Expectation of structural equation

Question

I am trying to learn about structural equations, and in this post here Correlation, regression and causal modeling I am having difficulties trying to prove the answer.

The problem is, given structural equations $$U=\epsilon_u$$ $$X=\delta U +\epsilon_x $$ $$Y=\beta X+\gamma U+\epsilon_y$$

Where all terms denoted by ϵ are mean zero and mutually independent and U, X and Y have been standardized (mean zero and unit variance). Suppose U is unobserved, then if we observe X=x: $$E[Y|X=x]=(\beta +\gamma \delta )x \:\:\:\:\:\:\:equation 1$$ If we set X=x, then: $$E[Y|X=x]=\beta x\:\:\:\:\:\:\:equation 2$$

How can I prove equation 1 and equation 2? I'm stuck at equation 1 because if U is unobserved, how do you do the expectation?

Answer 1

Edit: Carlos Cinelli's answer is the right one.

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In this part we don't need to worry about unobservancy. We know DGP, so we know everything. We calculate expectations, as for standard probability calculus random variables.

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In the first situation we can calculate $U$ as a function of X:

$U = \frac{(X - \epsilon_x)}{\delta}$

So then when we derive $Y$, it would be:

$Y = \beta X + \frac{\gamma}{\delta}(X - \epsilon_x) + \epsilon_y$

and after some simplification:

$Y = (\beta + \frac{\gamma}{\delta})X - \frac{\gamma}{\delta} \epsilon_x + \epsilon_y$

When we take expectation, we use assumption, that every $\epsilon$ has expectation zero and $X$ is equal to x. Then:

$\mathbb E(Y|X=x) = (\beta + \frac{\gamma}{\delta})x$

(So we can see, that the author of mentioned post made a mistake.)

The problem, that $U$ is not observed only has the consequences, that we can not calculate $\mathbb E(Y|X=x, U=u)$ using observed data. We would wish to do so in order to statistically estimate parameter $\beta$ instead of calculating observable $E(Y|X=x)$, which gives us some different value.

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In second situation it is much easier, however when we set value of X, we change DGP, because we performed an experiment (which is denoted as $do(X=x)$):

• $U=\epsilon_u$
• $X=x$
• $Y=\beta X+\gamma U+\epsilon_y$

In this situation we just substitute for $X$ and $U$:

$Y=\beta x+\gamma \epsilon_u+\epsilon_y$

And calculate the expectation:

$\mathbb E(Y|do(X=x)) = \beta x$

In this situation we do not care about unobserved $U$, because we achieve desired effect, when we estimate this expectation with only $x$.

Answer 2

Cure's answer is incorrect.

The formula of the conditional expectation of a bivariate gaussian is:

$$ E[Y\mid X=x] = E[Y] + \frac{Cov(Y,X)}{Var(X)}(x-E[X])\\ $$

Since the variables are standardized, we have that $E[Y] = E[X] = 0$ and that $Var(X) =1$.

And $Cov(Y, X) = \beta Var(X) + \gamma Cov(X, U) = \beta + \gamma\delta$. Thus,

$$E[Y|X=x]=(\beta +\gamma \delta )x$$

As in the original answer.