Total effects
Question for the stats experts:
There are three equations, with the same set of independent variables (which we combine in
the matrix X.)
[tex] risk1=X\gamma+e_1 [/tex]
[tex] risk2=X\zeta+e_2 [/tex]
[tex] cc=X\beta+\delta_1 risk1 + \delta_2 risk2 + e_3 [/tex]
we can rewrite cc as:
[tex] cc=X\beta+\delta_1 (X\gamma+e_1) + \delta_2 (X\zeta+e_2) + e_3 [/tex]
[tex] =X\beta+\delta_1 (X\gamma+e_1) + \delta_2 (X\zeta+e_2) + e_3 [/tex]
[tex] =X\beta+X(\delta_1\gamma) + X(\delta_2\zeta) +e_1\delta_1+e_2\delta_2+ e_3 [/tex]
Assume that the covariances between [tex]e_1,e_2[/tex] and [tex]e_3[/tex] are all zero and let [tex]r=e_1\delta_1+e_2\delta_2+ e_3[/tex]. Then we have the following equation
[tex] cc=X(\beta+\delta_1\gamma+\delta_2\gamma)+r [/tex]
Can this be estimated by least squares? Does it yield an estimate of the “total effect” as refered to in the path analysis/structural equation modelling literature?
2 Comments so far
The answer is yes, and yes.
This is an example of “controlling for intermediate outcomes” that pervades the empirical literature. The consensus among “causal inference” scholars seems to be that one should simply not control for them. Path Analysis/Structural Equations Modelling types try to model how the effect comes by.
Causal Inference has an upper hand now, but to tell you the truth I don’t really know why in this particular instance. It probably has to do with an assumption that further modelling makes the analysis less robust. But there are hidden assumptions in this statement that I would like to see fleshed out.