# Zero regression coefficient when correlations are not zero

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 $X\cdot Y$ is zero after taking into account Y? In other words, would a regression of $X$ on $Y$ and $X\cdot Y$ possibly result in a zero coefficient on $X\cdot Y$?

• what about Y = 1 / X (over a reasonably defined range) ? – Andre Holzner Aug 14 '11 at 7:10
• @Andre, simple code in R says that $y=1/x$ is not a solution: x <- runif(100,0.5,0.7); y<-1/x; summary(lm(x~y+I(x*y)-1)) – mpiktas Aug 16 '11 at 11:10

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).

• could you please add how did you get this data set. – mpiktas Aug 16 '11 at 11:08
• @mpiktas, I generated random correlated data, then I altered some of the values until received the wanted result (b for XY = 0). As for the "theoretical prerequisite", it is deduced from the well-known formula for b in the form it exists for OLS regression. – ttnphns Aug 16 '11 at 13:01
• if there is a code to reproduce this example, then it would be very good to include it in the text. – mpiktas Aug 16 '11 at 13:50