When the LATE Equals the ATE

Why the Complier Effect Need Not Be the Population Effect

An instrumental variable can identify an average treatment effect for compliers: the individuals whose treatment receipt changes when the instrument changes. The population average treatment effect, by contrast, averages over everyone in the target population. These averages refer to different groups, so they need not be equal. A particularly clear sufficient condition for equality is that treatment has the same effect on every individual. This tutorial proves that result and shows what changes when effects vary across people.

The question is not whether the instrument is relevant: relevance helps identify the complier effect. The question is whether the effect among compliers also represents the average effect across the whole population. That depends on how treatment effects are distributed across compliance types.

Definitions and Conditions

Let \(Z\in\{0,1\}\) be a binary instrument, \(D\in\{0,1\}\) a binary treatment, and \(Y\) the outcome. Write \(D(z)\) for the treatment an individual would receive if the instrument were set to \(z\). Under exclusion, write \(Y(d)\) for the outcome that individual would have if treatment were set to \(d\), regardless of the instrument value. The individual treatment effect is \(\Delta=Y(1)-Y(0)\).

The standard binary-instrument interpretation requires assumptions. Consistency means that observed treatment and outcome agree with the corresponding potential values. Instrument independence means \(Z\) is independent of the potential treatment receipts and outcomes, including \(D(0),D(1),Y(0),Y(1)\). Exclusion means the instrument affects the outcome only through treatment. Monotonicity means \(D(1)\geq D(0)\) for every individual, ruling out defiers. Relevance means the instrument changes the treatment probability, so \(P(D=1\mid Z=1)-P(D=1\mid Z=0)\ne 0\). These assumptions, together with a well-defined treatment and outcome, allow the Wald ratio to identify LATE.

Under monotonicity, individuals belong to three compliance types: compliers, always-takers with \(D(1)=D(0)=1\), and never-takers with \(D(1)=D(0)=0\). Let their population shares be \(p_C,p_A,p_N\), respectively, so \(p_C+p_A+p_N=1\). For each type \(g\), define its average treatment effect as \(\tau_g=E[\Delta\mid g]\). In particular, \(\tau_C=\mathrm{LATE}\) when \(p_C>0\).

Why Constant Individual Effects Guarantee Equality

The result follows directly from the definitions. If the individual treatment effect is the same value \(\tau\) for everyone in the target population, then averaging that effect over the whole population gives \(\tau\). Averaging it over the subset of compliers also gives \(\tau\). The instrument assumptions are needed to identify the complier effect from observed contrasts; the equality itself is a fact about the potential-outcome effects in the population.

The qualification “individual” matters. A constant effect across individuals is stronger than saying that an estimated regression coefficient, a conditional mean effect, or a group-level average is constant. Averages may be equal even when individuals have different effects, but that equality requires an additional relationship among the groups’ mean effects.

Deriving the Relationship Under Heterogeneity

Under monotonicity, the three compliance types partition the population. The law of total expectation therefore expresses the population mean effect as a share-weighted average of the type-specific mean effects. This decomposition makes precise how the population average can differ from the complier average.

This is the key justification. Since \(\mathrm{LATE}=\tau_C\), the ATE equals LATE if and only if

Constant individual effects make every type-specific mean equal, so both terms are zero. But equality can also arise if the weighted differences cancel. Conversely, if the effects for other types are, on balance, above the complier effect, the ATE is above LATE; if they are below, the ATE is below LATE. Effects can also vary within a compliance type: the decomposition uses each type’s average effect, not a claim that everyone in that type has the same effect.

Interpretation, Diagnostics, and Limitations

The equality theorem is useful because it states a sufficient condition under which an instrument’s complier effect also describes the population average effect. It does not say that a valid instrument makes the two estimands equal. Independence, exclusion, monotonicity, and relevance support the LATE interpretation; they do not impose constant treatment effects across compliance types.

A strong first stage is not evidence by itself that LATE equals ATE. It concerns how much treatment receipt changes with the instrument, not how the potential treatment effects of compliers compare with those of always-takers and never-takers. Likewise, a credible argument for random instrument assignment does not establish the effect-homogeneity condition.

In practice, constancy of individual treatment effects is generally not verified by comparing observed outcomes alone, because each individual reveals only one of \(Y(1)\) and \(Y(0)\). Researchers should therefore distinguish a result that follows under an explicitly defended constant-effect model from a result identified by the instrumental-variable assumptions. If effect homogeneity is not credible, the IV estimate should be interpreted as a complier effect, without silently relabeling it as the population average.

A common mistake is to reason, “The Wald ratio is valid, so it estimates the ATE.” The correction is that validity under the standard binary-instrument assumptions identifies LATE. To infer ATE equality, one needs a further condition, such as a constant individual treatment effect, or the exact weighted-balance condition in the decomposition. The balance condition is a mathematical characterization, not something guaranteed by instrument validity.

The result also depends on a defined target population and on the compliance-type partition used here. If the assumptions needed for the Wald interpretation fail, the ratio need not identify the complier effect in the first place. Even when those assumptions are plausible, the equality result alone does not tell us how precisely either effect is estimated from finite data.