11 | 9 |
Define the instrument-indexed outcomes by \(Y(z,d)=Y(d)\) for every person and each \(z,d\). This construction satisfies exclusion: for each person, the outcome at a given treatment level is the same under either instrument value. Independence is justified here by the stated random assignment, not by the outcome table.
The average treatment receipt under \(Z=0\) across the four people’s potential receipts is
Under \(Z=1\), it is
Thus the population first-stage contrast is \(0.75-0.50=0.25\), or 25 percentage points. Relevance holds in this example. Random assignment supports independence, and the stipulated potential outcomes satisfy exclusion; the first-stage calculation addresses relevance. Each condition has been checked for a different reason.
Why Exclusion Is Substantive
The exclusion restriction is about unobserved counterfactual comparisons: for a given person and treatment level, it compares \(Y(1,d)\) with \(Y(0,d)\). Ordinarily, only one of those outcomes can be observed. Consequently, the assumption is not generally verified by comparing observed outcome averages across instrument groups. Such averages combine any direct effect with differences in treatment receipt and may also reflect differences in who received each instrument value.
Assessment therefore requires a defensible account of how the instrument could affect the outcome. Ask whether assignment changes access, expectations, co-interventions, behavior, or measurement in ways that could affect the outcome without operating through the specified treatment. If it does, exclusion may fail. Defining treatment narrowly can also create a problem: a pathway through a related component that is omitted from \(D\) is still a pathway outside the stated treatment.
A common mistake is to reason, “The instrument and outcome differ across groups, so exclusion is false.” That conclusion does not follow: a difference may be caused entirely by the instrument changing treatment receipt, which exclusion permits. The converse mistake is to say, “The first stage is large, so exclusion is credible.” Relevance only concerns treatment receipt; it does not rule out a direct outcome pathway. The correction is to assess exclusion by asking whether \(Y(1,d)=Y(0,d)\) at each fixed \(d\), and to evaluate independence and relevance separately.
Check Your Understanding
Answer each question using the distinction between a fixed-treatment comparison and an observed comparison across instrument groups.
- State the exclusion restriction using \(Y(z,d)\) for a binary instrument and treatment.
- If \(Y(1,0)=Y(0,0)\) but \(Y(1,1)\ne Y(0,1)\), does exclusion hold? Explain.
- Under exclusion, can the observed outcome differ between people assigned \(Z=1\) and people assigned \(Z=0\)? Give the reason.
- Why does a nonzero first-stage contrast not by itself establish exclusion?
- In a randomized-instrument setting, which assumption concerns assignment being independent of potential outcomes: exclusion, independence, or relevance?