# Coefficient of $Y$ on $X$ and Coefficient of $X$ on $Y$ [duplicate]

Under what circumstances the coefficients from simple linear regression of $$Y$$ on $$X$$ is equal to that of $$X$$ on $$Y$$? Will it hold when the standard deviations of $$X$$ and $$Y$$ are the same? I would very appreciate it if anyone can give me any idea of it! Thanks.

• This is becoming a FAQ: see stats.stackexchange.com/search?q=regression+x+y.
– whuber
Commented Oct 13, 2020 at 12:58
• See my answer to Effect of switching response and explanatory variable in simple linear regression where I show tat if $X$ is always plotted on the horizontal axis and $Y$ on the vertical axis, then the two regression lines are not the same except when all the data points lie on a straight line. If the response is on the vertical axis and the explanatory variable on the horizontal axis, the two lines have the same slope when the variances are the same. But one line passes through $(\mu_X,\mu_Y)$ the other through $(\mu_Y,\mu_X)$. Commented Oct 13, 2020 at 20:54

Yes. The least squares estimator when regressing $$Y$$ on $$X$$ is
$$\hat\beta = \frac{\hat\sigma_{xy}}{\hat\sigma^2_x}$$
The only way to get the same value by regressing $$X$$ on $$Y$$ is if $$\hat\sigma^2_x = \hat\sigma^2_y$$. If so, $$\hat\beta$$ coincides with the correlation between $$X$$ and $$Y$$.
$$\hat\rho = \frac{\hat\sigma_{xy}}{\hat\sigma_x \hat\sigma_y}$$