<p>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.  </p>

<p>Anyway, the issue is analyzed in <a href="http://en.wikipedia.org/wiki/G._S._Maddala" rel="nofollow">Maddala's</a> Econometrics <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000871.html" rel="nofollow">textbook</a>.</p>

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

<p>Writing</p>

<p>$$\hat y = \hat a + \hat b x,$$
and </p>

<p>$$\hat x = \hat a' + \hat b'y$$</p>

<p>one can quickly find that </p>

<p>$$\hat b \hat b' = r^2_{xy}$$</p>

<p>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.  </p>

<p>One can also establish a relation between the two constant terms easily.</p>

CC BY-SA 3.0
2015-02-19T13:36:34.990Z
2016-09-06T17:19:07.647Z
2016-09-06T17:19:07.647Z
Alecos Papadopoulos
<p>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?</p>

<p>Many thanks</p>

CC BY-SA 3.0
2015-02-18T17:10:05.340Z
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
<regression>
rotating and exchanging x for y's in regression
bob
# rotating and exchanging x for y's in regression

(Asked by: username_0 on 2015-02-18)

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

**Tags:** regression
> **Comments:**
> **username_1**: 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. $w = \psi x + \phi y)$?  

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## 1 answers

### Answer by username_2 on 2015-02-19. Score: 5

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](http://en.wikipedia.org/wiki/G._S._Maddala) Econometrics [textbook](http://eu.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000871.html).

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.


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