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Qstn: X and Y have a correlation of 0.9. Does there exist another variable correlated to both X and Y with correlation -0.9.

My answer: yes. Since X and Y are correlated,

Y = a + bX ; b>0

Let there exist a variable Z correlated to both X and Y.

Then, Z = a1 + b1X ; b1<0

  Y = a2 + b2Z ; b2<0

Therefore,

Y = a2 + a1b2 +b1b2X ;b1b2>0

Please tell me if my reasoning is right?

kjetil b halvorsen
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Harry
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  • Does Y = a + bX necessarily follow from the fact that X and Y are correlated? – Adrian Dec 09 '15 at 16:50
  • Yes. Y = a+ bX is the regression equation. b, the regression coefficient depends on the sign of the correlation coefficient. – Harry Dec 09 '15 at 17:07
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    When $Z$ is correlated with $X$ and $Y$, it does not follow that $Z$ must be a linear combination of $X$ and $Y$. However, you could *look* for such a $Z$ and if you find one, then your question will have been answered positively. – whuber Dec 09 '15 at 17:49
  • Is the part relating to the linear relation right? I have no idea on which equation to use for the non linear part? – Harry Dec 09 '15 at 19:55
  • Doesn't the question relate to linear relationship alone as it mentions "correlation coefficient"? Otherwise, wouldn't the question have said "correlation ratio "? – Harry Dec 09 '15 at 20:02
  • If $Y=a+bX$ where $a$ and $b$ are constants, then $\rho_{X,Y} = 1$, not $0.9$ as you want it to be. In short, your reasoning hits a dead end right at the start. – Dilip Sarwate Dec 09 '15 at 23:34
  • According to [this answer](http://stats.stackexchange.com/a/72798/6633), there exist three random variables $X,Y,Z$ that have _identical_ covariances $\rho$, that is, $\rho_{X,Y} = \rho_{X,Z} = \rho_{Y,Z} = \rho$ for _any_ $\rho \in [-0.5,1]$ that you want to choose. So, choose $\rho = 0.9$. Now consider that random variables $X, Y$ and $-Z$ enjoy the property that $\rho_{X,Y} = 0.9; \rho_{X,-Z} = \rho_{Y,-Z} = -0.9.$ – Dilip Sarwate Dec 10 '15 at 04:16
  • Thank you. I did not understand as to why X,Y and -Z should be taken instead of the original variables. So, would the regression equation remain the same as before? – Harry Dec 10 '15 at 08:30
  • If you use $X,Y,Z$, then $\rho_{X,Y} = \rho_{X,Z} = \rho_{Y,Z} = 0.9$, but it is _you_ who insisted that the third variable must have correlation $-0.9$ with both $X$ and $Y$. $-Z$ serves that role perfectly. If you don't like negative signs attached to random variables, then **define** $W = -Z$, then note that $\rho_{X,Y} = 0.9; \rho_{X,W} = \rho_{Y,W} = -0.9$, and so $X,Y,W$ are the three random variables you are looking for. – Dilip Sarwate Dec 10 '15 at 22:00
  • Thank you. Can i answer this question using the property that the correlation matrix should be positive semi definite? – Harry Dec 11 '15 at 02:04
  • Although correlation is *measure* of linear relationship, neither its calculation nor its interpretation require that the relationship be linear (not even approximately). You can construct examples of such nonlinear relationships using code I provided at http://stats.stackexchange.com/a/152034. – whuber Dec 11 '15 at 14:46

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