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From the diagram, I need to find the effect of education on those students whose salary is $Y=1$. I was given the hint to use $E[Y_1 − Y_0|Y = 1]$.

My attempt:

I tried to expand the above equation and got as follows:

$E[Y_1 − Y_0|Y = 1] = E[Y|Y=1]-t(0-E[X|Y=1])$

After this I'm stuck and need help to solve this.

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1 Answer 1

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Ah, I see you are studying Pearl's Causal Inference in Statistics: A Primer, co-written with Glymour and Jewell. Excellent choice!

Theorem 4.3.2 says that if $\tau$ is the total effect of $X$ on $Y$, which is $ab$ in this case, then, for any evidence $Z=e,$ we have $$E[Y_{X=x}|Z=e]=E[Y|Z=e]+\tau(x-E[X|Z=e]).$$ So, in our case, if we want to compute $E[Y_1-Y_0|Y=1],$ we must evaluate \begin{align*} E[Y_1-Y_0|Y=1] &=E[Y|Z=e]+ab(1-E[X|Z=e])-E[Y|Z=e]-ab(0-E[X|Z=e])\\ &=E[Y|Y=1]+ab(1-E[X|Y=1])-E[Y|Y=1]+abE[X|Y=1]\\ &=1+ab-abE[X|Y=1]-1+abE[X|Y=1]. \end{align*} Can you finish?

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