Does it make sense to use correlations with residuals to infer causality? My question concerns the following situation: say we have three variables, A, B and X. All three variables are moderately and positively correlated between them. Assume that, for a priori reasons, I know either A or B causes X. I want to differentiate between "A causes B that causes X" and "B causes A that causes X" (assume also I'm not worried about other possibilities, e.g. a third variable causes A and B).
What I want to know is if using the residuals is a sound approach (I have a feeling it isn't, but I'm not being able to express it formally).
Imagine that the residuals of a regression of X on B are correlated with A, but the residuals of the regression of X on A are not correlated with B. Would that support the hypothesis "A causes B that causes X"? Or, a weaker assertion, would that mean that (I'm not sure this is the right way of saying it) all the variance explained by B is "contained" in the variance explained by A (that is, B is redundant)?
If that is not a sound approach, then what would be a nicer way of differentiating between A and B? Partial correlations? Principal component analysis?
I hope it makes sense. Thanks a lot.
 A: What you describe IS if fact partial correlation. I do not think you can use it to establish that there is, or is not, causality. 
A: This is a relevant question because, implicitly, this is what is done with path analysis. Using a mixture of latent and observed variables, two nodes in a graphical model are inferred to be causally related if their residuals are correlated. These residuals are calculated by fitting the linear model which conditions on all parent nodes in the network. So when you ask about "correlations with residuals", it's important to underscore these are not just some higgledy piggledy residuals gathered from fitting a kitchen-sink model controlling for every node in a network and variable in a dataset. These "parent nodes" must be confounders: and they must be the complete set of confounders for the assumptions of causality to be met. These assumptions (and more) are outlined in Judea Pearl's text, "Causality" 2nd ed.
An important point: if I fit the linear regression model for $Y$ having conditional mean as a linear combinations of $X_1, X_2, \ldots, X_p$, the inference for, say, the $X_1$ coefficient being non-zero controlling for $X_2, \ldots, X_p$ is identical to inference on the bivariate correlation of $Y$'s residuals after controlling for $X_2, \ldots, X_p$ and $X_1$'s residuals after controlling for $X_2, \ldots, X_p$.
In summary, the procedure is equivalent to linear regression with fully observed variables and path analysis with a mixture of observed and latent variables, and is thus subject to all the relevant conditions of causality.
