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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?

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    $\begingroup$ When both variables are standardized, the slope is the same as pearson's correlation which is always smaller than 1 in magnitude. See here $\endgroup$
    – Demetri Pananos
    Commented May 19, 2021 at 0:24
  • $\begingroup$ But why is that the case? $\endgroup$
    – user3138766
    Commented May 19, 2021 at 0:40
  • $\begingroup$ It's a consequence of the Cauchy-Schwarz Inequality (or any equivalent inequality). $\endgroup$
    – whuber ♦
    Commented May 19, 2021 at 13:32

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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 $s_x=s_y=1$.

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