Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Simple linear regression ▷ Regression line includes center of mass

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 $\beta_0$.

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Metadata: ID: P275 | shortcut: slr-comp | author: JoramSoch | date: 2021-10-27, 12:52.