Counterfactual Expectation Calculation

In summary, these steps are:1. Use Theorem 4.3.2 to find the expected salary of workers at skill level $Z=z$ had they received $x$ years of college education.2. Use the information in Section 3.8.2 and 3.8.3 to express all regression coefficients in terms of structural parameters, and show that $$E[Y_x|Z=z]=abx+\frac{bz}{1+a^2}.]$$3. Use the information in 2 to determine the $U_1$ and $U_2$ that correspond to $Z=z.$4. Reverse regression coefficients to obtain the variances
  • #1
Ackbach
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$\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 $E[X|Z=z]=E[Z]+R_{XZ}(z-E[Z]).$ 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 $$E[Y_x|Z=z]=abx+\frac{bz}{1+a^2}.]$$

View attachment 9643

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?

Thanks for your time!

Note: I have cross-posted this at Cross-Validated:

https://stats.stackexchange.com/questions/457740/counterfactual-expectation-calculation
 

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  • #2
I have obtained access to the full solutions manual, after contacting Wiley about it. I will not type up the solution in full, but simply note a few critical pieces of information I was missing in order to answer this question:

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 $\sigma_z^2=a^2\sigma_x^2+\sigma_{U_2}^2.$

This is sufficient information to obtain the desired result.
 

FAQ: Counterfactual Expectation Calculation

What is counterfactual expectation calculation?

Counterfactual expectation calculation is a statistical method used to estimate the expected outcome of a particular event or situation by comparing it to a hypothetical scenario in which certain factors or variables are changed. It is often used in research and analysis to assess the potential impact of different decisions or actions.

How is counterfactual expectation calculation different from other statistical methods?

Unlike other statistical methods that focus on analyzing past data, counterfactual expectation calculation involves creating a hypothetical scenario and comparing it to the actual situation. This allows for a more nuanced understanding of causality and can help identify the potential effects of different interventions or changes.

What are some common applications of counterfactual expectation calculation?

Counterfactual expectation calculation is commonly used in fields such as economics, public policy, and social sciences to evaluate the effectiveness of policies and interventions. It can also be used in marketing and business to assess the impact of different strategies and decisions on consumer behavior and market outcomes.

What are some challenges associated with counterfactual expectation calculation?

One of the main challenges of counterfactual expectation calculation is the difficulty in accurately identifying and measuring all relevant variables and factors. This can lead to biased or inaccurate results. Additionally, creating a truly comparable hypothetical scenario can be complex and time-consuming.

How can counterfactual expectation calculation be used to inform decision-making?

By providing an estimate of the expected outcome of a particular decision or action, counterfactual expectation calculation can help inform decision-making processes. It can also be used to compare the potential outcomes of different options and identify the most effective course of action. However, it is important to consider the limitations and potential biases of this method when using it to make decisions.

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