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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.

Thanks!

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  • $\begingroup$ What if $X = Z$? $\endgroup$ – BruceET May 19 '15 at 15:55

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