Range of regression coefficient

Question

Consider the regression $Y\sim X$ which gives us coefficient $\beta$, now if we do $X\sim Y$, what is the range of the new coefficient?

Solution

We have $$ \beta = \frac{\text{Cov}(X,Y)}{\text{Var}(X)} = \rho\frac{\sigma_Y}{\sigma_X} $$ where $\rho$ is the correlation; $\sigma_X,\sigma_Y$ are the standard deviations.

This means $$ \beta_{\text{new}} = \frac{\text{Cov(X,Y)}}{\text{Var}(Y)} = \rho\cdot\frac{\sigma_X}{\sigma_Y} = \rho^2\cdot\frac{1}{\beta} $$

And because $0\le\rho^2\le 1$, we have $$ \beta_{\text{new}} \in [0, 1/\beta]. $$