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Core IV conditions alone do not identify the population ATE

Statement refuted

The binary-IV core conditions—consistency, joint instrument independence, exclusion and relevance—by themselves identify the population ATE, without monotonicity or a homogeneity restriction.

Facts & Assumptions

Given: Two full-data laws with the same induced observed law, both satisfying the core IV conditions, but with different ATEs. Verify that this is a counterexample to identifiability.

[F1]

Consistency, exclusion, joint instrument independence and relevance are the IV conditions defined in the setup. (Binary instrumental-variable potential-outcome setup)

[F2]

The population ATE is $E [Y (1) - Y (0)]$ . (Binary-instrument principal strata and LATE)

[F3]

An estimand is a target function on the model, and identifiability means equal observed laws force equal target values. (Estimands on a statistical model, Identifiability of an estimand)

[F4]

The Wald ratio is the observed outcome contrast divided by the first-stage contrast. (Binary-instrument Wald ratio)

Counterexample

Let the unit type probabilities be $0.2$ always-takers, $0.3$ never-takers, $0.4$ compliers and $0.1$ defiers. Independently assign fair binary $Z$ by taking the product of this type law with a fair Bernoulli law. In both models use treatment pairs $(A (0), A (1)) = (1, 1), (0, 0), (0, 1), (1, 0)$ for those four types, and let $Y (a, z) = Y (a)$ .

Specify potential outcome pairs $(Y (0), Y (1))$ as follows:

Model $M_{0}$ : always $(0, 5)$ , never $(0, 0)$ , complier $(0, 1)$ , defier $(0, 3)$ .
Model $M_{1}$ : always $(10, 5)$ , never $(0, 10)$ , complier $(0, 1)$ , defier $(0, 3)$ .

In both models set $A = A (Z)$ and $Y = Y (A, Z)$ . No monotonicity or constant treatment effect is imposed.

1.1givenF1constructalgebra

In the product law, $Z$ is independent of all type-specific potential values, and both assignment atoms have positive probability. The observed treatment and outcome satisfy consistency by construction; $Y (a, z) = Y (a)$ is exclusion. The first-stage contrast is $E [A ∣ Z = 1] - E [A ∣ Z = 0] = (0.2 + 0.4) - (0.2 + 0.1) = 0.3 \neq 0$ , so relevance holds in both models.

2.1givenstep 1.1construct

For $Z = 0$ , both models give the same observed pairs: always-takers have $(A, Y) = (1, 5)$ , never-takers and compliers have $(0, 0)$ , and defiers have $(1, 3)$ . For $Z = 1$ , both give $(1, 5)$ to always-takers, $(0, 0)$ to never-takers and defiers, and $(1, 1)$ to compliers. Since type probabilities and the law of $Z$ also agree, the entire observed laws of $(Z, A, Y)$ agree.

3.1F2F3F4step 2.1algebra∎

The reduced-form contrast in either model is $Δ_{Y} = (0.2 \cdot 5 + 0.4 \cdot 1) - (0.2 \cdot 5 + 0.1 \cdot 3) = 0.1$ . Thus the common Wald ratio is $0.1 / 0.3 = 1 / 3$ by [F4]. However, $ATE (M_{0}) = 0.2 \cdot 5 + 0.3 \cdot 0 + 0.4 \cdot 1 + 0.1 \cdot 3 = 1.7$ , whereas $ATE (M_{1}) = 0.2 \cdot (- 5) + 0.3 \cdot 10 + 0.4 \cdot 1 + 0.1 \cdot 3 = 2.7$ . Therefore [F2] has different values at two parameters with the same observed law, and [F3] shows the ATE is not identifiable under only these core conditions.

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, Technical Points 16.2 and 16.7, printed pp. 212 and 224–225 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), LATE and ATE discussion (standard reference, not scraped)
Judea Pearl, Causal Inference in Statistics: An Overview, §3.5.2–§3.5.4, journal pp. 122–126 (standard reference, not scraped)