Direct Effects That Violate Exclusion

When the Instrument Changes Outcomes Apart From Treatment

An instrumental-variable argument needs more than an instrument that changes treatment uptake. The instrument must not also change the outcome through a route that bypasses treatment. That requirement is the exclusion restriction. If an encouragement provides useful information, an eligibility notice changes income, or an offer affects stigma or access even for people whose treatment status does not change, exclusion may fail. The outcome contrast between instrument groups then combines treatment effects with those additional effects.

The central consequence is specific: even if the instrument is randomized, relevant, and monotone, the Wald ratio need not equal the local average treatment effect (LATE) for compliers. It can contain a direct-effect contribution, which may inflate, attenuate, or even reverse the ratio. The issue is not simply whether the instrument is associated with outcomes; a valid instrument is expected to affect outcomes by changing treatment. The concern is an effect that remains when treatment is held fixed.

A direct effect here means any effect of changing \(Z\) on \(Y\) that does not operate through the treatment \(D\). For example, a program offer could increase knowledge among everyone who receives it, whether or not they enroll. If knowledge changes the measured outcome, the offer has an outcome pathway beyond enrollment. Similarly, an eligibility designation might alter disposable income or access to services without changing the treatment whose effect is being estimated.

To derive the consequence for the Wald ratio, assume a binary instrument and treatment, random assignment of \(Z\) independent of the potential treatment statuses and outcomes, monotonicity \(D_i(1)\ge D_i(0)\), and a nonzero first stage. Monotonicity rules out defiers. These assumptions identify the proportions of always-takers (A), never-takers (N), and compliers (C) in the population. Do not assume exclusion for the derivation below.

How Direct Effects Enter the Wald Ratio

For each compliance type, compare the outcome under \(Z=1\) with the outcome under \(Z=0\). Always-takers and never-takers keep the same treatment status across instrument values, so their outcome contrasts are direct effects at their fixed treatment levels. Compliers change treatment from 0 to 1. Their contrast includes both a treatment effect and a direct effect. Using the treatment effect at \(Z=1\), the complier contrast can be written as:

Random assignment makes the observed mean outcome difference by instrument value equal to the corresponding average potential-outcome contrast. Write \(p_A\), \(p_N\), and \(p_C\) for the proportions of the three compliance types. The first-stage difference is \(E[D\mid Z=1]-E[D\mid Z=0]=p_C\): only compliers change treatment. Define \(\tau_C(1)=E[Y_i(1,1)-Y_i(0,1)\mid C]\). The reduced-form outcome difference is then:

Dividing by the first stage gives the decomposition:

This is the key justification: the ratio equals a complier treatment contrast at \(Z=1\) plus a direct-effect term divided by the complier proportion. Under exclusion, every \(\delta_i(d)\) is zero, and the treatment contrast at \(Z=1\) is the usual complier treatment effect. If exclusion fails, the added term is generally not zero. It can be large when the direct effects are modest but compliers are rare, because the term is divided by \(p_C\). A different decomposition can use the treatment contrast at \(Z=0\) and direct effects at \(D=1\); either way, the observed ratio mixes pathways unless the relevant direct-effect contribution vanishes.

Examples of Direct-Effect Pathways

The pathway may be concrete even when it is difficult to measure. An information campaign can change behavior or knowledge without changing the treatment of interest. An offer with a cash payment can alter an income-related outcome even for someone who does not take up the treatment. A public eligibility label can create stigma, and a referral or authorization can improve access to other services. Each is a reason to ask what the instrument itself changes, not merely whether it predicts treatment.

What to Check, and What the Data Cannot Settle

The exclusion restriction concerns potential outcomes under instrument values that are not both observed for the same person. It is therefore not directly verified by comparing observed outcomes alone. Diagnostics can identify plausible pathways or contradictions, but passing a diagnostic does not establish exclusion.

• Map the instrument's actual consequences. List what assignment or receipt of the instrument changes besides treatment, including information, payments, labels, referrals, or access to other services. Ask whether each could affect the outcome while \(D\) is held fixed.
• Look for effects among people whose treatment status does not change. If a credible design or subgroup identifies an instrument effect on outcomes for non-takers, that is evidence against exclusion. In ordinary data, however, compliance types are not directly observed, so a simple comparison of observed non-takers does not automatically identify such an effect.
• Use negative-control outcomes carefully. An effect on an outcome that treatment could not plausibly affect may reveal a direct pathway or another design problem. A null result is not proof of exclusion: the chosen outcome may be insensitive, or direct effects may be specific to the primary outcome.
• Separate the reduced form from the Wald ratio. Report the instrument-outcome contrast and the instrument-treatment contrast. A weak first stage makes the ratio especially sensitive to even small direct effects, since their contribution is divided by the first stage.

A common mistake is to reason, “The instrument was randomized, so its effect on the outcome must be causal, and the Wald ratio is therefore the treatment effect.” Randomization supports a causal interpretation of the instrument's total effect, but that total effect includes every path from \(Z\) to \(Y\), not only the path through \(D\). The correction is to state the exclusion restriction separately and justify why assignment does not change the outcome at fixed treatment.

When a direct pathway is credible, the reduced-form effect of assignment can still be a meaningful policy effect of offering the instrument, under the design's other assumptions. It is not automatically the effect of treatment uptake. Alternatives include reporting the assignment effect without relabeling it as a treatment effect, or conducting a sensitivity analysis that specifies plausible bounds for direct effects and shows how those bounds alter the ratio. Such analyses depend on their stated bounds; they do not make exclusion true. If the direct effect is unknown and unrestricted, the observed reduced form and first stage alone cannot separate it from the treatment effect.