Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 4 / b / ii / Solution

A sufficient causal condition is conditional exchangeabilitytogether with consistency of potential outcomes and positivity in causal inference. For the ordinary-least-squares coefficient itself to equal one common causal effect, also require the correctly specified additive conditional-mean modelThen $\beta$ is the homogeneous treatment effect, and the adjusted coefficient consistently estimates both conditional effects and the average treatment effect in the analyzed population.