# Counterfactual Expectation Calculation

$$\newcommand{\doop}{\operatorname{do}}$$ Problem: (This is from Study question 4.3.1 from Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell.) Consider the causal model in the following figure and assume that $$U_1$$ and $$U_2$$ are two independent Gaussian variables, each with zero mean and unit variance.

Find the expected salary of workers at skill level $$Z=z$$ had they received $$x$$ years of college education. [Hint: Use Theorem 4.3.2, with $$e:Z=z,$$ and the fact that for any two Gaussian variables, say $$X$$ and $$Z,$$ we have $$E[X|Z=z]=E[X]+R_{XZ}(z-E[Z]).$$ Use the material in Sections 3.8.2 and 3.8.3 to express all regression coefficients in terms of structural parameters, and show that $$E[Y_x|Z=z]=abx+\frac{bz}{1+a^2}.]$$

Here, $$X$$ is education, $$Z$$ is skill, and $$Y$$ is salary. The accompanying SEM is \begin{align*} X&=U_1\\ Z&=aX+U_2\\ Y&=bZ. \end{align*}

My Work So Far: We are called on to compute $$E[Y_x|Z=z].$$ Now Theorem 4.3.2 states: Let $$\tau$$ be the slope of the total effect of $$X$$ on $$Y,$$ $$\tau=E[Y|\doop(x+1)]-E[Y|\doop(x)]$$ then, for any evidence $$Z=e,$$ we have $$E[Y_{X=x}|Z=e]=E[Y|Z=e]+\tau(x-E[X|Z=e]).$$ For our problem, with $$e:Z=z,$$ we have $$E[Y_{X=x}|Z=z]=E[Y|Z=z]+\tau(x-E[X|Z=z]).$$ Not sure where to go from there.

Now I know that this is a non-deterministic counterfactual problem, which means the process should be:

1. Abduction: Update $$P(U)$$ by the evidence to obtain $$P(U|E=e).$$
2. Action: Modify the model, $$M,$$ by removing the structural equations for the variables in $$X$$ and replacing them with the appropriate functions $$X=x,$$ to obtain the modified model, $$M_x.$$
3. Prediction: Use the modified model, $$M_x,$$ and the updated probabilities over the $$U$$ variables, $$P(U|E=e),$$ to compute the expectation of $$Y,$$ the consequence of the counterfactual.

So, for abduction, am I right in thinking that the only evidence we're using right now is $$Z?$$ In that case, we want to determine the $$U_1$$ and $$U_2$$ that correspond to $$Z=z.$$ We have the two equations \begin{align*} X&=U_1\\ z&=aX+U_2, \end{align*} or \begin{align*} X&=U_1\\ z-aX&=U_2. \end{align*} Without knowing the pre-condition value of $$X,$$ it's not clear how to continue. How do I continue? I'm also really not understanding the hint. Any thoughts about the hint?

Note, I have cross-posted this at

• Thanks for writing this up. There's a small error in the question, it should read $E[X|Z=z]=E[X]+R_{XZ}(z-E[Z])$ (too short to edit) – mmdanziger Nov 2 '20 at 10:26
• You're quite right. Fixed. Thanks much! – Adrian Keister Nov 2 '20 at 13:18

1. $$E[x|z]=\beta_{xz}\,z,$$ because of the model, and the relationship between $$x$$ and $$z.$$ Here $$\beta_{xz}$$ is the regression coefficient, as in $$X=\beta_{xz}Z.$$
2. Reversing regression coefficients requires knowing the variances: $$\beta_{xz}\sigma_z^2=\beta_{zx}\sigma_x^2.$$
3. The slope of the total effect, $$\tau,$$ you can read off the diagram as $$\tau=ab.$$
4. Variances add like this: if $$Z=aX+U_2,$$ then $$\sigma_z^2=a^2\sigma_x^2+\sigma_{U_2}^2.$$