Under the <partially linear model>,
$$
\mathbb E[Y\mid A=1,X]-\mathbb E[Y\mid A=0,X]
=\beta_2.
$$
Inserting this constant conditional contrast into the identification formula from part i gives
$$
\beta_1=\mathbb E[\beta_2]=\beta_2.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 3 / ii / Solution

A partially linear model combines a linear coefficient with an unrestricted nuisance function, for example $\mathbb E[Y\mid A,X]=\beta A+g(X)$.


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