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.

With heterogeneous effects, LATE can differ from ATE

Example

Take four equally likely types: an always-taker with treatment effect $4$ , a never-taker with effect $4$ , and two compliers with effects $1$ and $3$ . Let all $Y (0) = 0$ , let $Y (1)$ equal those effects, and let $Z$ be an independent fair binary assignment. The model has consistency, exclusion, instrument independence, relevance and monotonicity.

Facts & Assumptions

Given: The four equally likely types and randomized assignment described in the Example. Verify LATE, ATE and the Wald ratio.

[F1]

ATE and the complier event-conditioned mean are defined from the potential outcomes and principal strata. (Binary-instrument principal strata and LATE)

[F2]

Under the stated IV assumptions, the Wald ratio equals the complier average effect. (The Wald ratio identifies the complier average treatment effect)

Verification

1.1givenF1algebra

The complier event has probability $1 / 2$ and its two effects are $1$ and $3$ , so $LATE = 2$ . Averaging all four effects gives $ATE = (4 + 4 + 1 + 3) / 4 = 3$ .

2.1F2step 1.1algebra∎

The always-taker and never-taker outcomes do not change across instrument arms. The two compliers each have probability $1 / 4$ , so their treatment effects give reduced-form contrast $\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot 3 = 1$ ; the first-stage contrast is $1 / 2$ . The Wald ratio is therefore $2$ , equal to the LATE by [F2] but not to the population ATE.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, §§16.3–16.4 and Technical Point 16.7, printed pp. 214–225 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), LATE and ATE discussion (standard reference, not scraped)