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.

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#### Best Answer

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. $ac>0$.

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