LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

IV observed contrasts as potential-outcome contrasts

Statement

In an IV setup satisfying consistency, exclusion, and joint instrument independence, let $τ = Y (1) - Y (0)$ . Then $E [A ∣ Z = 1] - E [A ∣ Z = 0] = E [A (1) - A (0)]$ and $E [Y ∣ Z = 1] - E [Y ∣ Z = 0] = E [(A (1) - A (0)) τ]$ . No monotonicity or relevance condition is needed for these equalities; positivity of both instrument atoms and integrability are part of the setup.

Facts & Assumptions

Given: The IV setup of Binary instrumental-variable potential-outcome setup, consistency $A = A (Z)$ and $Y = Y (A, Z)$ , exclusion $Y (a, z) = Y (a)$ , and joint independence $Z ⊥ (A (0), A (1), Y (0), Y (1))$ . Show the two displayed contrasts.

[A1]

The setup supplies binary $Z, A (z)$ and integrable $Y (a, z)$ , with positive $P (Z = z)$ for $z = 0, 1$ . (Binary instrumental-variable potential-outcome setup)

[A2]

The setup assumes consistency, exclusion and joint instrument independence. (Binary instrumental-variable potential-outcome setup)

[F1]

Measurable functions of independent random elements remain independent. (Measurable coordinatewise functions preserve independence)

[F2]

For independent integrable real random variables, the expectation of their product factors as the product of their expectations. (Expectations factor over finite products of independent random variables)

[F3]

Finite sums, scalar multiples and products of measurable real functions are measurable whenever pointwise defined. (Arithmetic and lattice operations preserve measurability whenever they are defined)

[F4]

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

Proof

1.1givenA1A2F1F3

Put $W = (A (0), A (1), Y (0), Y (1))$ and $I_{z} = 1_{{Z = z}}$ . The functions $A (z)$ and $Y (0) + A (z) (Y (1) - Y (0))$ are measurable functions of $W$ by [F3], and are integrable because $A (z)$ is binary and $Y (0), Y (1)$ are integrable. Independence in [A2] and [F1] make each such function independent of $Z$ .

2.1A1F2step 1.1

By [F2], for each of these integrable functions $H (W)$ we have $E [I_{z} H (W)] = E [I_{z}] E [H (W)] = P (Z = z) E [H (W)]$ . Dividing by the positive $P (Z = z)$ and using the event-conditioned mean in [A1] gives $E [H (W) ∣ Z = z] = E [H (W)]$ .

3.1A1A2step 2.1

On ${Z = z}$ , consistency and exclusion in [A2] give $A = A (z)$ and $Y = Y (A (z)) = Y (0) + A (z) (Y (1) - Y (0))$ . Thus step 2.1 yields $E [A ∣ Z = z] = E [A (z)]$ and $E [Y ∣ Z = z] = E [Y (0) + A (z) τ]$ .

4.1F4step 3.1algebra∎

Subtracting the $z = 0$ identities from the $z = 1$ identities in step 3.1, and using [F4], gives $E [A (1) - A (0)]$ and $E [(A (1) - A (0)) τ]$ , respectively.

Depends on

Used by

Dependency tree · two levels

31 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, Technical Point 16.7, printed pp. 224–225 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), Randomized Encouragement assumptions and LATE proof (standard reference, not scraped)
Judea Pearl, Causal Inference in Statistics: An Overview, §3.5.2, journal pp. 122–123 (standard reference, not scraped)