# rotating and exchanging x for y's in regression

I was just wondering what happens generally if i send all my x points to y's and y's to x's (i.e reflect along the y=x line) - if I change the x's and y's will my old error minimizing line still be the error minimizing line after reflection? How about if I rotate points?

Many thanks

• A two variable (1 RHS, 1 LHS) setup like this: $y = \alpha + \beta x + \epsilon$? What do you mean by rotate, something like this? $z = \zeta + \iota w + \xi$, where z and w are linear combinations of x and y (e.g. $w = \psi x + \phi y)$? – BKay Feb 18 '15 at 17:10

This is a somewhat "dated" subject in introductory econometrics, I suspect because, in econometrics the models come from theories and arguments that try to a priori establish causality and not just association.

Anyway, the issue is analyzed in Maddala's Econometrics textbook.

In the 2001 3d ed. the issue for simple regression is presented in ch. 3. inside section 3.4, under the title "Reverse Regression".

Writing

$$\hat y = \hat a + \hat b x,$$ and

$$\hat x = \hat a' + \hat b'y$$

one can quickly find that

$$\hat b \hat b' = r^2_{xy}$$

i.e. the product of the two is equal to the squared correlation between $x$ and $y$. If this correlation coefficient is close to unity then the two regression lines will be close.

One can also establish a relation between the two constant terms easily.