# Relationship between correlations

#### oracle133

##### New Member
Hi,

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.

#### Jake

What can you say about corr(x, x-z)? Not much. Depending on the relative variances of x and z, this correlation could be strongly positive, strongly negative, or anything in between. This is easiest to see if we ignore the denominator of the correlation and just focus on the covariance (beginning with your correct right-hand-side expression):

$$\sigma^2_X-\sigma_{XZ}=\sigma^2_X-\rho_{XZ}\sigma_X\sigma_Z=\sigma_X(\sigma_X-\rho_{XZ}\sigma_Z)$$.

So corr(x, x-z) is equal to the product of $$\sigma_X$$ and $$\sigma_X-\rho_{XZ}\sigma_Z$$. The term $$\sigma_X$$ is necessarily positive, and we have assumed that $$\rho_{XZ}$$ is positive as well. But the whole term $$\sigma_X-\rho_{XZ}\sigma_Z$$ could still be a big positive number (if $$\sigma_X>>\sigma_Z$$) or a big negative number (if $$\sigma_Z>>\sigma_X$$), which would make the covariance (and hence the correlation) strongly positive or strongly negative, respectively.