I've read that centering two normal (or symmetrical) variables $X$ and $Z$ affects correlation of centered $X$ with interaction term $X\cdot Z$ in such way, that this correlation $cor(X-EX, X\cdot Z)$ is $0$. I am not sure ... (here I use the numerator of correlation, which is covariance)
When I'm doing my own calculations I get stuck here:
$cov(X,X\cdot Z) = E(X\cdot X \cdot Z) - E(X)\cdot E(X\cdot Z) = E(X^2\cdot Z) - E(X)\cdot E(X\cdot Z)$
because without any information about independence between $X$ and $Z$ it's over. Even knowing that these two variables are normal it gives me nothing. At least me :-) The independence between $X$ and $Z$ would give me only that
$cov (X, X\cdot Z) = E(X^2)\cdot E(Z)-E(X)\cdot E(X)\cdot E(Z) = E(Z)\cdot var(X)$
It's not $0$. But the book 'says' explicitly:
$cov(X\cdot Z,X) = var(X)\cdot E(Z) + cov(X,Z)\cdot E(Z)$
If $X$ and $Z$ are centered, then $EX$ and $EZ$ are both zero, and the covariance between $X$ and $XZ$ is zero as well. Thus the correlation between $X$ and $XZ$ is also zero. The same holds for the correlation between $Z$ and $XZ$
So did I missed something (and the book is right) or ... is my thinking correct?
The book is "Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences" by Cohen, Cohen, Aiken, West.