Standardized bivariate linear regression coefficients > 1?

Question

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?

When both variables are standardized, the slope is the same as pearson's correlation which is always smaller than 1 in magnitude. See here — Demetri Pananos, May 19 '21 at 00:24
It's a consequence of the Cauchy-Schwarz Inequality (or any equivalent inequality). — whuber, May 19 '21 at 13:32

score 2 · Accepted Answer · answered May 19 '21 at 00:51

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 $s_x=s_y=1$.

Standardized bivariate linear regression coefficients > 1?

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