Suppose that $X$, $Y$, and $W$ are random variables. Also consider the random variable $Z=XY$. I am interested in comparing the Pearson correlation coefficients $\rho(Z,W)$ to the correlation coefficients $\rho(X,W)$ and $\rho(Y,W)$. I have a data set where the correlation coefficients seem to roughly obey
$$\rho(Z,W) =\rho(XY,W) \stackrel{?}{=} \rho(X,W) + \rho(Y,W)$$
I put the question mark over the equality because I am wondering if this is a consequence of the math or something interesting about my data.
Is there any relationship between the correlation coefficient of a product of two random variables with a third r.v. and the correlation coefficient of each r.v. with the third r.v.?
This may not be true in general (though if so that is interesting). If it is only true in a limited set of cases, what assumptions must be made about the r.v.s (independence, distribution, etc)? If it is not true, why (counterexamples, general reasoning why such a thing shouldn't be true)? In my particular case I think that $X$ and $Y$ are correlated and normal, and that $W$ is log-normal (and clearly correlated to $X$, $Y$, and $Z$).
Best Answer
It is certainly not true in general. Covariance is a bi-linear operator, so the product term is unlikely to decompose into something constructive. Here is a counterexample:
Assume $X$ is not degenerate. Take $Y=1/X$ a.s., $W=X$ a.s.
$\rho(XY,W)=\rho(1,X)=0$
$\rho(X,W)=\rho(X,X)=1$
$\rho(Y,W)=\rho(1/X,X)\ne -1$ as $X$ and $1/X$ are not linearly related (a.s.)
So clearly the equality you suggested does not hold.
Note that $cov(X+Y,W)=cov(X,W)+cov(Y,W)$. Assuming that all the variables have unit variance (so that correlations are equal to covariances), the following must hold:
$\rho(XY,W)=\rho(X+Y,W)$. This can probably provide some insight into the necessary relationship between $X$ and $Y$.