Does the regression line go through the center of mass in >1 dimension?

Question

It is known and I have found proof that (like here https://math.stackexchange.com/questions/635670/show-that-the-least-squares-line-must-pass-through-the-center-of-mass) in Simple Linear Regression, the regression line always goes through the center of mass. But I have not found any proof that generalizes this to the case where we have a generic number of covariables $p$.

Does it hold? And if yes is there a proof?

Answer 1

Decompose the dependent variable as fitted values and residuals, $$ y=\hat y+e=X\hat\beta+e $$ If the regression is such that residuals sum to zero (typically because it contains a constant), we have, with $i$ a vector of ones, $$ \frac{i'y}{n}=\frac{i'X}{n}\hat\beta+\frac{i'e}{n}=\frac{i'X}{n}\hat\beta $$ or $$ \bar y=\bar X\hat\beta $$