slr-comp.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	theorem	sources	proof_id	shortcut	username
proof	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2021-10-27 05:52:00 -0700	The regression line goes through the center of mass point	Statistical Models	Univariate normal data	Simple linear regression	Regression line includes center of mass	authors year title in pages url Wikipedia 2021 Simple linear regression Wikipedia, the free encyclopedia retrieved on 2021-10-27 https://en.wikipedia.org/wiki/Simple_linear_regression#Numerical_properties	P275	slr-comp	JoramSoch

Theorem: In simple linear regression, the regression line estimated using ordinary least squares includes the point $M(\bar{x},\bar{y})$.

Proof: The fitted regression line is described by the equation

$$ \label{eq:slr-ols-regline} y = \hat{\beta}_0 + \hat{\beta}_1 x \quad \text{where} \quad x,y \in \mathbb{R} ; . $$

Plugging in the coordinates of $M$ and the ordinary least squares estimate of the intercept, we obtain

$$ \label{eq:slr-ols} \begin{split} \bar{y} &= \hat{\beta}_0 + \hat{\beta}_1 \bar{x} \\ \bar{y} &= \bar{y} - \hat{\beta}_1 \bar{x} + \hat{\beta}_1 \bar{x} \\ \bar{y} &= \bar{y} ; . \end{split} $$

which is a true statement. Thus, the regression line goes through the center of mass point $(\bar{x},\bar{y})$, if the model includes an intercept term $\beta_0$.