Conditionally on $X$, a valid <instrumental variable> $Z$ must satisfy three core conditions.

* <Instrument relevance>: $Z$ changes the conditional distribution of $A$, for example $\operatorname{Cov}(Z,A\mid X)\ne0$ on a set of positive probability.
* <Instrumental-variable independence>: $Z$ is independent of latent outcome causes and the relevant <potential outcomes>, for example $Z\mathrel\perp\{Y(a):a\}\mid X$.
* 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-221/2/a/solution