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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 1 / 1 / Solution

<Instrumental-variable monotonicity> is
$$
A(1)\geq A(0)\quad\text{for every subject}.
$$
It excludes defiers who would quit without the incentive but continue smoking when offered it. The possible <principal strata> are then never-takers $(0,0)$, compliers $(0,1)$, and always-takers $(1,1)$.

By random assignment, consistency, and exclusion,
$$
\begin{aligned}
\mathbb E[Y\mid Z=1]-\mathbb E[Y\mid Z=0]
&=\mathbb E\{Y(A(1))-Y(A(0))\}\\
&=\mathbb E[(Y(1)-Y(0))(A(1)-A(0))].
\end{aligned}
$$
The second identity follows by checking the two possible binary exposure values. Under monotonicity, $A(1)-A(0)$ is the indicator of being a complier. Therefore the numerator is
$$
\mathbb P(\text{complier})
\mathbb E[Y(1)-Y(0)\mid\text{complier}],
$$
while
$$
\mathbb E[A\mid Z=1]-\mathbb E[A\mid Z=0]
=\mathbb E[A(1)-A(0)]
=\mathbb P(\text{complier}).
$$
Their ratio is the <complier average treatment effect>, also called the <local average treatment effect>.

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

{title2=LATE}
{wiki}

The local average treatment effect is the average <causal effect> among compliers, the units whose treatment status changes in the direction encouraged by the instrument.

= Complier average treatment effect
{synonym}

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