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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 221

The <instrumental variable> graph is
$$
Z\longrightarrow A\longrightarrow Y,
\qquad
U\longrightarrow A,
\qquad
U\longrightarrow Y,
$$
with no arrow $Z\to Y$ and no common cause of $Z$ with $A$ or $Y$.

In <potential outcome> notation, a valid instrument requires:

* <Instrumental-variable independence>: $Z\perp\{A(0),A(1),Y(0),Y(1)\}$, strengthened to all relevant joint potential outcomes as needed. Coin flipping makes the incentive assignment independent of quitting behavior and blood pressure under either assignment.
* <Exclusion restriction>: $Y(z,a)=Y(a)$. The incentive can affect blood pressure only by changing whether the subject quits, not through stress, income, or another direct route.
* <Instrument relevance>: $\mathbb P\{A(1)\ne A(0)\}>0$, or at least $\mathbb E[A(1)-A(0)]\ne0$. The monetary incentive must change quitting probability.
* <Consistency of potential outcomes> and no interference: observed quitting and blood pressure equal the potential values under the assigned encouragement and received exposure, and one subject's assignment does not affect another's outcome.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 4 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 4 / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 4 / i / Solution

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