# Regression betas of X on Y and Y on X are both less than one? [duplicate]

Intuitively, I can't really wrap my head around this. If I regress y on x and the beta is less that one, shouldn't the beta from a regression of x on y be greater than one. Mathematically, I know the relationship:

i.e. beta = cov(x,y)/var(x)


so that as long as var(y) and var(x) are greater than cov(x,y), both betas will be less than one. But intuitively, this doesn't seem to make sense to me. The context I'm looking that is in stock returns. If I get a beta of less than 1, doesn't that mean the stock that is the independent variable is less volatile than the dependent? And if that's the case should the inverse regression yield a beta greater than one?

What am I missing? Is there an easy way to conceptualize this?

• Search for "regression to the mean."
– whuber
Commented Nov 18, 2019 at 19:16
• @whuber I'd already read extensively through the question thread that's linked. While they are similar questions, I was hoping for a more intuitive explanation that can be explained to the layman. Maybe, the answer to my question is very obvious but that posted thread still left me confused, which is why I posted here. Apologies. Commented Nov 18, 2019 at 19:49
• You seem to be thinking that the two regressions will be the same line, but they are not - unless the variables are perfectly linearly related. Imagine two completely unrelated variables. Both regression lines will have slope coefficient 0. Commented Nov 19, 2019 at 0:29