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The monotone IV first stage is the complier proportion

Statement

Under consistency, joint instrument independence and monotonicity in the binary IV setup, $E [A ∣ Z = 1] - E [A ∣ Z = 0] = P (C),$ where $C = {A (1) > A (0)}$ is the complier event. If the instrument is relevant, then $P (C) > 0$ .

Facts & Assumptions

Given: A binary IV setup with consistency, joint instrument independence, monotonicity and relevance. Prove the first-stage identity and positivity of the complier stratum.

[A1]

Relevance means the observed first-stage contrast is nonzero. (Binary instrumental-variable potential-outcome setup)

[F1]

Independent random elements have independent generated sigma-algebras; events from these sigma-algebras satisfy $P (B \cap D) = P (B) P (D)$ . (Independent random elements, Independent sigma-algebras and independent events)

[F2]

Under monotonicity the defier event is null, and $A (1) - A (0)$ is $1$ on $C$ and $0$ on the other principal strata. (Binary monotonicity excludes defiers, Binary-instrument principal strata and LATE)

[F3]

The integral is linear on integrable functions. (The Lebesgue integral is linear on $L^{1} (μ)$ )

Proof

1.1givenF2

On the complier event $C$ , $(A (0), A (1)) = (0, 1)$ , so $A (1) - A (0) = 1$ ; on the always-taker and never-taker events the difference is $0$ . The defier event is null by [F2]. Hence $A (1) - A (0) = 1_{C}$ almost surely.

2.1givenF1F3step 1.1algebra

For $z = 0, 1$ , the event ${A (z) = 1}$ belongs to the sigma-algebra of the potential-value tuple. Joint instrument independence and [F1] give $P (A (z) = 1, Z = z) = P (A (z) = 1) P (Z = z)$ . Since $A = A (z)$ on ${Z = z}$ by consistency and $P (Z = z) > 0$ , the event-conditioned mean in the setup gives $E [A ∣ Z = z] = P (A (z) = 1, Z = z) / P (Z = z) = P (A (z) = 1) = E [A (z)]$ . Subtracting these identities and using [F3] and step 1.1 yields $E [A ∣ Z = 1] - E [A ∣ Z = 0] = E [A (1) - A (0)] = P (C)$ , without an exclusion assumption.

3.1A1step 2.1∎

Relevance [A1] says this first-stage contrast is nonzero. Since $P (C)$ is nonnegative, step 2.1 implies $P (C) > 0$ .

Depends on

Used by

The Wald ratio identifies the complier average treatment effect Theorem

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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), Compliance Types and LATE proof (standard reference, not scraped)