How to compare the effect of $X$ on $Y_{1}$ and $Y_{2}$, when $Y_{1}$ and $Y_{2}$ are correlated? I have a single IV ($X$) and two DVs ($Y_{1}$, $Y_{2}$). All variables are continuous. $Y_{1}$ and $Y_{2}$ are significantly and moderately correlated. I would like to know if $X$ is more strongly correlated to $Y_{1}$ or $Y_{2}$ (or vice versa). What I think I have to do is to "control for" or "partial out" the correlation between $Y_{1}$ and $Y_{2}$. 
First, I don't think I want a partial correlation, as that controls for the shared variance between the control variable (let's say it's $Y_{2}$ here) and both the IV ($X$) and the DV ($Y_{1}$). I only want to control for the shared variance between $Y_{1}$ and $Y_{2}$. 
Second, I don't think I want a semi-partial correlation in a multiple regression, as that only controls for the variance shared by the IV ($X$) and the control ($Y_{2}$). 
Third, I don't think I want a MANOVA, as I don't want to know what the effect of $X$ on $Y_{1}$ and $Y_{2}$ (linearly combined) is. I want to know how the two correlation differ.
I seem to want a semi-partial correlation in which the variance shared by $Y_{1}$ and $Y_{2}$ is shared. So, it seems that I should compute residuals for both $Y_{1}$ and $Y_{2}$. To do this, I'll regress $Y_{1}$ on $Y_{2}$ to save the residuals for $Y_{1}$ ($Y_{1}^{*}$), and vice versa for $Y_{2}$ ($Y_{2}^{*}$). Then, the correlation between $X$ and $Y_{1}^{*}$ will be the correlation between $X$ and $Y_{1}$ controlling for the variance shared between $Y_{1}$ and $Y_{2}$; vice versa for $X$ and $Y_{2}$. 
My questions:


*

*Is this right?

*Is there a simple way to do this in a statistical package, or do I have to compute the residuals first, then run my correlational or regression analysis with $X$?

*How do I  compare the effects of $X$ in $Y_{1}$ v. $Y_{2}$? If I just run two correlational analyses ($X-Y_{1}^{*}$ and $X-Y_{2}^{*}$), I can just do an r-to-Z test. But what if I run regressions instead? Can I compare betas in this case?
 A: When you have multiple dependent variables the solution is to use set correlation* (SC), canonical correlation analysis, or MANOVA. The SC is a generalization of both correlation analysis and MANOVA (Cohen et al., 2003). “Set correlation is the amount of shared variance ($R^2$) between two sets of variables” (Revelle, n.d.)
It involves three metrics of trace correlations: multivariate $R^2$, symmetric $T^2$, and asymmetric $P^2$. Symmetry refers to the relationship
$$R^2_{X, Y} = R^2_{Y, X}$$
Asymmetry refers to the situation when a variable is added to either side and a decrease in the correlation can be observed. $R^2_{Y, X}$ reflects the generalized variance of set Y accounted for by set X (Cohen et al., 2003).
To compute the set correlation use the R package psych
For the determinants of the correlation matrices:
$$R^2_{Y,X}=1-|R_{Y,X}|/(|R_Y||R_X|)$$
(eq. 16.2.1, p.610, Cohen et al., 2003)
This equation shows that unity minus the shared variance will give you an indication of how much of the variance is not shared.

Notes
As I finally arrived at work where I have my reference books, I was able to find an answer to your question. It should be noted that this is a fairly advanced approach - it can be found in the final, 16th, and last chapter of Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (Cohen, Cohen, West, & Aiken, 2003. Mahwah, NJ: Lawrence Erlbaum.) with the first author having introduced it himself in 1982. See:
Cohen, J. (1982). Set correlation as a general mulitivariate data-analytic method. Multivariate Behavioral Research, 17(3):301-341.  
*Care should be taken to avoid Type I errors, of which the likelihood increases with multiple tests (i.e. sets).
