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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, 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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