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I know that in general a structural causal model (SCM) can be written in terms of structural equations. And in a more qualitative but formal manner, we can rewrite a structural model in terms of DAG.

Now suppose we have a simple structural equation like this:

$$y = \beta_0 + \beta_1x_1 + \beta_2 x_2 + .. + \beta_n x_n + e$$

where $e$ is completly exogenous. We don't know anything else about the model.

How can we write this model in a DAG?

EDIT Moreover I have some sub-questions:

1) the SCM above imply that $E[y|do(x_1,...,x_n)] = \beta_0 + \beta_1x_1 + .. + \beta_n x_n$ , now is true that $E[y|do(x_1,...,x_n)] = E[y|x_1,...,x_n]$ regardless the causal nexus among $x$s?

2) if we known only a subsample of dependent/causal variables like $x_1,...x_k$ with $k<n$ then we have a problem that sound like omitted variables. Now exist a way for find the others variables ($x_{k+1},...,x_n$)?

2a)If it exist, the causal nexus among $x$s become relevant?

2b)If it not exist, is still possible to identify the causal parameters $\beta_1,...,\beta_k$?

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  • $\begingroup$ "As become this model in DAG form ?" This isn't standard English, so I'm not sure what you're asking. DAGs are nonparametric, but your equation is a specific parameterization, so if you were to draw a DAG for it, there would be many parameterizations consistent with the DAG. Also, it's not clear what the dependence relations are among the Xs; do they share common causes? Do they have a causal ordering? $\endgroup$ – Noah Jul 10 at 19:53
  • $\begingroup$ Firstly I'm sorry for my english. However the equation that I want to translate in DAG form is exactly the above one. Moreover, in general, for any DAG structure there are many parametric equation forms but for any parametric equations there is onli one DAG structure. It's correct? $\endgroup$ – markowitz Jul 11 at 9:30
  • $\begingroup$ About the relations among the Xs the problem is exactly that we don't know much. At maxim we can say that all Xs are causes, at least potentially, for y and that Xs are, at least potentially, correlated each others. Clear causal nexus among the Xs are absolutely don't known. $\endgroup$ – markowitz Jul 11 at 9:36
  • $\begingroup$ A simple DAG with arrows pointing from the $x$s to $y$ is all you could draw, but that wouldn't be any more descriptive than saying "all the $x$s cause $y$". A DAG provides less information than a parametric structural causal model. Furthermore, without any assumptions about the causal relations among the $x$s, this DAG couldn't be used for any causal analysis (i.e., to identify confounders, colliders, or instruments). $\endgroup$ – Noah Jul 12 at 16:46
  • $\begingroup$ I feared something like that. However is a matter of fact that sometimes econometric models tried to achieve causal conclusions with an econometric theory that, in causal term, go not beyond the above conditions. $\endgroup$ – markowitz Jul 16 at 20:56
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Your model statement specifies a class of DAGs, not a single DAG. That is, all DAGs in which $x_1, \dots, x_n$ are direct causes of $y$, and $e$ is exogenous are DAGs compatibles with your assumptions.

For instance, for simplicity, say we have only $x_1$ and $x_2$. Then, among several other alternatives, the following DAGs would be compatible with your model specification:

enter image description here enter image description here enter image description here

But the following DAG would not be compatible (since the error term of $Y$ is correlated with the error term with $x_2$, but note in this DAG the causal effect of $x_1$ is still identified):

enter image description here

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  • $\begingroup$ Do bidirectional dashed arrows represent related structural errors in DAGs? $\endgroup$ – markowitz Aug 13 at 11:54
  • $\begingroup$ @markowitz exactly $\endgroup$ – Carlos Cinelli Aug 13 at 20:24
  • $\begingroup$ Probably In this framework full understanding of the backdoor criterion is necessary (I'm studing). However It seems me correct to say that if we collect the data and regress $y$ on $x_1$ and $x_2$ in neither of the three case above we achieve both the correct causal effect of $x_1$ then $x_2$ on $y$. Its right? $\endgroup$ – markowitz Aug 26 at 12:44
  • $\begingroup$ Exist a DAG specification for which a simple regression $y = \beta_1 x_1 + \beta_2 x_2 + e$ return parameters where both have causal meaning ? $\endgroup$ – markowitz Sep 4 at 12:05
  • $\begingroup$ @markowitz in the three first models, all of them would return parameters with causal meaning-- the controlled direct effects, $E[y|do(x_1, x_2)] = \beta_1x_1 +\beta_2x_2 $. $\endgroup$ – Carlos Cinelli Sep 5 at 6:26

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