I am trying to figure out the relationship between correlations of variables where one of the variables defined as the difference between two other variables.
I have variables x and z, which are positively correlated. I define a new variable, x-z = y and find Corr(x,y). What I am interested in is what is the relationship between Corr(x,z) and Corr(x,y) = Corr(x, x-z).
I have worked out that Cov(x,y) = Cov(x,x-z) = Var(x) - Cov(x,z) (I think this is correct but not sure).
I am struggling with expression for the correlation. So far I have:
Corr(x,y) = Corr(x, x-z) = [Var(x) - Cov(x,z)]/[sqrtVar(x)]*sqrt(Var(x) + Var(z) - 2Cov(x,z)).
Basically what I am trying to figure out mathematically is, if I know that Cov(x,z) is positive, if I take the correlation between x and the difference between x and z does it necessarily follow that Corr(x, x-z) >= Corr(x,z) or something along those lines.