# Standardized bivariate linear regression coefficients > 1?

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 May 19, 2021 at 0:24
• But why is that the case? May 19, 2021 at 0:40
• It's a consequence of the Cauchy-Schwarz Inequality (or any equivalent inequality).
– whuber
May 19, 2021 at 13:32

$$\hat{\beta_1} = r_{xy} \dfrac{s_y}{s_x}$$
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 $$s_x=s_y=1$$.