The regression questions firms actually ask

Regression to the mean, $\mathbb{E}[y \mid x] - \mu_y = \rho\,\frac{\sigma_y}{\sigma_x}(x - \mu_x)$ — When $x$ and $y$ are imperfectly correlated and measured in the same units, an observation far from the mean in $x$ is expected to be closer to the mean in $y$ — by a factor of exactly $\rho$. It is not a force pulling things back; it is what imperfect correlation means. The tallest fathers have sons who are tall but less so, and last year’s best fund is expected to be above average this year but not top.

Regress $y$ on $x$ and you get $\rho\sigma_y/\sigma_x$; regress $x$ on $y$ and you get $\rho\sigma_x/\sigma_y$. The product is $\rho^2$, so the two slopes are reciprocals only when $|\rho| = 1$. Plotted on the same axes the two lines both pass through the means and pinch together as the correlation rises.

Holds when

• If $\sigma_x = \sigma_y$, both slopes equal $\rho$, which makes the regression-to-the-mean statement visible directly.
• Neither line is "the relationship" — each answers a different conditional expectation.