Why a Constant Treatment Effect Changes What an Instrument Identifies
An instrumental variable is often used when treatment uptake is not randomized: the instrument shifts who receives treatment, and the ratio of its effect on the outcome to its effect on treatment can recover a causal effect. Without additional assumptions, this ratio need not describe the average effect for the whole population. A strong alternative assumption changes the target: if every individual has the same treatment effect, a valid and relevant instrument identifies that common effect, even if some individuals move against the encouragement.
This result is useful because monotonicity of treatment uptake can be implausible in some settings. For example, an encouragement might increase uptake for some people but reduce it for others. Treatment-effect homogeneity does not make the instrument assumptions unnecessary; it supplies a different substantive route from the instrument-induced contrast to the population average treatment effect.
Potential Outcomes and the Assumptions
Consider a binary instrument \(Z\), with \(Z=1\) denoting the instrument or encouragement being present and \(Z=0\) its absence. Treatment \(D\) is binary, with \(D=1\) indicating receipt of treatment. Let \(D(z)\) denote the treatment an individual would receive under instrument value \(z\), and let \(Y(d)\) denote the outcome that individual would have under treatment status \(d\). The population average treatment effect is \(E[Y(1)-Y(0)]\).
We use the standard consistency and no-interference setup: each person's observed treatment is \(D=D(Z)\), and the observed outcome is the potential outcome corresponding to that treatment. The instrument has no direct effect on the outcome once treatment is fixed, as expressed by \(Y(d)\) rather than an outcome potential depending separately on \(z\). The assumptions needed for the result are:
- Instrument independence: \(Z\) is independent of \((Y(0),Y(1),D(0),D(1))\). In a randomized encouragement study, random assignment is one basis for this assumption.
- Exclusion: the instrument affects the outcome only through treatment. Thus, for a given treatment status \(d\), changing \(Z\) does not change the potential outcome \(Y(d)\).
- Relevance: \(E[D\mid Z=1]-E[D\mid Z=0]\ne 0\). The instrument must change the population's treatment uptake rate.
- Treatment-effect homogeneity: \(Y(1)-Y(0)=\tau\) for every individual, for a common constant \(\tau\).
The first three assumptions describe the instrument and its relationship to treatment and outcome. Homogeneity is a separate, strong assumption about the treatment effects themselves. Under it, the population average treatment effect is \(E[Y(1)-Y(0)]=\tau\). The instrument independence assumption concerns the joint collection of potential outcomes and potential treatments, not merely independence between \(Z\) and the observed treatment.
The Wald Ratio Under Homogeneous Effects
The numerator is called the reduced-form effect of the instrument on the outcome. The denominator is the first-stage effect of the instrument on treatment uptake. Both are differences between the \(Z=1\) and \(Z=0\) population means. The ratio is defined only when the first stage is nonzero.
To justify the result, homogeneity gives \(Y(d)=Y(0)+\tau d\) for \(d\in\{0,1\}\). By consistency and exclusion, \(Y=Y(0)+\tau D(Z)\). For each instrument value \(z\), instrument independence allows the conditional means of \(Y(0)\) and \(D(z)\) to equal their population means:
Subtract the expression for \(z=0\) from the expression for \(z=1\). The \(E[Y(0)]\) terms cancel. By consistency, \(E[D(z)]=E[D\mid Z=z]\), so the result is
Relevance makes the expression in braces nonzero. Dividing both sides by it gives the theorem. This argument also shows why instrument independence and exclusion matter: without them, the outcome contrast need not reduce to the effect of changing treatment uptake multiplied by \(\tau\).
Worked Example: Recovering an Effect in Outcome Units
Suppose a hypothetical randomized encouragement study compares people assigned to receive or not receive an encouragement to start a program. Treatment is program participation, and the outcome is a symptom score, measured in points, with lower scores indicating fewer symptoms. Suppose the population uptake rate is \(0.50\) in the encouraged arm and \(0.30\) in the other arm. The corresponding mean symptom scores are \(8.0\) and \(10.0\) points.
The first stage is \[ E[D\mid Z=1]-E[D\mid Z=0]=0.50-0.30=0.20. \] The reduced-form outcome difference is \[ E[Y\mid Z=1]-E[Y\mid Z=0]=8.0-10.0=-2.0\text{ points}. \] Thus the Wald ratio is \[ \frac{-2.0\text{ points}}{0.20}=-10.0\text{ points per treatment}. \]
If instrument independence, exclusion, relevance, and a constant individual treatment effect are valid, the identified population average treatment effect is \(-10.0\) symptom-score points for program participation versus nonparticipation. The negative sign indicates a reduction in the score. These assumed population contrasts are hypothetical; observed sample contrasts would estimate the population ratio rather than equal it exactly.
The assumptions require substantive justification. Random assignment of the encouragement supports instrument independence; the study design must also make it plausible that encouragement affects symptoms only by changing participation. The uptake difference of \(0.20\) satisfies relevance in this hypothetical population. Constant effects are not established by these four aggregate means: they are an additional assumption required for the population-average interpretation.
Why Monotonic Uptake Is Not Required Here
For each individual, the pair \((D(0),D(1))\) describes how treatment uptake responds to the instrument. A complier has \((0,1)\), a defier has \((1,0)\), and always-takers and never-takers have \((1,1)\) and \((0,0)\), respectively. Monotonicity rules out defiers by requiring \(D(1)\ge D(0)\) for everyone. The homogeneous-effects result does not impose that restriction.
The reason is visible in the proof: the instrument changes expected treatment by \(E[D(1)-D(0)]\), and, when every individual's effect is \(\tau\), its outcome contrast is \(\tau E[D(1)-D(0)]\). An individual's response to the instrument can contribute positively or negatively to the first stage, but the same \(\tau\) multiplies each contribution. The ratio therefore remains \(\tau\) if the net first stage is nonzero.
Worked Example: A Population With Both Compliers and Defiers
Suppose a hypothetical population consists of 40% compliers, 10% defiers, 30% never-takers, and 20% always-takers. Instrument assignment is independent of these response types and of the outcome potentials. Suppose exclusion holds and every individual's treatment effect is \(4\) outcome points.
When \(Z=0\), compliers and never-takers are untreated; defiers and always-takers are treated. The uptake rate is therefore \(0.10+0.20=0.30\). When \(Z=1\), compliers and always-takers are treated; defiers and never-takers are untreated. The uptake rate is \(0.40+0.20=0.60\). The first stage is \(0.60-0.30=0.30\), which is nonzero.
For example, let the population mean under no treatment be \(20\) points. Homogeneity means the outcome under treatment is \(4\) points higher for everyone. The outcome means in the two instrument arms are \[ E[Y\mid Z=0]=20+4(0.30)=21.2,\qquad E[Y\mid Z=1]=20+4(0.60)=22.4. \] The reduced form is \(22.4-21.2=1.2\) points, and the Wald ratio is \[ \frac{1.2\text{ points}}{0.30}=4\text{ points per treatment}. \]
Defiers are present, so monotonicity fails. Nevertheless, the ratio recovers the common treatment effect, which is also the population average treatment effect. This calculation relies on the stipulated independence, exclusion, and homogeneity assumptions; the presence of a nonzero first stage checks relevance.
This conclusion differs from the usual monotonicity-based complier interpretation. When treatment effects vary across individuals, a Wald ratio need not equal the population average treatment effect merely because the instrument is independent, excluded, and relevant. Under monotonicity, it has a complier-focused interpretation under the standard binary-instrument setup. Under homogeneity, the population average effect is identified instead because the effect is the same for compliers, defiers, and everyone else. Neither assumption should be silently substituted for the other.
From Population Identification to Sample Calculation
In data, the conditional means are estimated from the two instrument groups. The sample Wald estimate is the difference in sample outcome means divided by the difference in sample treatment proportions. It is a ratio of estimates; it does not make the assumptions true, and in a finite sample it need not equal the population parameter exactly. A near-zero estimated first stage also makes the ratio unstable: small changes in the estimated outcome difference can then produce large changes in the ratio.
Worked Example: Calculating the Sample Wald Estimate
In a hypothetical study, 500 participants are assigned to each instrument arm. In the \(Z=1\) arm, 210 receive treatment and the mean outcome is \(8.1\) points. In the \(Z=0\) arm, 140 receive treatment and the mean outcome is \(9.5\) points. Assume for purposes of interpretation that assignment is randomized, exclusion is credible, the population first stage is nonzero, and treatment effects are homogeneous.
The observed treatment proportions are \(210/500=0.42\) and \(140/500=0.28\), so the estimated first stage is \[ \widehat{E[D\mid Z=1]}-\widehat{E[D\mid Z=0]} =0.42-0.28=0.14. \] The observed outcome difference is \[ 8.1-9.5=-1.4\text{ points}. \] The sample Wald estimate is therefore \[ \widehat{\tau}_{W}=\frac{-1.4\text{ points}}{0.14} =-10.0\text{ points per treatment}. \]
The negative estimate indicates that treatment lowers the outcome by an estimated \(10.0\) points under the stated assumptions. The numerical calculation alone does not establish randomization, exclusion, or homogeneous effects. Nor does this example provide a confidence interval: quantifying sampling uncertainty requires an appropriate inferential procedure in addition to the identification argument.
Common Mistake: Treating the Wald Ratio as Automatically an ATE
- Wrong reasoning: “The instrument is relevant, so the Wald ratio is the population average treatment effect.” Relevance only says the instrument changes treatment uptake; it does not by itself establish independence, exclusion, or homogeneity.
- Correction: State the assumptions that justify the target. The population-average interpretation here requires instrument independence, exclusion, a nonzero first stage, and a constant individual treatment effect. If effects vary, the ratio does not automatically identify the population average.
- Wrong reasoning: “A defier means this result cannot apply.” The theorem does not require monotonicity. Under homogeneous effects, defiers are permitted; their uptake changes enter the first stage with the appropriate sign.
- Correction: Check the assumptions the result actually needs. Do not infer that homogeneity holds merely because the Wald ratio is calculable, and do not confuse an identifying assumption with a feature testable from the two arm means.
Homogeneity is powerful precisely because it is restrictive: it converts a contrast driven by instrument-induced changes in uptake into a common effect for the whole population. Its importance is not that it can be verified from the Wald calculation, but that it makes clear which substantive claim supports an average-effect interpretation when monotonicity is not imposed.
Check Your Understanding
Use the definitions and assumptions above to answer these questions.
- Write the population Wald ratio and identify which part is the reduced form and which part is the first stage.
- Which assumptions ensure that the outcome difference across instrument arms is the common treatment effect multiplied by the uptake difference?
- If the reduced-form difference is \(-3\) outcome points and the first stage is \(0.25\), what is the Wald ratio, with units?
- Why can the Wald ratio identify the population average treatment effect when defiers are present but effects are homogeneous?
- Does a nonzero first stage establish treatment-effect homogeneity? Explain.