I don't really have a motivation for this – but I was thinking about this and couldn't work it out.
Suppose I have a random variables $X$ and $Y$ which are correlated. Is it possible that the partial correlation between $X$ and $Xcdot Y$ is zero after taking into account Y? In other words, would a regression of $X$ on $Y$ and $Xcdot Y$ possibly result in a zero coefficient on $Xcdot Y$?
Best Answer
Yes, that is possible. Take these data for example
x y xy .2217 .5000 .1108 .3048 -.9787 -.2983 -1.6445 .3512 -.5775 -.2461 -.4866 .1197 -.3170 -.0954 .0302 -1.1603 1.8352 -2.1294 -.8720 .1372 -.1196 -1.7852 -.2160 .3856 1.0100 .0165 .0166 .3000 -.3251 -.0975 $XY$ is a product of $X$ and $Y$. Multiple regression of $X$ on $Y$ and $XY$ yields $b$ for $XY$ as 0 and $b$ for $Y$ as -.444. Constant is -.386.
Note the theoretical prerequisite for this: $bXY$ will be 0 if and only if $rX.XY$ (i.e. correlation bw $X$ and $XY$; "." here means "with") $= rX.Y * rY.XY$. Here, .280 = (-.361) * (-.776).
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