This question come from this discussion (Under which assumptions a regression can be interpreted causally? ). That discussion touch too arguments and is not possible to speak about all things there. So I pose the question here I give my answer too.
The Data Generating Process interpretation is matter of debate. For examples read here: What is a 'true' model? and What does a data-generating process (DGP) actually mean?
If we want make causal inference properly we have to intend the DGP as in Pearl literature, then his properties are encoded in a Pearl Structural Causal Models (SCM). So if the DGP is known we can consider DGP and SCM as synonyms, otherwise the SCM encode all we know/assume about the DGP. For an exhaustive exposition of SCM read here: do(x) operator meaning? (the Carlos Cinelli answer).
The linear true model is the more used object/name in econometrics literature in place of DGP. In econometric literature the role of causality is important even if many times is not properly treated (for example read: Under which assumptions a regression can be interpreted causally? and Is the linearity assumption in linear regression merely a definition of $\epsilon$? and references therein). Now, remaining simple and closest as possible to the econometrics literature the proper way for make causal inference is consider the true model as an linear SCM.
$y = X’ \theta + \epsilon$
we can interpret all three objects $[y,X, \epsilon]$ as random variables ($X$ is a vector). Read here for more details: linear causal model
Then, the following conditions hold:
In the SCM the sign $=$ stand for “:=” (definition). The causality, implied by definition/assumption, move from the right to the left. Given the variables involved, the SCM is not another representation for the joint probability distribution of them; the SCM is related but different thing. Indeed, in general, for any SCM is possible to find many joint distributions that caracterize the variables involved and, conversely, for any joint distribution of them is possible to find many SCM which these variables come from. However any SCM imply some restrictions for the joint distribution of the variables. These restrictions are the basis for any causal inference.
In our case (above), even if $y$ and $X$ can be observable variables we do not stay in a situation like regression case, where given $(y,X)$, as a consequence, the errors/residuals and parameters are given too (read here: Zero conditional expectation of error in OLS regression )
Indeed $\epsilon$ and $X$ are completely free random variables, and $\theta$s free parameters, and for this reason we can have both situations: $\epsilon$ is a structural causal error that can be exogenous $E[\epsilon|X]=0$ or not $E[\epsilon|X] \neq 0$. The only usual implicit assumption is that $\epsilon$ have zero mean; quite obvious assumption for any kind of errors. Note that about exogeneity the notation $E$ do not stand for usual expectation but for interventional expectation. More formally, and for avoid ambiguity, do-operator would be needed. Exogenous error $E[\epsilon|do(X)]=0$ or not $E[\epsilon|do(X)] \neq 0$. Read here for more about that: conditional and interventional expectation and again here do(x) operator meaning?
The above SCM can be interpreted as a decomposition where things that we put on the right and side part, represent causal assumptions (also the linearity is an implicit causal assumption here). In particular what we put in $X$ and what remain in $\epsilon$ is an assumption too and, then, exogeneity or not is a restriction/assumption about both.
It is easy to simulate $y$ starting from $X$ and $\epsilon$; the previous sign $:=$ stand for that. I talked about random variables in a single equation but the extension to random processes and/or system follow naturally.
People can says: "but in real data I can observe $y$ and $X$ not $\epsilon$". It’s true, indeed $\epsilon$, the structural causal error, is an unobservable variable and, at least in general, exogeneity is an untestable assumption about that.
Moreover people have to refrain them to “visualize” the structural error and its properties, exogeneity as first, from something like data fitting … this is precisely a pure statistical procedure that we have to avoid.
If some identification condition (that are causal assumptions too, like exogeneity) are assumed … is possible to arrive at testable (in statistical sense) implications.
This list is surely uncompleted and, even if I can try to defend what I said, I do not give any warranty about these. I stay here for learn. I’m happy if something above can be add and/or correct. The only condition I want is that all can be documented in causal inference literature.