Solved – Linear regression: explanation of slope constraints

regression

I am given regression equations, one showing linear regression of x on y, and the other y on x. Both refer to the same set of data-points.

4X - 5Y + 33 = 0
20X - 9Y - 107 = 0

  • Taking the first to be X (dependent) on Y (independent) => of the form X = a + bY.
    I end up with slope value of 1.25

  • Taking the second to be Y (dependent) on X (independent) => of the form Y = a + bX.
    I end up with slope value of 2.22

Now, these values somehow imply that the system of regression lines is invalid? How is that? I am unable to visualize this.

The book states the "rule" being that:

  1. Both coefficients (slopes) must be less than 1
  2. Both coefficients (slopes) must be of the same sign

Are these correct? What other constraints exist for a set of regression equations to be valid?

Best Answer

I am told both regression coefficients cannot exceed 1

Well, yes they can. There are no constraints about values of coefficients. Where did you get it from? The slope is simply the tangent of the angle between your line and OX axis, so it can get any value in $\mathbb{R}$

The only trivial "constraint" is that given simple linear regression from $\mathbb{R}$ to $\mathbb{R}$, if you look at the equations $$ y=ax+b$$ and $$ x=cy+e$$ Then at least one of the $\{a,c\}$ has an absolute value not greater then $1$.

Assuming that $a\neq 0$ $$y=ax+b \iff -ax=b-y \iff x=\frac{-y}{-a}+\frac{b}{-a} \iff x = \frac{y}{a} - \frac{b}{a} \iff x = \frac{1}{a}y - \frac{b}{a}$$

so $c=\frac{1}{a}$, and as a result, if $|a|>1$ then $|c|=\left |\frac{1}{a}\right | < 1$

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