1
$\begingroup$

x and y are two columns of financial data which have been standardized. Assuming one implements a simple linear regression on x and y, is it possible to observe a slope greater than 1?

I ran some numbers in Excel and cannot get the slope to ever exceed 1. Can someone please explain the mathematical reason why this is impossible?

$\endgroup$
3
  • 1
    $\begingroup$ When both variables are standardized, the slope is the same as pearson's correlation which is always smaller than 1 in magnitude. See here $\endgroup$ May 19, 2021 at 0:24
  • $\begingroup$ But why is that the case? $\endgroup$ May 19, 2021 at 0:40
  • $\begingroup$ It's a consequence of the Cauchy-Schwarz Inequality (or any equivalent inequality). $\endgroup$
    – whuber
    May 19, 2021 at 13:32

1 Answer 1

2
$\begingroup$

It is a well known result that the slope of a simple linear regresison is

$$\hat{\beta_1} = r_{xy} \dfrac{s_y}{s_x}$$

Here $r_{xy}$ is the sample correlation coefficient and $s_x, s_y$ are the sampel standard deviations of $x$ and $y$. The result is obtained immediately when you realize that standardization fixes $s_x=s_y=1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.