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Binary-instrument Wald ratio

Definition

For a setup of Binary instrumental-variable potential-outcome setup with integrable observed outcome $Y$ , define the reduced-form outcome contrast and first-stage treatment contrast by $Δ_{Y} : = E [Y ∣ Z = 1] - E [Y ∣ Z = 0], Δ_{A} : = E [A ∣ Z = 1] - E [A ∣ Z = 0] .$ Whenever $Δ_{A} \neq 0$ , the Wald ratio is the observed-data functional $ψ_{W} : = \frac{Δ_{Y}}{Δ_{A}} .$ This definition alone does not assert that the ratio equals a treatment effect.

Depends on

Binary instrumental-variable potential-outcome setup

Used by

Core IV conditions alone do not identify the population ATE Counterexample
A direct encouragement effect changes the Wald ratio Example
A finite randomized encouragement example Example
A small first stage magnifies reduced-form changes Example
Defiers can make the Wald ratio differ from the complier effect Example
With heterogeneous effects, LATE can differ from ATE Example
A constant additive effect is identified by the Wald ratio Theorem
The Wald ratio identifies the complier average treatment effect Theorem

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, §16.2, printed pp. 212–214 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), Proof of Identification of the LATE (standard reference, not scraped)
Judea Pearl, Causal Inference in Statistics: An Overview, §3.5.2, journal pp. 122–123 (standard reference, not scraped)