In <potential outcome> notation, a valid <instrumental variable> $Z$ must satisfy:

* <Instrument relevance>: the conditional law of $A$ changes with $Z$.
* <Instrumental-variable independence>: $Z$ is independent of the joint collection of potential treatments and outcomes, such as $\{A(z),Y(a):z,a\}$.
* The <exclusion restriction>: $Y(z,a)=Y(a)$, so $Z$ can affect $Y$ only through $A$.
* <Consistency of potential outcomes> and no <interference in causal inference> connect the potential variables to the observations.

In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, including
$$
Z_1\leftarrow M_1\to C\to Y
\qquad\text{and}\qquad
Z_1\leftarrow S\to Z_3\to Y.
$$
Equivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating <instrumental-variable independence>; it is not a valid marginal instrument.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-221/1/i/solution