Mediation - uniqueness of coefficients My question is general to mediation but perhaps best illustrated with example. I have the following situation from marketing - TV GRPs drives Paid Search clicks, and I suspect both of these independently drive visits to the marketer's website. I assume that there is no other independent variable that makes sense in the model.


*

*I regress site visits on TV GRPs and end up with a significant coefficient - this is the total effect of the IV TV GRP's on site visits - easy

*I regress Paid Search clicks on TV GRPs and again see the relationship is strong. From my business knowledge, I know the relationship is causal.

*I regress site visits on TV GRP's and PS clicks jointly. I see that the coefficient on PS clicks is highly significant and that on TV GRP's is mildly significant.


What does this mean? Since there is correlation (more specifically, a causal relationship between IV TV GRP's and mediator PS clicks), one cannot expect the regression coefficients in the third regression above to be unique or interpretable. There are infinite ways of combining TV GRP's and PS clicks to yield the outcome variable that are all equivalent since all we require is for the total effect of TV GRP's and the total value of the intercept term to more-or-less equal the coefficient and intercept from the first regression. There is no other variable that could possibly force a certain value for the coefficient of the mediator. How then do we conclude what the direct and indirect effect of TV GRPs on site visits are?
 A: There are always infinite ways of combining the values of predictors to arrive at an outcome (plus error). The point is that in any model you run, you are estimating the coefficients of the best fitting model that describes that relationship. It's true that the coefficients on TV GRPs and PS clicks could be anything and you'd still have an algebraically valid relationship between the total, indirect, and direct effects, but the specific coefficients estimated are based on an algorithm selected to produce the best fitting model. Assuming you use maximum likelihood or ordinary least squares to fit all the models, the estimates are guaranteed to be unbiased, consistent, and maximally efficient (if the distributional and functional form assumptions are correct).
Of course, there is uncertainty in your estimates, so one shouldn't place too much stock in the estimates without considering their standard errors. In addition, there is an emerging area of research in fungible estimates, which are estimates that only slightly worse fitting than the unbiased estimates but which might have (radically) different values. 
