Deriving the LATE Identification Result

From the Instrument to the Complier Effect

An instrumental-variable analysis compares outcomes across levels of an instrument, even though the instrument is not itself the treatment of interest. The key question is what that outcome contrast represents. Under specific assumptions, it is not generally the average treatment effect for everyone. It is the average causal effect among people whose treatment receipt is changed by the instrument: the compliers.

This tutorial derives that result. The earlier material on potential outcomes for binary instruments and treatments, the four compliance types, and the IV assumptions supplies the setup. The derivation makes explicit how independence, exclusion, and monotonicity combine to connect the reduced-form contrast to a causal effect.

Definition: Let \(Z\in\{0,1\}\) be a binary instrument, \(D\in\{0,1\}\) the treatment received, and \(Y\) the outcome. For each unit, \(D(z)\) is the treatment that would be received if the instrument were set to \(z\), and \(Y(d)\) is the outcome under treatment level \(d\). The observed variables obey \(D=D(Z)\) and, under exclusion, \(Y=Y(D)\). A complier has \(D(1)=1\) and \(D(0)=0\). The complier average causal effect, or LATE, is \(E[Y(1)-Y(0)\mid D(1)>D(0)]\).

Assumptions for the Identification Result

The result concerns population contrasts, not merely a calculation on a sample. Assume potential outcomes are well-defined and have finite means. Also assume consistency: each unit's observed treatment and outcome equal the potential values corresponding to the instrument assignment and treatment actually received. The following assumptions then give the identification result.

Independence: \(Z\) is independent of the joint vector \(\{Y(1),Y(0),D(1),D(0)\}\). Random assignment of the instrument is one way this may hold.
Exclusion: the instrument affects the outcome only through treatment. In potential-outcome terms, the outcome under instrument value \(z\) and treatment \(d\) is \(Y(d)\); it has no additional dependence on \(z\).
Monotonicity: \(D(1)\geq D(0)\) for every unit. Thus, no unit takes treatment when \(Z=0\) but would refuse it when \(Z=1\).
Relevance: the instrument changes the population probability of treatment, so \(E[D\mid Z=1]-E[D\mid Z=0]\neq 0\). Under independence and monotonicity, this means there is a positive proportion of compliers.

Monotonicity partitions the population into always-takers, with \(D(1)=D(0)=1\); never-takers, with \(D(1)=D(0)=0\); and compliers, with \(D(1)=1,D(0)=0\). The fourth logical type, defiers, has \(D(1)=0,D(0)=1\), and is ruled out by monotonicity. Write \(C\) for the event that a unit is a complier and \(p_C=P(C)\).

Why the Reduced Form Is a Scaled LATE

The reduced-form contrast is the difference in mean outcome between instrument groups: \(\Delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0]\). To see what it measures, first express the observed outcome under assignment \(z\) as \(Y(D(z))\), using consistency and exclusion. Independence ensures that the instrument groups have the same distribution of all potential outcomes and potential treatment receipts. Therefore, their outcome contrast can be evaluated using the same population distribution of potential values:

\Delta_Y=E[Y(D(1))]-E[Y(D(0))].

For always-takers, treatment is one under either instrument assignment, so their contribution to the difference inside this expectation is \(Y(1)-Y(1)=0\). For never-takers it is \(Y(0)-Y(0)=0\). For compliers it is \(Y(1)-Y(0)\). Consequently, only the complier group contributes to the population reduced form. Conditioning on compliance type gives the key identity:

\begin{aligned} \Delta_Y &=P(C)E[Y(1)-Y(0)\mid C]\\ &=p_C\times \text{LATE}. \end{aligned}

The first-stage contrast has a parallel derivation. Under independence, the treatment means in the two instrument groups identify the population means of \(D(1)\) and \(D(0)\). The difference \(D(1)-D(0)\) equals one for compliers and zero for always-takers and never-takers. Hence:

\Delta_D=E[D\mid Z=1]-E[D\mid Z=0] =E[D(1)-D(0)]=p_C.

If \(p_C>0\), division of the reduced-form identity by the first-stage identity identifies the complier average causal effect.

Theorem: Under consistency, finite outcome means, independence, exclusion, monotonicity, and a nonzero first stage, \(\displaystyle \text{LATE}=\frac{E[Y\mid Z=1]-E[Y\mid Z=0]}{E[D\mid Z=1]-E[D\mid Z=0]}\). The numerator is the reduced-form outcome contrast, and the denominator is the first-stage treatment contrast. The ratio is an effect among compliers, not automatically an effect for the entire population.

Worked Applications

Worked Example: Recovering LATE From Population Contrasts

Suppose a hypothetical population has 30% always-takers, 25% never-takers, and 45% compliers. Assume the instrument is independent of potential outcomes and treatment receipts, exclusion holds, and there are no defiers. Suppose the mean effect \(E[Y(1)-Y(0)\mid C]\) is 4 outcome points.

The first-stage contrast is the complier share: \(\Delta_D=0.45\). The reduced-form contrast is the complier share times their mean effect: \(\Delta_Y=0.45(4)=1.8\) outcome points. The Wald ratio is therefore \(\Delta_Y/\Delta_D=1.8/0.45=4\) outcome points per treatment.

The calculation is consistent with the assumed compliance shares: always-takers and never-takers do not change treatment receipt with the instrument, while 45% do. The identified effect is the average treatment effect among those compliers. The assumptions, rather than the arithmetic alone, justify that causal interpretation.

Worked Example: Calculating the Wald Estimate From Hypothetical Data

In a hypothetical randomized encouragement study, 500 people are assigned to each instrument group. In the encouraged group, 310 receive treatment and the mean outcome is 16.4 units. In the unencouraged group, 210 receive treatment and the mean outcome is 15.2 units. Assume the instrument assignment is independent, exclusion is credible, and encouragement cannot reduce any person's chance of treatment.

The treatment proportions are \(310/500=0.62\) and \(210/500=0.42\), so the estimated first-stage contrast is \(0.62-0.42=0.20\). The estimated reduced-form outcome contrast is \(16.4-15.2=1.2\) outcome units. The Wald estimate is

\widehat{\text{LATE}} =\frac{16.4-15.2}{310/500-210/500} =\frac{1.2}{0.20} =6.0\text{ outcome units per treatment}.

Under the stated assumptions, this estimates a mean causal effect of 6.0 outcome units for compliers. The estimated first stage is positive, as required for a nonzero denominator, and the stipulated direction of encouragement is consistent with monotonicity. These checks do not establish that independence or exclusion is true; they state what the causal interpretation depends on.

Worked Example: Why Noncompliers Do Not Add to the Reduced Form

Consider a hypothetical population with 20% always-takers, 50% never-takers, and 30% compliers. Suppose the complier average treatment effect is 3 outcome units. Assume independence, exclusion, monotonicity, and consistency.

The first-stage contrast is \(p_C=0.30\). Always-takers receive treatment under both instrument values, and never-takers receive no treatment under either value; their outcome contrasts across instrument groups are zero under exclusion. The compliers contribute \(0.30(3)=0.90\) outcome units to the population contrast. Thus the reduced form is \(0.90\), and the Wald ratio is \(0.90/0.30=3.0\) outcome units per treatment.

This example explains why a reduced-form effect smaller than the complier effect is not a contradiction: the instrument moves treatment for only 30% of the population. Dividing by that treatment shift rescales the outcome contrast to the complier effect.

Interpretation, Diagnostics, and Limits

The equality between the Wald ratio and LATE is an identification statement: it says which population causal effect the observed population contrasts equal under the assumptions. In a sample, the corresponding ratio is an estimate and is subject to sampling uncertainty. A large estimate or a positive first stage does not itself verify the assumptions.

Some empirical features can alert an analyst to problems. A first-stage contrast close to zero makes the Wald denominator small, so even modest sampling variation in either contrast can produce a highly variable ratio. A weak first stage does not by itself prove that the assumptions fail, but it makes the estimate unstable and its uncertainty important. Treatment rates moving in the opposite direction from the substantive assignment mechanism may prompt a check of coding and design. Neither observation can establish monotonicity at the individual level, because each person's pair \(D(1),D(0)\) is not jointly observed.

Independence and exclusion also require substantive justification, not just a favorable comparison of observed group characteristics. If assignment is associated with potential outcomes, the instrument groups need not represent the same potential-outcome distribution. If the instrument changes outcomes through a pathway other than treatment, the reduced form includes that pathway as well. In either case, the decomposition used in the derivation no longer yields the stated LATE. If defiers exist, the first stage is no longer simply the complier share, and the reduced form need not be the complier share times the complier effect.

The Wald ratio also differs from the reduced-form contrast itself. The reduced form answers how much the outcome mean differs by instrument assignment. The ratio scales that contrast by the change in treatment receipt, giving an effect per treatment change under the IV assumptions. A comparison of outcomes by treatment received alone is not the same calculation and does not, without additional assumptions, have this identification result. Likewise, the LATE need not equal a population-wide average causal effect: it is defined for the subgroup whose treatment receipt responds to this instrument.

The result is useful precisely because treatment receipt may not be randomized even when the instrument is. It converts the randomized instrument's outcome contrast into a causal effect for the affected subgroup. Its scope remains local to that subgroup and depends on the design and assumptions. It should not be presented as an effect for everyone unless a separate argument supports that extension.

Define the contrasts.
Compute the outcome contrast \(E[Y\mid Z=1]-E[Y\mid Z=0]\) and treatment contrast \(E[D\mid Z=1]-E[D\mid Z=0]\).

Connect the first stage to compliance.
With independence and monotonicity, the treatment contrast equals the proportion of compliers.

Interpret the ratio cautiously.
With exclusion and the other stated assumptions, the reduced form divided by the first stage identifies LATE; assess design credibility and first-stage strength.

Common pitfall: “The instrument raises the outcome by 1.2 units, so treatment raises the outcome by 1.2 units for everyone.” This confuses the reduced-form contrast with the treatment effect and overgeneralizes the target population. The correction is to divide the outcome contrast by the first-stage treatment contrast to obtain the Wald estimate, then interpret it as an effect among compliers only under the IV assumptions.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

Under monotonicity, what does the population first-stage contrast equal?
A hypothetical study has a reduced-form contrast of 0.8 outcome units and a first-stage contrast of 0.25. What is the Wald estimate, and for whom is it interpreted?
Why do always-takers and never-takers contribute zero to the reduced-form contrast under exclusion?
Which assumption rules out units whose treatment receipt falls when the instrument changes from zero to one?
Does a nonzero first stage demonstrate that independence and exclusion hold? Explain.

Solutions. (1) It equals the proportion of compliers, \(p_C\), given independence and monotonicity. (2) \(0.8/0.25=3.2\) outcome units per treatment; under the assumptions, this is LATE for compliers. (3) Each type has the same treatment receipt under both instrument values, so exclusion means its outcome is also unchanged by the instrument. (4) Monotonicity, stated as \(D(1)\geq D(0)\) for every unit. (5) No. A nonzero first stage establishes instrument relevance at the population level, but does not establish independence or exclusion.