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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A direct encouragement effect changes the Wald ratio

Example

Let the target population have equally likely types $c, n$ and let $Z$ be an independent fair binary assignment. Set $(A (0), A (1)) = (0, 1)$ for type $c$ and $(0, 0)$ for type $n$ . Define $Y (a, z) = 2 a + z$ , and observe $A = A (Z)$ and $Y = Y (A, Z)$ . The exclusion restriction fails because changing $z$ changes $Y$ even at fixed $a$ ; the treatment effect at either fixed $z$ is $2$ .

Facts & Assumptions

Given: The two-type population and structural equations in the Example. Compute the first stage, reduced form and Wald ratio.

[F1]

The setup defines consistency, exclusion and the binary event-conditioned means. (Binary instrumental-variable potential-outcome setup)

[F2]

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

Verification

1.1givenF1constructalgebra

The product assignment makes $Z$ independent of the type and hence of all potential values; consistency holds by construction. Under $Z = 1$ , half the population has $A = 1$ and the mean outcome is $(3 + 1) / 2 = 2$ ; under $Z = 0$ , $A = 0$ for both types and the outcome mean is $0$ . Thus $Δ_{A} = 1 / 2$ and $Δ_{Y} = 2$ .

2.1F1F2step 1.1algebra∎

By [F2], the Wald ratio is $2 / (1 / 2) = 4$ , although the treatment contrast $Y (1, z) - Y (0, z)$ equals $2$ for every unit and each fixed $z$ . The discrepancy comes from the violated exclusion restriction, not treatment-effect heterogeneity.

Depends on

Used by

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Dependency tree · two levels

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, Technical Point 16.1 and §16.5, printed pp. 210–222 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), exclusion restriction section (standard reference, not scraped)
Judea Pearl, Causal Inference in Statistics: An Overview, §3.5.2 and §3.5.4, journal pp. 122–126 (standard reference, not scraped)