Proof That the Wald Ratio Identifies Complier LATE

Why the Wald Ratio Has a Specific Target

An instrument changes treatment take-up for some individuals but not necessarily for everyone. The Wald ratio divides the difference in mean outcomes across instrument assignments by the difference in treatment rates across those assignments. The ratio is therefore not automatically the average treatment effect for the whole population. Under the instrumental-variable assumptions, it identifies an average effect for the individuals whose treatment is changed by the instrument: the compliers.

The key to the result is to examine the two assignment contrasts at the level of potential outcomes. Independence lets us interpret the observed contrasts as contrasts in potential assignment outcomes. Exclusion removes any direct effect of assignment on the outcome. Monotonicity then implies that individuals whose treatment does not change contribute zero to both contrasts. Relevance ensures that the remaining group has positive probability, so the ratio is defined.

Setup and Assumptions

Let \(Z\) be a binary instrument, with values 0 and 1, and let \(D\) be a binary treatment. For each individual, \(D(z)\) is the treatment that would be received under instrument value \(z\), and \(Y(d)\) is the outcome that would be observed under treatment value \(d\). Assume these potential outcomes are well-defined, that one individual's assignment does not affect another individual's outcome, and that observed treatment and outcome satisfy consistency: \(D=D(Z)\) and, under exclusion, \(Y=Y(D)\).

The four assumptions play different roles. Independence connects assignment-specific observed means to potential outcomes. Exclusion specifies which potential outcome is observed when treatment is set. Monotonicity rules out individuals who would take treatment under \(Z=0\) but not under \(Z=1\). Relevance rules out a zero denominator. For the identification result, these assumptions concern the population; sample contrasts are estimates of the population contrasts, not the assumptions themselves.

With binary \(Z\) and \(D\), monotonicity leaves three possible treatment-response types: compliers, who take treatment only under \(Z=1\); always-takers, who take it under either assignment; and never-takers, who take it under neither assignment. Relevance does not say that everyone complies. It says that the assignment changes treatment take-up for a positive proportion of the population.

From Assignment Contrasts to Potential Outcomes

Define the population reduced-form assignment contrast and the first-stage treatment contrast as

By consistency and exclusion, an individual assigned \(Z=z\) has observed treatment \(D(z)\) and observed outcome \(Y(D(z))\). Independence makes the distribution of the potential variables the same across the two assignment groups. Consequently, the assignment-specific observed means equal the corresponding population means of potential assignment outcomes:

These equalities do not claim that the same person is observed under both assignments. They use the independence assumption to identify each assignment group's observed mean with a mean of potential outcomes in the population.

Proof: Only Compliers Contribute

Let \(I_C\) equal 1 for a complier and 0 otherwise. Under monotonicity, \(D(1)-D(0)\) is 1 for a complier and 0 for an always-taker or never-taker. Thus \(D(1)-D(0)=I_C\) for every individual. For the outcome contrast, if treatment does not change, the outcome under the two assignments is the same under exclusion. If treatment changes, monotonicity says the individual is a complier, and the outcome difference is \(Y(1)-Y(0)\). In one identity covering all cases,

Taking expectations gives the two central expressions:

Relevance says \(\Delta_D\ne 0\). Since monotonicity implies \(\Delta_D=P(C)\geq 0\), relevance therefore gives \(P(C)>0\). For any integrable variable \(A\), the indicator identity \(E[I_C A]=P(C)E[A\mid C]\) holds when \(P(C)>0\). Apply it to \(A=Y(1)-Y(0)\). The result is

This proof also covers the case \(P(C)=1\). Then \(I_C=1\) for every individual, the first stage is 1, and the outcome contrast is \(E[Y(1)-Y(0)]\). The Wald ratio is the average treatment effect among compliers because the entire population consists of compliers. No conditional expectation given a zero-probability noncomplier group is needed anywhere in the derivation.

Interpretation and a Common Mistake

The result is an identification statement: under the assumptions, the population ratio of assignment contrasts equals a particular causal estimand. It does not say that the Wald ratio equals the ATE in every population. The ATE averages effects over compliers and noncompliers, while this ratio weights only individuals whose treatment status responds to the instrument. Those two averages coincide in special cases, such as when everyone complies or when the groups' average effects happen to be equal, but neither equality follows from the IV assumptions alone.

A common mistake is to reason, “The instrument is relevant and independent, so the Wald ratio is the treatment effect for everyone.” The correction is that relevance says only that treatment rates differ by assignment, while independence makes the assignment contrast interpretable. Exclusion and monotonicity then yield the complier-specific result; none of these assumptions requires the complier effect to equal the effects for always-takers or never-takers.

The target population is instrument-specific: it consists of those who would change treatment status under the two instrument values being compared. This is why the causal interpretation must name compliers rather than silently generalizing to the full population. The next step in using the result is to calculate assignment contrasts from a concrete compliance-type distribution while keeping that target clear.