What type of multivariate linear regression is this? I'm trying to reproduce a result from a book (see bottom) and it doesn't work. I would like to do some further readings about this method but he doesn't specifically give the method other than a formula.
I've already tried identifying the method using Wikipedia with no success.
This is the linear model: $Z = cX + dY$
He provides an equation for $c$ :
$$
c = \frac{{\rm corr}(X,Z) - {\rm corr}(Y,Z){\rm corr}(X,Y)}{1-{\rm corr}(X,Y)^2}
$$
The weight $d$ is calculated equivalently. He then writes that $c$ and $d$ can be used to calculate the squared error. Using trial and error, I figured out that the correlation coefficient ${\rm corr}()$ is very likely Spearman's $\rho$ (At least that's the method he used so far to calculate correlation coefficients.). Additionally, he mentions that the means of $X$ and $Y$ are assumed to be vanishing.
I'm relatively new to linear regression, so at first I thought it's least squares but this equation doesn't look like it to me.
Does anyone recognize this method and can give me a name, so I can read more about it?
This all comes from a popular science book on football/soccer statistics. The book is in German. The formula can be found in Appendix A7.3 on pp. 297. The particular example, I'm trying to reproduce can be found on pp. 140.
 A: The equation given for $c$ is suspiciously like the equation for a semi-partial correlation1:
$$
r_{Z(X|Y)} = \frac{{\rm corr}(X,Z) - {\rm corr}(Y,Z){\rm corr}(X,Y)}{\sqrt{1-{\rm corr}(X,Y)^2}}\ ,
$$
except that your denominator does not include the square root.  That might be a typo2.  As a result, I wonder if the author isn't talking about the following structural equations model (SEM) with $Z$ caused by $X$ and $Y$, which are themselves correlated:  

This is a rather low-powered usage of SEM, it's just that you are analyzing a correlation matrix according to a specified underlying pattern and finding the path coefficients (i.e., $c$ and $d$) that will optimally reproduce the observed pattern of correlations using the specified path model.  Because you are working with the correlation matrix, the variables will all have mean zero.  The paths turn out to be the semi-partial correlations because you have specified that $X$ and $Y$ are correlated, but $Z$ is simply a function of $X$ and $Y$, their inter-correlation notwithstanding.  
1. To learn more about semi-partial correlations, see this website or my answer here: What's the order of correlation?
2. If it's not a typo, I have no idea what this might be. 
A: The formula cited is simply the formula for the beta coefficient, or the standardized regression coefficient. Thus, the book is simply stating the formula for calculating standardized versions of c and d in the regression equation stated.
Any stats program should be able to provide standardized regression coefficients. However, often unstandardized regression coefficients will be provided as default and some option will need to be selected to return the standardized values. Perhaps this is why you could not get the same results.
