In the Chen and Pearl (2013) article there are several critics about econometrics textbooks. Currently I try to understand more about it. In particular the Authors written (pag 4, footnote 5):
From a causal analytic perspective, $X$ is exogeneous if $E[Y |X] = E[Y |do(X)]$ (Pearl, 2000). However, for purposes of this paper, we will use the aforementioned defnition in which $X$ is exogenous if it is independent of $e$. Note that if $X$ is independent of $e$ then $E[Y |X] = E[Y |do(X)]$. The converse may not hold. For example, when $e$ is a vector of factors with cancelling influences on $Y$ .
If I understand correctly "e", at least in general, is a structural error and not regression residual.
The structural model is $y=bX+ e$.
Then if $X$ is independent of $e$, the regression residual of a regression (OLS world) write with the same form of structural model are equal to structural error and the slope parameter have causal meaning. (conditional expectation is always true for OLS regression)
My question is: all structural model apart, if the residuals regression are completely independent of regressors, not only uncorrelated (true by costruction), parameters have a causal meaning? Or residual say nothing in itself about causality even under independance?