This is actually one of the problems in Gujarati's Basic Econometrics 4th edition (Q3.11) and says that the correlation coefficient is invariant with respect to the change of origin and scale, that is $$\text{corr}(aX+b, cY+d) = \text{corr}(X,Y)$$ where $a$,$b$,$c$,$d$ are arbitrary constants.

But my main question is the following: Let $X$ and $Y$ be paired observations and suppose $X$ and $Y$ are positively correlated, i.e. $\text{corr}(X,Y)>0$. I know that $\text{corr}(-X,Y)$ would be negative based on intuition. However if we take $a=-1, b=0, c=1, d=0$, it follows that $$\text{corr}(-X,Y) = \text{corr}(X,Y) >0$$ which does not make sense.

I would appreciate if someone can point out the gap. Thanks.

  • 5
    $\begingroup$ If the book really says what you say it says, it's wrong; you need $ac>0$ $\endgroup$
    – Glen_b
    Commented Oct 21, 2015 at 9:03
  • $\begingroup$ @Glen_b Yea I do think the book state it wrongly, unless I am blind since I do not really see any conditions imposed on the constants. $\endgroup$
    – Daniel
    Commented Oct 21, 2015 at 14:33
  • 1
    $\begingroup$ It may be that scale is understood as a positive quantity. $\endgroup$
    – Xi'an
    Commented Oct 21, 2015 at 15:30
  • $\begingroup$ @Xi'an It could be, but I do not think it is stated in the book. But thanks a lot for the edit and the answer by the way :) $\endgroup$
    – Daniel
    Commented Oct 22, 2015 at 5:08

2 Answers 2


Since $$\text{corr}(X,Y)=\frac{\text{cov}(X,Y)}{\text{var}(X)^{1/2}\,\text{var}(Y)^{1/2}}$$ and $$\text{cov}(aX+b,cY+d)=ac\,\text{cov}(X,Y)$$ the equality$$\text{corr}(aX+b, cY+d) = \text{corr}(X,Y)$$only holds when $a$ and $c$ are both positive or both negative, i.e. $ac>0$.


We know that:

  1. COR(a X + b, c Y + d) = V(a X + b, c Y + d)/(V(a X + b) V(c Y + d))^0.5.
  2. V(a X + b, c Y + d) = V(a X, c Y) = a c V(X, Y).
  3. V(a X + b) = V(a X) = a^2 V(X).
  4. V(c Y + d) = V(c Y) = c^2 V(Y).

Combining all of the above:

  • COR(a X + b, c Y + d) = a c V(X, Y) / (a^2 c^2 V(X) V(Y))^0.5 = a c COR(X, Y) / (|a| |c|) = sign(a) sign(c) COR(X, Y).

So COR(a X + b, c Y + d) equals COR(X, Y) for a c >= 0, and -COR(X, Y) for a c < 0.


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