How to test causation in econometrics? It's always the case that we need to test possible channels through which independent variable ($x$) affect dependent variable ($y$). For example, education and health status are two possible channels for intergenerational income mobility, i.e. parents' income can affect children's income through either education or health status.
My question is: is there any credible method to test and compare these competitive channels? In other words, can we answer how many explained variation in $y$ is due to $x_1$ and how many due to $x_2$?
In some literature, the author just runs two extra regressions besides the original one ($y$ on $x$): $x_1$ on $x$ and $y$ on $x_1$ and multiply their coefficients together: $dy/dx=dy/dx_1*dx_1/dx$. However, I doubt this will stand if there exist more channels, as there might be severe multicollinearity if $y$ is regressed on multiple channels $x_1$, $x_2$...
Answers with examples from econometric literature will be most appreciated, while any comments and discussion are welcomed!
 A: I want to suggest reading an interview with Angus Deaton, the most recent Nobel Laureate in economics, for a frank assessment of the issues raised by the OPs "channel" question regarding their "test and comparison"...here's the link: 
https://medium.com/@timothyogden/experimental-conversations-angus-deaton-b2f768dffd57#.jr9a1ea8w
And here's a quote:

People turned to RCTs (random control trials) because they got tired
  of all the arguments over observational studies about exogeneity and
  instruments and sample selectivity and all the rest of it. But all of
  those problems come back in somewhat different forms in RCTs. So I
  don’t see a difference in terms of quality of evidence or usefulness.
  There are bad studies of all sorts.

Deaton's point is an honest assessment of the difficulties of untangling ("testing and comparing") confounded information. It is also a point that has been made by many others in other contexts. For instance, the excellent Cosima Shalizi, in a paper critiquing the social network analyses of James Fowler and Nicholas Christakis (http://smr.sagepub.com/content/40/2/211.abstract), notes that several processes analyzed by social theorists are generically confounded:

Homophily, or the formation of social ties due to matching individual
  traits; social contagion, also known as social influence; and the
  causal effect of an individual’s covariates on his or her behavior or
  other measurable responses.

Similarly in studies of aging, it has been noted that the challenges associated with untangling the confounding effects of age, cohort, and temporal period effects are virtually insuperable.
Econometrics is no different in this regard. The metrics or "channels" may have changed, but the difficulties of reliably and accurately decomposing confounding effects related to education, health, poverty, status, wealth, income, etc., remain. The irony of Deaton's point throughout the interview is that RCTs -- the naively imagined "magic bullet" and gold standard for many -- are not able to resolve the problems. As one poster to this thread noted, at that point, "theory" becomes your best guide. Of course, multiple, widely differing theories can all provide an adequate fit to the same data.
A: If you have a regression model $y=b_0+b_1 x_1 +b_2 x_2 + e$, then $\sigma^2_y=b_1^2\sigma^2_{x_1}+b_2^2\sigma^2_{x_2}+b_1b_2\sigma_{x_1,x_2}+b_1\sigma^2_e$. In this regard you could see $\sigma^2_{x_1}\beta^2_1$ as the part of variance $\sigma^2_y$ channeled through variable $x_1$, IF the covariance $\sigma_{x_1,x_2}$ between your variables is low.
It's an analogy to a spectral density, but in spectral analysis your waves are oethogonal. In econometrics your variables are rarely orthogonal. However, you try not to have multi collinearity, so this way to separate the channels still has some meaning.
