LATE Under Heterogeneous Treatment Effects

Why Heterogeneous Effects Change the IV Target

An instrumental variable does not generally move treatment for everyone. With a binary instrument and monotone treatment response to the instrument, it changes treatment receipt only for compliers. If the treatment effect also varies across people, the outcome contrast induced by the instrument therefore averages effects among those compliers—not effects across the entire population. The size and direction of the difference depend on who complies and on the effects in each compliance group.

The central result is not that the instrumental-variable estimate is biased whenever effects are heterogeneous. Under the IV assumptions, the Wald ratio identifies a well-defined causal effect: the average treatment effect for compliers. The important interpretive limit is that this local average need not equal the population average treatment effect.

Definition: Let \(Z\in\{0,1\}\) be the instrument and \(D\in\{0,1\}\) treatment receipt. Let \(D(z)\) denote the treatment a person would receive if the instrument were set to \(z\), and let \(Y(d)\) denote the outcome if treatment were set to \(d\). The individual treatment effect is \(\tau=Y(1)-Y(0)\). A complier has \(D(1)=1\) and \(D(0)=0\). The local average treatment effect (LATE) is \(E[Y(1)-Y(0)\mid D(1)>D(0)]\), the average individual treatment effect among compliers.

For the derivation below, use the standard IV assumptions: independence of the instrument from the potential treatments and outcomes; exclusion, so the instrument affects the outcome only through treatment; monotonicity, \(D(1)\geq D(0)\) for every person; and relevance, so the instrument changes the probability of treatment. Also assume consistency and no interference, so observed treatment and instrument values correspond to the relevant potential outcomes and one person's assignment does not change another person's outcome.

With a binary instrument and monotonicity, people are never-takers, always-takers, or compliers. Never-takers have \(D(1)=D(0)=0\), always-takers have \(D(1)=D(0)=1\), and compliers have \(D(1)=1,D(0)=0\). Defiers, who would do the reverse, are excluded by monotonicity. Treatment effects may vary both across these groups and among people within a group.

Why the Wald Ratio Averages Complier Effects

Write the reduced-form outcome contrast as the difference in mean outcomes by instrument assignment, and the first stage as the corresponding difference in mean treatment receipt. Under independence and exclusion, the expected outcome for instrument value \(z\) can be written as \(E[Y(D(z))]\). Independence permits this potential-outcome contrast to represent the observed contrast by instrument group. Therefore:

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

The second equality follows because \(D(z)\) is binary: if a person receives the same treatment under both instrument values, that person's contribution is zero; if treatment changes from zero to one, the outcome difference is \(Y(1)-Y(0)\). By monotonicity, \(D(1)-D(0)\) is one for compliers and zero for everyone else. Thus the reduced form equals the complier share multiplied by their average treatment effect. The same assumptions give the first stage:

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

Relevance requires this first stage to be nonzero. Dividing the reduced form by the first stage then yields:

\frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]} = \frac{P(C)E[Y(1)-Y(0)\mid C]}{P(C)} = E[Y(1)-Y(0)\mid C],

where \(C\) denotes membership in the complier group. This is the identification result: under these assumptions, the population Wald ratio identifies the LATE even when treatment effects are heterogeneous. It does not identify each complier's effect separately; it identifies their average.

Worked Example: A Complier Effect Above the Population Average

Consider a hypothetical population in which compliers make up 40%, always-takers 20%, and never-takers 40%. Suppose the average treatment effects are 5 outcome units for compliers, 1 for always-takers, and 2 for never-takers. Assume the instrument satisfies independence, exclusion, and monotonicity, and has a nonzero first stage.

The population ATE averages effects across all three groups, weighted by their population shares:

$$ ATE=(0.40)(5)+(0.20)(1)+(0.40)(2)=2.00+0.20+0.80=3.00 $$

The instrument changes treatment only for compliers, so its reduced-form outcome contrast is \((0.40)(5)=2.00\) outcome units. Its first-stage contrast is the increase in treatment receipt among the 40% who comply: \(0.40\). The Wald ratio is therefore:

\text{Wald ratio}=\frac{2.00}{0.40}=5.00\text{ outcome units per treatment}

The Wald ratio is the complier average effect, 5.00 outcome units per treatment. The population ATE is 3.00 outcome units per treatment. Both calculations are correct; they average effects over different groups. The IV result does not imply that treating the whole population would produce an average effect of 5.00.

Heterogeneity Within the Complier Group

The LATE is itself an average, not a claim that every complier has the same effect. If some compliers benefit more than others, the instrument-induced outcome contrast combines those effects in proportion to how common each kind of complier is. This weighting is by the composition of the complier group, not by the size of each person's treatment effect.

Worked Example: Averaging Unequal Effects Among Compliers

In a hypothetical population, 30% are compliers with an average effect of 2 outcome units, and another 20% are compliers with an average effect of 8 outcome units. The remaining 20% are always-takers with an average effect of 1, and 30% are never-takers with an average effect of 4. Assume the IV conditions hold.

The complier share is \(0.30+0.20=0.50\). The reduced-form contrast is the population-weighted sum of treatment effects for the two complier subgroups:

\text{Reduced form}=(0.30)(2)+(0.20)(8)=0.60+1.60=2.20

The first stage is \(0.50\), because half the population changes treatment receipt in response to the instrument. Hence:

LATE=\frac{2.20}{0.50}=4.40\text{ outcome units per treatment}

The same result follows by averaging within the complier group: \(\big((0.30)(2)+(0.20)(8)\big)/0.50=4.40\). By contrast, the population ATE is:

$$ ATE=(0.30)(2)+(0.20)(8)+(0.20)(1)+(0.30)(4)=0.60+1.60+0.20+1.20=3.60 $$

Thus the complier average is 4.40 outcome units per treatment, while the population average is 3.60. The larger effects among the high-effect compliers raise the LATE, but the ATE also includes effects for always-takers and never-takers.

Different Instruments Can Target Different Compliers

The complier group is defined relative to an instrument: it consists of people whose treatment receipt that instrument changes. Two valid instruments can therefore move treatment for different people. When effects vary across people, their Wald ratios can differ even if both instruments satisfy the IV assumptions for the population and treatment under study. Such a difference can reflect different complier populations, not necessarily a contradiction in the calculations.

Worked Example: Two Valid Instruments, Two Local Effects

Suppose, hypothetically, the same target population contains three groups: 20% whose treatment receipt responds only to instrument A and whose average treatment effect is 2 outcome units; 20% whose receipt responds only to instrument B and whose average effect is 8; and 60% whose receipt responds to neither instrument and whose average effect is 4. Assume each instrument, considered in its own assignment design, is independent of the potential outcomes and potential treatments, satisfies exclusion and monotonicity, and has a nonzero first stage.

For instrument A, the first stage is 0.20 and the reduced form is \((0.20)(2)=0.40\). Its Wald ratio is \(0.40/0.20=2.00\) outcome units per treatment. For instrument B, the first stage is also 0.20, but its reduced form is \((0.20)(8)=1.60\), giving \(1.60/0.20=8.00\) outcome units per treatment. Each ratio is a LATE for its own complier group.

For comparison, the population ATE is \((0.20)(2)+(0.20)(8)+(0.60)(4)=0.40+1.60+2.40=4.40\) outcome units per treatment. Neither instrument's LATE equals that population average in this example. The difference between the instruments arises because they shift treatment for groups with different average effects.

Interpretation, Diagnostics, and Limits

A valid IV analysis should report its target in context: for example, “the estimated average effect of treatment among people whose treatment receipt is changed by this instrument.” Naming the instrument matters, because the identity of that group depends on the instrument and the setting. An IV estimate is not automatically an effect for everyone who could receive treatment, for all treated people, or for a population defined only by observed characteristics.

The empirical first stage can be checked by estimating the difference in treatment rates across instrument groups. A very small first stage raises a weak-instrument concern: the ratio can be highly unstable, and its usual large-sample approximation may be poor. A strong first stage, however, does not prove independence, exclusion, or monotonicity. Those assumptions require substantive justification; observed data alone generally cannot verify them.

The main alternatives answer different questions under different assumptions. A randomized treatment comparison can identify a population average effect when treatment itself is randomly assigned and outcomes are observed as intended. Matching or regression methods can target effects for a broader population when their own treatment-assignment assumptions are justified. IV methods instead use variation in treatment generated by the instrument and, under the stated assumptions, identify the complier average. These approaches should not be treated as interchangeable ways to estimate one automatically common quantity.

A common mistake is to say, “The instrument is randomized, so the Wald ratio is the ATE.” Random assignment of the instrument supports independence, but it does not make the instrument change treatment for everyone. The denominator is the share whose treatment receipt responds under monotonicity, and the numerator reflects effects for those responders. The correction is to interpret the ratio as the LATE unless additional grounds establish that it represents a broader target.

Another limitation is that the assumptions and the IV estimate do not reveal the treatment effects of always-takers or never-takers. Consequently, the population ATE cannot generally be recovered merely by relabeling a LATE. For policy decisions affecting groups beyond the compliers, this distinction can matter: the estimated local effect may not predict the average effect of a policy that reaches a different mix of people.

Key takeaway: With independence, exclusion, monotonicity, and relevance, the Wald ratio equals the average treatment effect among compliers. Heterogeneous effects do not invalidate that interpretation; they make clear why it need not be the population ATE.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If 25% of a population are compliers and their average treatment effect is 6 outcome units, while the other groups never change treatment receipt, what are the reduced-form contrast and first stage under the IV assumptions? What is the Wald ratio?
Suppose the reduced form is 1.5 outcome units and the first stage is 0.30. What does the Wald ratio identify, and what is its value?
Why can a valid instrument with a large first stage still identify an effect different from the population ATE?
Two valid instruments have Wald ratios of 3 and 7 outcome units per treatment. Give one interpretation consistent with the IV assumptions and heterogeneous treatment effects.

Solutions. (1) The reduced form is \((0.25)(6)=1.50\) outcome units and the first stage is \(0.25\), so the Wald ratio is \(1.50/0.25=6.00\) outcome units per treatment, the LATE. (2) The ratio is \(1.5/0.30=5.00\) outcome units per treatment; under the assumptions it identifies the complier average effect. (3) A large first stage says many people respond to the instrument, not that all population groups respond or have the same effects. The ratio averages effects among compliers, whereas the ATE averages effects across the full population. (4) The instruments may change treatment receipt for different complier groups whose average treatment effects differ; each ratio can be a valid LATE for its own group.