Complier-Specific Effects with Multiple Instruments

Why Multiple Instruments Can Target Different Effects

An instrumental-variable estimate is not automatically an effect for every person whose treatment could change. Under the usual assumptions, a binary instrument identifies an average treatment effect among the people whose treatment status that instrument changes: its compliers. With multiple instruments, the people induced to take treatment by one instrument need not be the people induced by another. If treatment effects vary across people, the corresponding complier-specific effects can differ even when both instruments are valid.

The key is to define a complier group relative to a particular contrast. When instruments are considered one at a time, the contrast for one instrument should specify the values held fixed for the others. Otherwise, “the compliers for instrument A” can be ambiguous: A may change treatment for one group when B is low and for a larger group when B is high.

Potential Outcomes and Instrument-Specific Compliers

Let \(D\) be a binary treatment, with \(D=1\) denoting treatment and \(D=0\) control, and let \(Y\) be the outcome. Write \(Y(1)\) and \(Y(0)\) for a unit’s potential outcomes under treatment and control. The individual treatment effect is \(Y(1)-Y(0)\). Let the instrument vector be \(Z=(Z_1,\ldots,Z_K)\), with each instrument binary, and write \(D(z)\) for the treatment that would be received under instrument assignment \(z\).

For instrument \(j\), choose a fixed setting \(s\) for all the other instruments. The two assignments being contrasted are \(Z_j=0\) and \(Z_j=1\), with \(Z_{-j}=s\) in both cases. The instrument-specific potential treatments are \(D(0,s)\) and \(D(1,s)\). The group whose treatment is changed by this contrast is therefore defined for that instrument and that setting \(s\), not for the instrument in isolation from the contrast.

Definition: For instrument \(j\) at setting \(s\) of the other instruments, a unit is a complier if \(D(1,s)>D(0,s)\). It is an always-taker if both potential treatments equal 1, and a never-taker if both equal 0. A defier has \(D(1,s)<D(0,s)\). Under binary treatment and the monotonicity assumption \(D(1,s)\geq D(0,s)\) for every unit, there are no defiers for this contrast.

For example, with instruments A and B, the group \(D(1,0)>D(0,0)\) consists of the compliers for A when B is fixed at 0. The group \(D(1,1)>D(0,1)\) consists of the compliers for A when B is fixed at 1. These groups can differ. Monotonicity is a claim about each stated contrast; it does not mean that everyone responds to every instrument in the same way.

Assumptions and the Instrument-Specific LATE

To interpret a contrast as a causal effect, assumptions must apply to the particular instrument contrast being used. A clean version is a randomized joint assignment of the instruments. The relevant conditions are:

Consistency and no interference: the observed treatment and outcome equal the potential treatment and outcome under the assignments actually received, and one unit’s assignment does not change another unit’s outcome.
Independence: the randomized instrument vector is independent of the potential outcomes and potential treatments. In particular, within \(Z_{-j}=s\), assignment of \(Z_j\) is independent of those potential quantities.
Exclusion: the instruments affect the outcome only through treatment. Thus the potential outcome under instrument assignment \(z\) and treatment \(d\) is \(Y(d)\), not a separate function of \(z\).
Monotonicity: \(D(1,s)\geq D(0,s)\) for every unit in the contrast. This rules out defiers for that contrast.
Relevance: the contrast changes the probability of treatment, so \(E[D\mid Z_j=1,Z_{-j}=s]-E[D\mid Z_j=0,Z_{-j}=s]\neq 0\).

Under these conditions, the contrast identifies the local average treatment effect (LATE) among the compliers for that instrument and setting. Independence can be justified by random assignment; in a nonrandomized setting it requires a corresponding conditional independence argument, which is not supplied merely by having several instruments.

Theorem: Under consistency, no interference, independence, exclusion, monotonicity, and relevance for the contrast \((j,s)\), the conditional Wald ratio equals the average treatment effect among that contrast’s compliers:

\frac{E[Y\mid Z_j=1,Z_{-j}=s]-E[Y\mid Z_j=0,Z_{-j}=s]} {E[D\mid Z_j=1,Z_{-j}=s]-E[D\mid Z_j=0,Z_{-j}=s]} = E[Y(1)-Y(0)\mid D(1,s)>D(0,s)].

The denominator is the first-stage difference in treatment probability. The numerator is the reduced-form difference in mean outcome. Their ratio is not, in general, the average treatment effect for the full population, nor the effect for all people who happen to receive treatment.

Why the Wald Ratio Selects Compliers

The result follows by writing each assignment contrast as an average over the population. Independence lets the observed conditional means represent means of the relevant potential quantities. By exclusion, the outcome under each instrument assignment is the outcome under the treatment that assignment induces. Thus the population outcome contrast is

E\!\left[Y\{D(1,s)\}-Y\{D(0,s)\}\right].

Always-takers receive treatment under both assignments, and never-takers receive control under both, so neither group contributes to this difference. Compliers switch from control to treatment, contributing \(Y(1)-Y(0)\). Consequently,

\Delta_Y = E\!\left[(Y(1)-Y(0))\,\mathbf{1}\{D(1,s)>D(0,s)\}\right], \qquad \Delta_D = P\{D(1,s)>D(0,s)\}.

Here \(\Delta_Y\) is the reduced-form contrast, \(\Delta_D\) is the first-stage contrast, and \(\mathbf{1}\{\cdot\}\) equals 1 when its condition holds and 0 otherwise. Since the complier share is positive by relevance, dividing the outcome contrast by the treatment contrast gives the mean treatment effect within the complier group. This derivation also shows why monotonicity matters: if defiers are present, they contribute in the opposite direction, so the ratio need not be a complier average.

Worked Example: Two Instruments, Two Complier Groups

Consider a hypothetical population of 1,000 people, split equally into groups A and B. Treatment uptake follows \(D=z_A\) in group A and \(D=z_B\) in group B. Suppose assignment of the two instruments is randomized, neither instrument has a direct effect on the outcome, and the treatment effects are 2 outcome units for group A and 8 outcome units for group B. For clarity, suppose everyone’s untreated outcome is 10 units.

Hold B at 0 and compare A=1 with A=0. In group A, 500 people switch from untreated to treated; in group B, treatment remains 0. The treatment probability therefore rises from 0 to \(500/1{,}000=0.5\). Mean outcome rises from 10 to \(10+(500/1{,}000)(2)=11\). The Wald ratio is

\frac{11-10}{0.5-0}=\frac{1}{0.5}=2\text{ outcome units}.

This contrast identifies the effect for group A, the A-compliers at B=0. Now hold A at 0 and compare B=1 with B=0. The 500 people in group B switch treatment status, so the treatment probability again rises by 0.5. Mean outcome rises from 10 to \(10+(500/1{,}000)(8)=14\). Thus

\frac{14-10}{0.5-0}=\frac{4}{0.5}=8\text{ outcome units}.

Both instruments satisfy the stated validity conditions, yet their complier-specific effects differ. The reason is not a contradiction or invalidity: the instruments induce treatment for different groups, whose treatment effects differ.

Worked Example: A Complier Group Can Depend on the Other Instrument

In a second hypothetical population, half the people are type G and half type H. Their treatment uptake is \(D=z_A\) for type G and \(D=z_Az_B\) for type H. Assume randomized assignment, exclusion, and untreated outcomes of 10 units. Let the treatment effect be 2 units for G and 8 units for H. Each instrument is monotone in each of the contrasts below.

First compare A while holding B at 0. When A changes from 0 to 1, type G changes treatment from 0 to 1, while type H remains untreated. The first stage is 0.5. The outcome contrast is \((0.5)(2)=1\), so the Wald ratio is \(1/0.5=2\) outcome units: the effect for G.

Now compare A while holding B at 1. Both types change treatment from 0 to 1 when A changes from 0 to 1. The first stage is 1, and the outcome contrast is \((0.5)(2)+(0.5)(8)=5\). The Wald ratio is \(5/1=5\) outcome units. This contrast’s A-compliers include both G and H, so its LATE differs from the A-at-B=0 LATE.

For comparison, hold A at 1 and change B from 0 to 1. Type G is already treated at either B value; type H switches from untreated to treated. The first stage is 0.5, the outcome contrast is \((0.5)(8)=4\), and the Wald ratio is \(4/0.5=8\) outcome units. If instead A is held at 0, changing B does not change treatment for either type: the first stage is zero, so that contrast is not relevant and its Wald ratio is undefined.

This example checks relevance separately for each contrast. A is relevant at both stated B settings, but B is relevant at A=1 and irrelevant at A=0. “Valid instrument” does not mean “relevant in every conditional comparison.”

Worked Example: A Marginal Contrast Can Include Multiple Settings

Continue the hypothetical population in the previous example, and suppose B is randomized independently of A with probability 0.5 for each value. Consider the marginal contrast for A, averaging over the randomized distribution of B rather than fixing B at one value. For type G, changing A always changes treatment, so G contributes \(0.5\) of the population to the A-complier group. For type H, A changes treatment only when B=1, so H contributes \(0.5\times0.5=0.25\). The marginal first stage is therefore \(0.5+0.25=0.75\).

The corresponding outcome contrast is the contribution of each complier share times its treatment effect: \((0.5)(2)+(0.25)(8)=1+2=3\) outcome units. The marginal Wald ratio is

\frac{3}{0.75}=4\text{ outcome units}.

This is the mean treatment effect among those induced by A when B is distributed as in this randomized population: it includes all G and the H individuals who have B=1. It is neither the A-at-B=0 effect of 2 nor the A-at-B=1 effect of 5. The exact target depends on how the contrast handles the other instrument.

Interpretation and a Common Mistake

A complier-specific effect is a causal effect for a defined subgroup, but the subgroup is defined by potential responses to an instrument contrast. It is generally not directly observable which individuals are compliers, because each person reveals treatment under only the instrument assignment actually received. Randomization and the assumptions identify the group’s average effect through the ratio; they do not label each person’s latent type.

A common mistake is to say, “Both instruments are valid, so their estimates should equal the same treatment effect.” That conclusion silently assumes that the instruments identify the same group or that treatment effects are constant across the groups they induce. Validity supports a causal interpretation for each relevant contrast; it does not guarantee a shared complier population. When effects are heterogeneous, different valid instruments can correctly identify different local effects.

For any reported multiple-instrument LATE, specify which instrument changes, the values held fixed (or the distribution averaged over) for other instruments, the relevant first stage, and the resulting complier population. Without those details, the target effect is not fully described.

Check Your Understanding

Answer each question using the instrument contrast named in it.

For binary treatment, what potential-treatment inequality defines compliers for instrument \(j\) when the other instruments are fixed at \(s\)?
Why does the outcome contrast receive no contribution from always-takers or never-takers under exclusion?
In the second worked example, why does the A-complier group differ between B=0 and B=1?
If a conditional first-stage difference is zero, what does that imply for the Wald ratio in that contrast?
Can two valid instruments identify different LATEs? State the role of treatment-effect heterogeneity in your answer.