Three Effects, Three Target Populations
An instrumental-variable estimate can be causally meaningful without describing the average effect for everyone in a population or for everyone who received treatment. Under the standard IV assumptions, the Wald ratio identifies the average treatment effect for compliers: individuals whose treatment receipt changes with the instrument. The average treatment effect (ATE) and average treatment effect on the treated (ATT) average over different groups, so they can differ from this local average treatment effect (LATE).
The distinction matters when effects vary across people. A treatment may help compliers modestly, help always-takers substantially, and have little effect for never-takers. An IV estimate then answers a specific causal question about compliers, not a universal question about the treatment’s effect.
Here \(D(1)\) denotes the treatment an individual would receive if \(Z=1\), and \(D(0)\) denotes the treatment they would receive if \(Z=0\). Under monotonicity, there are three compliance types: always-takers, with \(D(1)=D(0)=1\); compliers, with \(D(1)=1,D(0)=0\); and never-takers, with \(D(1)=D(0)=0\). The LATE is also often called the complier average causal effect.
The ATT is defined by actual treatment receipt in the population under consideration. In an encouragement design, actual receipt depends in part on the instrument assignment. Consequently, the ATT can depend on how many people are assigned to each instrument level. That does not change the definition of LATE, which conditions on compliance type rather than realized receipt.
When the Wald Ratio Has a Local Interpretation
For a binary instrument and binary treatment, the Wald ratio is the difference in mean outcomes across instrument groups divided by the difference in treatment rates across those groups. Its LATE interpretation requires more than a nonzero denominator. A standard sufficient set of assumptions is consistency and no interference, instrument independence from potential outcomes and potential treatment receipts, exclusion of a direct effect of \(Z\) on the outcome, monotonicity, and relevance. Relevance means the instrument changes the probability of treatment; monotonicity rules out defiers, whose treatment receipt would move opposite to the instrument.
The identifying result is specific: the ratio averages effects among the individuals whose treatment is changed by \(Z\). It does not, without further assumptions, turn into an average over always-takers, never-takers, or the full population. In particular, knowing that the instrument is independent does not mean that the people induced to take treatment are representative of everyone.
Why the Target Averages Differ
Let \(\pi_A,\pi_C,\pi_N\) denote the population shares of always-takers, compliers, and never-takers, and let \(\tau_A,\tau_C,\tau_N\) denote their average treatment effects. Under the three-type classification, the ATE is a population-share-weighted average of the type-specific effects. The LATE is simply the complier effect. The ATT instead weights effects by the shares of each type among people who actually received treatment.
For a randomized instrument, let \(q=P(Z=1)\). Under monotonicity, always-takers receive treatment under either instrument assignment, compliers receive it only when \(Z=1\), and never-takers do not receive it. Thus, among treated individuals, the population masses of always-takers and compliers are \(\pi_A\) and \(q\pi_C\), respectively. If the population contains at least some treated individuals, the ATT is therefore
This expression is a weighted average of effects among the treated types, not the population average and not necessarily the complier average. It also makes clear that the treated group’s composition can change with \(q\). This formula assumes that instrument assignment is randomized with probability \(q\) and that the type-specific average effects are defined in the population receiving that assignment mechanism.
Worked Examples
Worked Example: Calculating All Three Targets
Consider a hypothetical population with 20% always-takers, 30% compliers, and 50% never-takers. Suppose their average treatment effects, in test-score points, are 4, 2, and 1, respectively. The instrument is randomized with \(P(Z=1)=0.5\). Calculate LATE, ATE, and ATT.
LATE. Compliers have an average effect of 2 points, so \(\text{LATE}=2\) points.
ATE. Weight each type’s effect by its population share:
ATT. All always-takers are treated, while half of the compliers are treated on average because \(q=0.5\). The treated population mass is \(0.20+(0.5)(0.30)=0.35\). Its average treatment effect is
The three answers are not all distinct: LATE is 2 points, ATE is 1.90 points, and ATT is about 3.143 points. The ATT is largest here because always-takers, who have a larger effect, make up a substantial share of those treated.
Worked Example: Interpreting an IV Contrast
In a second hypothetical population, 25% are always-takers, 40% are compliers, and 35% are never-takers. The average treatment effects are 1, 3, and \(-1\) health-score points, respectively. Suppose untreated mean outcomes are 10 points for always-takers, 8 for compliers, and 6 for never-takers. Treatment adds the stated type-specific effect. The instrument is randomized, exclusion holds, and there are no defiers. With half the population assigned to each instrument level, calculate the outcome contrast, first stage, Wald ratio, ATE, and ATT.
When \(Z=0\), always-takers receive treatment, but compliers and never-takers do not. The mean outcome is
When \(Z=1\), both always-takers and compliers receive treatment. The mean outcome is
The reduced-form outcome contrast is \(9.25-8.05=1.20\) points. The treatment rate is \(0.25\) when \(Z=0\), and \(0.25+0.40=0.65\) when \(Z=1\), so the first stage is \(0.65-0.25=0.40\). Therefore,
That result equals the complier effect, as required by the stated assumptions. It is not the ATE: \((0.25)(1)+(0.40)(3)+(0.35)(-1)=1.10\) points. With \(q=0.5\), the treated mass is \(0.25+(0.5)(0.40)=0.45\), and
The Wald ratio, ATE, and ATT answer different questions: the effect among compliers, the effect across the population, and the effect among actual recipients, respectively.
Worked Example: How the Treated Group Can Change
Suppose a third hypothetical population has 10% always-takers, 50% compliers, and 40% never-takers. Their respective average treatment effects are 5, 2, and 0 outcome units. Compare the ATT under two randomized instrument assignment probabilities, \(q=0.2\) and \(q=0.8\). The type shares and effects stay fixed.
The population ATE is fixed:
For \(q=0.2\), the treated mass is \(0.10+(0.2)(0.50)=0.20\). Thus
For \(q=0.8\), the treated mass is \(0.10+(0.8)(0.50)=0.50\), giving
Increasing the proportion assigned to encouragement brings more compliers into the treated group, reducing the relative weight of the higher-effect always-takers. The LATE and population ATE do not change in this calculation, but the ATT does because the composition of actual treatment recipients changes.
Justification and Practical Interpretation
The weighting formulas follow from partitioning the population by compliance type. Each individual belongs to one type under the stated binary-instrument and monotonicity setup. Taking the average treatment effect over that partition gives the ATE formula: each type-specific mean effect is multiplied by that type’s population share, and the shares sum to one. Conditioning instead on \(D=1\) changes the weights. Always-takers are treated regardless of \(Z\), while a complier is treated only when \(Z=1\), which occurs for a fraction \(q\) under randomized assignment. Never-takers contribute no treated individuals. Dividing the treated types’ weighted effects by their total treated mass gives the ATT formula.
The Wald ratio has a different weighting argument: under independence, exclusion, and monotonicity, the instrument-induced outcome change comes from compliers, while always-takers’ and never-takers’ treatment status does not change. The outcome contrast therefore reflects the complier effect multiplied by the complier share; the first stage is that share. Their ratio is the complier effect. This identifies LATE, not the other two averages.
| Estimand or approach | Target | What must be justified |
|---|---|---|
| LATE from an instrument | Compliers whose treatment receipt responds to the instrument | IV assumptions, including relevance and monotonicity |
| ATE | The full target population | A design or assumptions that identify effects across that population |
| ATT | Individuals who actually received treatment | A design or assumptions identifying effects for the treated group |
| Randomized treatment comparison | Often the population assigned to treatment, depending on the estimand and implementation | Random assignment and clear handling of incomplete uptake |
| Matching or regression adjustment | May target ATE or ATT, depending on the analysis | Conditional exchangeability and adequate overlap, among other conditions |
Diagnostics can assess parts of an IV analysis but cannot make the target populations interchangeable. Check that the first stage is meaningfully different from zero and report its size; a weak or nearly absent first stage makes the ratio unstable and the local interpretation practically uninformative. Examine whether the instrument assignment mechanism is credible and whether exclusion is plausible in the application. These assumptions are not all verified by a statistical test. Also state who the compliers plausibly are, when that can be defended, and avoid presenting a complier estimate as a population-wide effect without an additional argument.
A common mistake is to say, “The instrument is randomized, so the Wald ratio is the ATE.” Randomization supports independence of the instrument, but it does not make every person’s treatment status respond to the instrument. Under the IV assumptions, the Wald ratio concerns compliers. To interpret it as ATE or ATT requires further conditions, such as restrictions on treatment-effect heterogeneity or a design that identifies those other targets. A numerical coincidence between LATE and ATE in one setting is not a general identification result.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A population is 20% always-takers, 30% compliers, and 50% never-takers. Their average treatment effects are 4, 2, and 1 units. What are the ATE and LATE?
- In the population in Question 1, the instrument is randomized with \(q=0.5\). Calculate the ATT.
- Why can an IV estimate equal the LATE without equaling the ATT?
- In a randomized encouragement design with a valid instrument, does a positive first stage by itself establish that the Wald ratio is the ATE? Explain.
Solutions. (1) \(\text{ATE}=(0.20)(4)+(0.30)(2)+(0.50)(1)=1.90\) units; \(\text{LATE}=2\) units. (2) \(\text{ATT}=[(0.20)(4)+(0.5)(0.30)(2)]/[0.20+(0.5)(0.30)]=1.10/0.35\approx3.143\) units. (3) The IV assumptions identify the average effect among compliers, while ATT averages among everyone who actually received treatment; those groups and their effect averages need not match. (4) No. A positive first stage establishes relevance in direction and size, not representativeness of compliers. Under the other required IV assumptions, the Wald ratio identifies LATE, not automatically ATE.