ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Defiers can make the Wald ratio differ from the complier effect

Example

Take a population with type probabilities $0.2$ always-takers, $0.3$ never-takers, $0.4$ compliers, and $0.1$ defiers. Independently randomize $Z$ with probabilities $1 / 2, 1 / 2$ . Let $Y (0) = 0$ for all types and let $Y (1) = 0, 0, 1, 3$ for always-takers, never-takers, compliers and defiers, respectively. Set $Y (a, z) = Y (a)$ and observe by consistency. Exclusion and instrument independence hold, but monotonicity fails.

Facts & Assumptions

Given: The finite type distribution in the Example. Compute the two observed contrasts, the Wald ratio and the complier mean effect.

[F1]

Consistency, exclusion, joint instrument independence, event-conditioned means and the four principal strata are defined in the IV setup. (Binary instrumental-variable potential-outcome setup, Binary-instrument principal strata and LATE)

[F2]

The observed outcome contrast equals the expectation of the potential outcome difference weighted by $A (1) - A (0)$ under the core IV assumptions. (IV observed contrasts as potential-outcome contrasts)

[F3]

The Wald ratio divides the outcome contrast by the first-stage contrast. (Binary-instrument Wald ratio)

Verification

1.1givenF1F2algebra

The first-stage contrast is $P (C) - P (D) = 0.4 - 0.1 = 0.3$ . By [F2], the outcome contrast is $E [(A (1) - A (0)) (Y (1) - Y (0))] = 0.4 \cdot 1 - 0.1 \cdot 3 = 0.1$ ; always-takers and never-takers contribute zero.

2.1F1F3step 1.1algebra∎

The Wald ratio is $0.1 / 0.3 = 1 / 3$ by [F3]. The complier effect is $1$ , whereas the defier contribution enters the numerator with a negative sign. Thus the LATE conclusion cannot be used without monotonicity.

Depends on

Used by

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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 and LATE proof (standard reference, not scraped)