 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-207/1/1/solution

Let

Z

indicate a recent birthday event,

A

indicate a social gathering, and

Y

denote subsequent household COVID-19 infection. The standard instrumental variable conditions are:

Instrument relevance: $Z$ changes the probability or intensity of $A$ .
Instrumental-variable independence: conditional on chosen baseline covariates, birthday timing is independent of unmeasured causes of infection and of the relevant potential outcomes.
The exclusion restriction: $Z$ affects $Y$ only through the social gathering $A$ .
For a local average treatment effect, instrumental-variable monotonicity excludes households that would hold a gathering without a birthday event but would suppress it because of one.

The usual consistency in causal inference, well-defined exposure, and absence of relevant interference in causal inference are also needed to interpret the result causally.

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