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The monotone IV reduced form is a complier-weighted effect

Statement

Under consistency, exclusion, joint instrument independence and monotonicity in the binary IV setup, with $τ = Y (1) - Y (0)$ and $C = {A (1) > A (0)}$ , $E [Y ∣ Z = 1] - E [Y ∣ Z = 0] = E [1_{C} τ] .$ If $P (C) > 0$ , the right-hand side is $P (C) E [τ ∣ C]$ .

Facts & Assumptions

Given: A binary IV setup with consistency, exclusion and joint instrument independence, and monotonicity. Prove the reduced-form identity.

[F1]

The observed reduced-form contrast equals $E [(A (1) - A (0)) (Y (1) - Y (0))]$ under consistency, exclusion and joint instrument independence. (IV observed contrasts as potential-outcome contrasts)

[F2]

Under monotonicity the defier event is null and $A (1) - A (0) = 1_{C}$ almost surely. (Binary monotonicity excludes defiers)

[F3]

For an event $C$ of positive probability, the event-conditioned mean is $E [τ ∣ C] = E [1_{C} τ] / P (C)$ . (Binary instrumental-variable potential-outcome setup)

Proof

1.1F1F2algebra

By [F1], the observed reduced-form contrast is $E [(A (1) - A (0)) τ]$ . By [F2], replacing the treatment contrast by $1_{C}$ almost surely gives $E [1_{C} τ]$ .

2.1F3step 1.1algebra∎

If $P (C) > 0$ , the definition in [F3] rearranges to $E [1_{C} τ] = P (C) E [τ ∣ C]$ . If $P (C) = 0$ , then $1_{C} = 0$ almost surely, so both sides of the first identity in the Statement are $0$ ; the conditional mean is simply not asserted in that case.

Depends on

Used by

The Wald ratio identifies the complier average treatment effect Theorem

Dependency tree · two levels

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, Technical Point 16.7, printed pp. 224–225 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), Proof of Identification of the LATE (standard reference, not scraped)