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A constant additive effect is identified by the Wald ratio

Statement

Let $Θ$ be a nonempty set of full-data laws satisfying the binary IV setup, consistency, exclusion and joint instrument independence. Each $θ \in Θ$ induces an observed-data law $P_{θ}$ for $(Z, A, Y)$ , and let $ψ_{A T E} (θ) = E_{θ} [Y (1) - Y (0)]$ . Suppose relevance holds and there is a constant $τ_{0} \in R$ such that $Y (1) - Y (0) = τ_{0}$ almost surely under every $θ \in Θ$ . Then $ψ_{W} (P_{θ}) = \frac{E_{θ} [Y ∣ Z = 1] - E_{θ} [Y ∣ Z = 0]}{E_{θ} [A ∣ Z = 1] - E_{θ} [A ∣ Z = 0]} = τ_{0} = ψ_{A T E} (θ) .$ Consequently, the ATE is identifiable in the induced observed-data model. Monotonicity is not assumed.

Facts & Assumptions

Given: The IV setup with consistency, exclusion, joint instrument independence, relevance, and a constant additive treatment effect $τ = Y (1) - Y (0) = τ_{0}$ almost surely. Prove the Wald formula identifies the population ATE.

[A1]

The individual treatment effect is the same constant $τ_{0}$ almost surely under each law in $Θ$ .

[F1]

The outcome and treatment observed contrasts equal their potential-outcome contrasts under consistency, exclusion and joint instrument independence. (IV observed contrasts as potential-outcome contrasts)

[F2]

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

[F3]

An estimand is a function from the model's parameter set to a target space. (Estimands on a statistical model)

[F4]

Identifiability means equal observed sampling laws force equal target values. (Identifiability of an estimand)

[F5]

The integral of an integrable sum or scalar multiple is the corresponding sum or scalar multiple of the integrals. (The Lebesgue integral is linear on $L^{1} (μ)$ )

Proof

1.1givenF1

By [F1], the reduced-form contrast is $E [(A (1) - A (0)) τ]$ and the first-stage contrast is $E [A (1) - A (0)]$ . No monotonicity condition is used in these identities.

2.1A1F5step 1.1algebra

By [A1] and linearity of expectation [F5], $E [(A (1) - A (0)) τ] = τ_{0} E [A (1) - A (0)]$ .

3.1givenA1F2step 2.1algebra

Relevance makes $E [A (1) - A (0)] \neq 0$ , so division in [F2] gives $ψ_{W} = τ_{0}$ . Also $E [Y (1) - Y (0)] = E [τ_{0}] = τ_{0}$ , proving the displayed ATE equality.

4.1F2F3F4step 3.1∎

For any $θ, η \in Θ$ with $P_{θ} = P_{η}$ , the observed contrasts and hence the observed-data functional $ψ_{W}$ agree. Step 3.1 identifies each ATE with that same ratio, so $ψ_{A T E} (θ) = ψ_{A T E} (η)$ . By [F3] and [F4], this is identifiability of the ATE on the induced observed-data model.

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, §16.3 and Technical Point 16.3, printed pp. 214–216 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), IV assumptions and LATE sections (standard reference, not scraped)
Judea Pearl, Causal Inference in Statistics: An Overview, §3.5.2, journal pp. 122–123 (standard reference, not scraped)