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The Wald ratio identifies the complier average treatment effect

Statement

Let $Θ$ be a nonempty class of full-data laws satisfying the binary IV setup, consistency, exclusion, joint instrument independence, relevance and monotonicity. Each $θ \in Θ$ induces the observed-data law $P_{θ}$ of $(Z, A, Y)$ . Define the estimand $ψ_{C} (θ) = E_{θ} [Y (1) - Y (0) ∣ C]$ , where $C = {A (1) > A (0)}$ . Then the Wald ratio is defined and $ψ_{W} (P_{θ}) = \frac{E_{θ} [Y ∣ Z = 1] - E_{θ} [Y ∣ Z = 0]}{E_{θ} [A ∣ Z = 1] - E_{θ} [A ∣ Z = 0]} = ψ_{C} (θ) .$ Thus the LATE is identifiable in the induced observed-data model. This does not assert that the LATE equals the population ATE.

Facts & Assumptions

Given: The binary IV setup with consistency, exclusion, joint instrument independence, relevance and monotonicity. The observed law is the law of $(Z, A, Y)$ induced by each full-data law in $Θ$ . Prove the Wald equality and identifiability of $ψ_{C}$ .

[F1]

The first-stage contrast equals $P (C)$ under monotonicity and is strictly positive under relevance. (The monotone IV first stage is the complier proportion)

[F2]

The reduced-form contrast equals $P (C) E [Y (1) - Y (0) ∣ C]$ when $P (C) > 0$ . (The monotone IV reduced form is a complier-weighted effect)

[F3]

The Wald ratio is the reduced-form contrast divided by the nonzero first-stage contrast. (Binary-instrument Wald ratio)

[F4]

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

[F5]

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

Proof

1.1givenF1F2

By [F1], $Δ_{A} = P (C) > 0$ . By [F2], $Δ_{Y} = P (C) E [Y (1) - Y (0) ∣ C]$ .

2.1F3step 1.1algebra

Dividing the second identity in step 1.1 by the positive first identity gives $Δ_{Y} / Δ_{A} = E [Y (1) - Y (0) ∣ C]$ . By [F3], this quotient is the Wald ratio, so it equals the LATE.

3.1F3F4F5step 2.1∎

If $P_{θ} = P_{η}$ , then both observed contrasts, and therefore their ratio, are equal for $θ$ and $η$ . Step 2.1 identifies each LATE with that common ratio, so $ψ_{C} (θ) = ψ_{C} (η)$ . By [F4] and [F5], the LATE is identifiable in this observed-data model.

Depends on

Used by

A finite randomized encouragement example Example
With heterogeneous effects, LATE can differ from ATE Example

Dependency tree · two levels

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, §16.4 and Technical Point 16.7, printed pp. 217–225 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), Compliance Types, LATE and its complete proof (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)