Infer one link of a causal structure, from observations Suppose I have continuous random variables $X,Y,Z$ with the following causal structure:
                                                      
I hypothesize a simple regression model for each r.v., specifically,
\begin{aligned}[l]
Y &= a_1 X + \cal{N}(\mu_1,\sigma_1^2),\\ Z &= a_2 X + a_3 Y + \cal{N}(\mu_2,\sigma_2^2)
\end{aligned}
I have many observations sampled from the joint distribution $(X,Y,Z)$ and would like to infer the parameters.  I am particularly interested in inferring $a_3$, i.e., the link from $Y$ to $Z$.
What method is appropriate for this?  I could imagine using multilinear approximation to fit a regression estimate $Z \sim \alpha_1 X + \alpha_2 Y + \beta$, and then using $\alpha_2$ as my estimate for $a_3$; is this a good approach?
 A: Based on your DAG, and under linearity, we have a SEM with two structural equations:
$Z = \alpha_3 Y + \alpha_2 X + \epsilon_1$
$Y = \alpha_1 X + \epsilon_2$
Here $\alpha_{1/2/3}$ are the direct causal effects.
Now, we can see that
$Z = \alpha_3 \alpha_1 X + \alpha_2 X +  \alpha_3 \epsilon_2 + \epsilon_1 = \alpha_4 X  +  \epsilon_3$
where $\alpha_4 = \alpha_3 \alpha_1 + \alpha_2$ represents the total causal effect of $X$ on $Z$
and $\epsilon_3 = \alpha_3 \epsilon_2 + \epsilon_1 $
Now I add some needed (causal) assumptions more. In the initial two structural equations the structural errors are exogenous  ($E[\epsilon_1 | Y, X]=0$ and $E[\epsilon_2 | X]=0$) and them are independent.
So, as consequence, in the last structural equations the structural error $\epsilon_3$  is exogenous too ($E[\epsilon_3 | X]=0$)
Then, you can perform three useful regressions
$Z = \theta_1 X + u_1$
$Y = \theta_2 X + u_2$
$Z = \theta_3 Y + \theta_4 X + u_3$
here  $\theta_1$ identify $\alpha_4$, $\theta_2$ identify $\alpha_1$, $\theta_3$ identify $\alpha_3$ (what you looking for) and $\theta_4$ identify $\alpha_2$.
Note that not all regressions are "good". For example if we run this regression
$Z = \theta_5 Y + u_4$
the coefficient $\theta_5$ do not identify any parameter of the SEM. Indeed $\theta_5$ is biased for $\alpha_3$ ($X$ play as omitted/confounder variable).
