What LATE Does and Does Not Represent

Introduction

An instrumental-variable estimate can have a clear causal interpretation without describing the effect for everyone. With a binary instrument and treatment, the standard Wald ratio identifies a local average treatment effect (LATE): an average treatment effect for people whose treatment status is changed by the instrument. That population is defined by the particular instrument and its effect on treatment receipt.

This distinction matters when interpreting results. An effect for instrument-induced compliers is not automatically the average effect for the whole population, nor does it tell us the effect for people who would always take treatment or would never take it. To make those distinctions precise, we will state the relevant assumptions and relate the Wald ratio to compliance types.

What LATE Refers To

Let $Z\in\{0,1\}$ be a binary instrument and $D\in\{0,1\}$ a binary treatment. For each unit, $D(z)$ is the treatment status that would occur if the instrument were set to $z$, and $Y(d)$ is the outcome that would occur if treatment were set to $d$. The individual treatment effect is $Y(1)-Y(0)$. The principal stratum, or compliance type, is determined by the pair $(D(0),D(1))$.

Definition: A complier has $D(0)=0$ and $D(1)=1$: the unit receives treatment when encouraged by $Z=1$, but not when $Z=0$. The local average treatment effect is $$ \mathrm{LATE}=E[Y(1)-Y(0)\mid D(0)=0,D(1)=1]. $$ It is the mean treatment effect among compliers for this instrument and population.

The other possible types are always-takers, with $D(0)=D(1)=1$, and never-takers, with $D(0)=D(1)=0$. A defier has $D(0)=1$ and $D(1)=0$: the unit does the opposite of the instrument’s encouragement. These labels describe potential treatment responses, not observed behavior alone. For a person observed with $Z=1$ and $D=1$, for example, the data do not by themselves reveal whether that person is a complier or an always-taker.

The usual LATE interpretation relies on assumptions about the instrument. In the unconditional version used here, instrument independence means $Z$ is independent of the potential treatments and outcomes. The exclusion restriction means the instrument affects the outcome only through treatment, so the relevant outcome under assignment $z$ is $Y(D(z))$, not an outcome with a separate direct effect of $z$. Relevance requires that the instrument change the probability of treatment. Monotonicity rules out defiers: $D(1)\geq D(0)$ for every unit. Consistency links observed treatment and outcome to the corresponding potential values.

Under these assumptions, the Wald ratio is the difference in mean outcomes by instrument status divided by the difference in mean treatment rates:

\frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]} =\mathrm{LATE}, \qquad E[D\mid Z=1]-E[D\mid Z=0]>0.

Why the Ratio Is Local

The key is that, under monotonicity, changing $Z$ changes treatment only for compliers. Always-takers receive treatment under either instrument value, while never-takers receive no treatment under either value. Given independence and exclusion, the instrument-induced difference in mean outcomes therefore comes only from compliers. The first stage—the difference in treatment rates—also measures the proportion of compliers.

\begin{aligned} E[D\mid Z=1]-E[D\mid Z=0] &=P(\text{complier}),\\ E[Y\mid Z=1]-E[Y\mid Z=0] &=P(\text{complier})\, E[Y(1)-Y(0)\mid \text{complier}]. \end{aligned}

Dividing cancels the complier proportion, leaving the average effect within that stratum. This identifies an average for the people whose treatment responds to this instrument. It does not make the instrument a random sample of the population’s treatment effects, and it does not reveal the treatment effects of the other types.

Worked Example: A Complier Effect Different From the Population Average

Consider a hypothetical population in which 25% are always-takers, 50% are compliers, and 25% are never-takers. Suppose the mean treatment effects in these three groups are, respectively, 20, 4, and 22 outcome units. Assume instrument independence, exclusion, relevance, and monotonicity hold.

First stage. Under $Z=1$, both always-takers and compliers receive treatment, for a treatment rate of $0.25+0.50=0.75$. Under $Z=0$, only always-takers receive treatment, for a rate of $0.25$. Thus the first stage is $0.75-0.25=0.50$, equal to the complier proportion.

Reduced form. Only compliers change treatment when $Z$ changes, so the outcome difference by instrument status is the complier share times the complier mean effect:

0.50(4)=2.00\text{ outcome units}.

The Wald ratio is therefore $2.00/0.50=4.00$ outcome units. It identifies the mean effect for compliers, not for every person.

The population average treatment effect instead averages effects across all three types:

\mathrm{ATE} =0.25(20)+0.50(4)+0.25(22) =5.00+2.00+5.50 =12.50\text{ outcome units}.

Here the LATE is 4.00 outcome units while the ATE is 12.50 outcome units. Both are averages of treatment effects, but they average over different populations. Knowing the LATE alone does not determine the ATE; the effects among always-takers and never-takers also matter.

What LATE Does Not Say

The word “local” is about the population to which the average refers, not a guarantee that the effect is small or geographically local. The complier group is defined by potential responses to the instrument. A different encouragement, eligibility rule, or study population can induce a different set of compliers. Consequently, a LATE has no automatic interpretation as the effect for all people who might receive treatment under a policy.

It is not generally the ATE. The ATE averages $Y(1)-Y(0)$ over the whole population. LATE averages it only over compliers. They coincide under additional conditions, such as a common treatment effect across compliance types, but validity of the instrument alone does not imply those conditions.
It is not the effect for always-takers or never-takers. Under monotonicity, those types do not change treatment with the instrument, so the Wald comparison does not isolate their treatment effects.
It is not every complier’s individual effect. LATE is a conditional mean. Individual effects among compliers can vary, and the average need not be the effect for any particular person.
It is not automatically transportable. Applying the estimate to another setting requires a reason to expect the relevant effects and complier population to support that interpretation there.

Worked Example: An Average Does Not Describe Each Complier

Suppose a hypothetical instrument satisfies independence, exclusion, relevance, and monotonicity. Among four compliers, the individual treatment effects are 1, 3, 5, and 7 outcome units. These are the only units whose treatment changes with $Z$; always-takers and never-takers may also be present.

The complier average is

\mathrm{LATE} =\frac{1+3+5+7}{4} =\frac{16}{4} =4\text{ outcome units}.

If the four compliers are equally represented, the first stage is proportional to four compliers and the reduced form sums their four effects in the corresponding population comparison. The ratio yields their mean of 4.00, but none of these four individual effects is 4.00. Thus reporting a LATE of 4 outcome units does not mean that every complier gains 4 units, or that a particular person’s effect is known. It summarizes the complier group.

Why Monotonicity Matters

The standard interpretation depends on ruling out defiers. If defiers exist, increasing $Z$ makes some units less likely to receive treatment. Their contributions to the reduced form and first stage have the opposite sign from those of compliers. The Wald ratio then need not equal the average effect for compliers.

Worked Example: A Wald Ratio When Defiers Are Present

In a hypothetical population, suppose 40% are compliers, 10% are defiers, 20% are always-takers, and 30% are never-takers. Assume independence and exclusion, and suppose the mean treatment effect is 5 outcome units for compliers and 2 for defiers. For this calculation, monotonicity does not hold because the defier group is nonempty.

Compliers increase treatment receipt when $Z$ changes from 0 to 1, while defiers decrease it. Always-takers and never-takers do not change treatment. The first stage is therefore

$$ 0.40-0.10=0.30. $$

The reduced form adds the compliers’ effect contribution and subtracts the defiers’ contribution:

0.40(5)-0.10(2)=2.00-0.20=1.80\text{ outcome units}.

The Wald ratio is $1.80/0.30=6.00$ outcome units. That is neither the complier mean effect of 5 nor the defier mean effect of 2. In this example it is a signed combination induced by the different directions of treatment response. Without monotonicity, the usual claim that the Wald ratio is a complier average is not justified.

Assumptions and Interpretation in Practice

A numerical Wald estimate is not, by itself, evidence that its assumptions hold. Independence concerns how the instrument is assigned or generated; exclusion rules out a direct pathway from the instrument to the outcome; relevance concerns a nonzero treatment-rate change; and monotonicity rules out units who respond in the opposite direction. These assumptions are substantive claims about the design and behavior, not features that can be verified just by dividing two sample differences.

Even when the assumptions are plausible, describe the estimand precisely. State the treatment, outcome, instrument, population, and the group of compliers induced by that instrument. For example, an estimate in hours of employment should be described as the average change in employment hours caused by treatment among compliers in the stated population—not as the effect “on employment” for everyone. A strong first stage may improve precision, but it does not turn a complier effect into an ATE.

Key takeaway: Under independence, exclusion, relevance, and monotonicity, the binary-instrument Wald ratio identifies the average treatment effect for instrument-induced compliers. It does not generally identify effects for the full population, always-takers, never-takers, or individual compliers.

Common Mistake

Wrong reasoning: “The instrument is valid, so its estimate is the average treatment effect for everyone.” Validity assumptions support a causal interpretation, but they do not make the compliers representative of all units in their treatment effects.

Correction: State the estimand as the complier average treatment effect, and distinguish it from the ATE unless additional assumptions establish that they are equal. Also avoid inferring individual effects from a group average or attributing the estimate to compliance types whose treatment status the instrument does not change.

Check Your Understanding

Answer each question by referring to the population and assumptions involved.

In potential-treatment notation, what condition defines a complier?
Why does the first stage equal the complier proportion under monotonicity?
If the LATE is 4 outcome units, what additional information would be needed to calculate the ATE across always-takers, compliers, and never-takers?
Can a LATE of 4 imply that every complier’s individual effect is 4? Explain.
In the presence of defiers, why may the Wald ratio fail to be the complier average treatment effect?