linear causal model

Question

Currently I’m focused on linear causal model expressed as a structural equation like this:

$y = \beta_1 x_1 + \beta_2 x_2 + … + \beta_k x_k + u$

where $E[u|x_1,x_2,…,x_k]=0$ (exogenous error)

we don’t know nothing about about causal nexus and/or statistical dependencies among $x$s. However all variables involved ($x$s) are measurable and no others are relevant for explanation of $y$. The structural parameters $\beta_i$ are unknown constants.

I know that several DAG are compatible with the specification above (see this strongly related question for some examples: Causality: Structural Causal Model and DAG). Now the specification above is quite general but, if I have understood correctly, the following related statements are right:

1) The structural coefficients $\beta_i$ represent the direct causal effect of $x_i$ on $y$ (for $i=1,…k$) and we have $E[y|do(x_1,…,x_k)]=E[y|x_1,…,x_k]= \beta_1 x_1 + \beta_2 x_2 + … + \beta_k x_k$. Therefore those effect are identified. In other words all the direct effects are computable by the regression written.

2) If there aren’t causal nexus among $x$s and $x$s are statistical independent each others we have also that $E[y|do(x_i)]=E[y|x_i]=\beta_i x_i$ for $i=1,…k$. If some dependencies exist this conclusion is no more true.

3) If there aren’t causal nexus among $x$s the direct causal effect of $x_i$ on $y$ coincide with their total causal effects. Moreover the total is the effect that in experimental language is known as average causal effect (ACE) or average treatment effect on the treated (ATT); then what is usually intended as causal effect in econometrics and what backdoor criterion refers on.

4) If there are causal nexus among $x$s but we don’t now what they are, we cannot know what combination of structural parameters give us the total effects. Therefore is not possible to identify them.

5) if we know all the causal nexus among $x$s and there aren’t unobserved common cause or, equivalently, there are no related structural errors, then the causal effect (total and direct) are identifiable.

I made some mistakes? If yes can you give me some easiest as possible counterexample and, then, the correct statements?

EDIT: I edited the post deleting the two final sub-questions. I hope that now it sounds good for moderators.

Answer 1

By structural I will understand that the structural equation is encoding the averge response of Y when the x are manipulated, that is:

$$ E[Y|do(x_1, \dots, x_k)]= \beta_1x_1 + \dots + \beta_kx_k $$

So answering your questions:

1. That's correct. The proof is simple, since

$$ E[Y|x_1, \dots, x_k] = \beta_1x_1 + \dots + \beta_kx_k + E[u|x_1, \dots, x_k] = \beta_1x_1 + \dots + \beta_kx_k $$

As you said, these are the controlled direct effects of each $x_i$ when holding the other $x_j$ fixed.

2. If there are no causal effects among the $X$ and they are not confounded, then these coefficients are also the total effects. To see this, draw a DAG with all $X$ pointing to $Y$ and no arrow between the $X$. Note that to identify the total effect with $E[Y|x_i]$ alone you need that $X_i$ is unconfounded without conditioning on all the other $X$ as well.

3. Correct.

4. Correct. For an example, imagine the graph $X_1 \rightarrow X_2$, $X_2\rightarrow Y$ and $X_1 \rightarrow Y$. Here $X_2$ is a mediator, and the total and direct effects of $X_1$ on $Y$ are different. But you could just flip the positions of $X_1$ and $X_2$ and now $X_2$ is a confounder for $X_1$, and the total and direct effects of $X_1$ on $Y$ are the same.

5. Correct. If you know the DAG and the model is Markovian (all errors are independent) then all causal effects (direct and indirect) are identified.