regression and causation

Question

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?

Answer 1

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?

No, let me show you a simple counter-example. Consider the SCM:

$$ Y = U_{y}\\ X = U_{x} $$

Where the density of the structural error terms $f(U_{y},U_{x})$ is a bivariate gaussian with mean zero, unit variance and covariance $\sigma_{u_{x}u_{y}}$. Here, trivially, neither $X$ nor $Y$ cause each other, that is $E[Y|do(x)] =E[Y]$ and $E[X|do(y)]=E[X]$.

Yet, the regression coefficient of $X$ on $Y$ is $\sigma_{u_{x}u_y}$. Now since the regression residual is constructed as $\epsilon = Y-\sigma_{u_{x}u_y}X$, it is normally distributed (linear combination of normals). Also by construction, $\epsilon$ is uncorrelated with $X$, since $cov(\epsilon, X) = cov(Y, X) - \sigma_{u_{x}u_y} = 0$. But this also implies they are independent, since $\epsilon$ and $X$ are joint normal (no correlation implies independence in multivariate normal distributions).

So here I showed you an example where the residual is completely independent of the regressor, yet the regression coefficient has no causal meaning. But more generally, you should internalize the following mantra: "no causes in, no causes out". It's impossible to make causal inference without causal assumptions, so whenever you wonder whether it's possible for a statistical quantity to have causal meaning, but your set of assumptions make reference only to the joint probability distribution, you can be assured the answer is "no".