Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 2 / a

Table of contents
- Solution a

Solution

Conditionally on

X

, a valid instrumental variable

Z

must satisfy three core conditions.

Instrument relevance: $Z$ changes the conditional distribution of $A$ , for example $Cov (Z, A ∣ X) \neq = 0$ on a set of positive probability.
Instrumental-variable independence: $Z$ is independent of latent outcome causes and the relevant potential outcomes, for example $Z ⊥ {Y (a) : a} ∣ X$ .
The exclusion restriction: $Z$ affects $Y$ only through $A$ , written $Y (a, z) = Y (a)$ .

Together with consistency of potential outcomes and positivity in causal inference, these assumptions make variation in

A

induced by

Z

causally interpretable. In the displayed graph, relevance is the edge

Z \to A

, independence is the absence of a path from

Z

to

U

after conditioning on

X

, and exclusion is the absence of a direct

Z \to Y

edge.

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