Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Simple linear regression ▷ Ordinary least squares

Theorem: Given a simple linear regression model with independent observations

$\label{eq:slr} y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,$

the parameters minimizing the residual sum of squares are given by

$\label{eq:slr-ols} \begin{split} \hat{\beta}_0 &= \bar{y} - \hat{\beta}_1 \bar{x} \\ \hat{\beta}_1 &= \frac{s_{xy}}{s_x^2} \end{split}$

where $\bar{x}$ and $\bar{y}$ are the sample means, $s_x^2$ is the sample variance of $x$ and $s_{xy}$ is the sample covariance between $x$ and $y$.

Proof: The residual sum of squares is defined as

$\label{eq:rss} \mathrm{RSS}(\beta_0,\beta_1) = \sum_{i=1}^n \varepsilon_i^2 = \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i)^2 \; .$

The derivatives of $\mathrm{RSS}(\beta_0,\beta_1)$ with respect to $\beta_0$ and $\beta_1$ are

$\label{eq:rss-der} \begin{split} \frac{\mathrm{d}\mathrm{RSS}(\beta_0,\beta_1)}{\mathrm{d}\beta_0} &= \sum_{i=1}^n 2 (y_i - \beta_0 - \beta_1 x_i) (-1) \\ &= -2 \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i) \\ \frac{\mathrm{d}\mathrm{RSS}(\beta_0,\beta_1)}{\mathrm{d}\beta_1} &= \sum_{i=1}^n 2 (y_i - \beta_0 - \beta_1 x_i) (-x_i) \\ &= -2 \sum_{i=1}^n (x_i y_i - \beta_0 x_i - \beta_1 x_i^2) \end{split}$

and setting these derivatives to zero

$\label{eq:rss-der-zero} \begin{split} 0 &= -2 \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i) \\ 0 &= -2 \sum_{i=1}^n (x_i y_i - \hat{\beta}_0 x_i - \hat{\beta}_1 x_i^2) \end{split}$

yields the following equations:

$\label{eq:slr-norm-eq} \begin{split} \hat{\beta}_1 \sum_{i=1}^n x_i + \hat{\beta}_0 \cdot n &= \sum_{i=1}^n y_i \\ \hat{\beta}_1 \sum_{i=1}^n x_i^2 + \hat{\beta}_0 \sum_{i=1}^n x_i &= \sum_{i=1}^n x_i y_i \; . \end{split}$

From the first equation, we can derive the estimate for the intercept:

$\label{eq:slr-ols-int} \begin{split} \hat{\beta}_0 &= \frac{1}{n} \sum_{i=1}^n y_i - \hat{\beta}_1 \cdot \frac{1}{n} \sum_{i=1}^n x_i \\ &= \bar{y} - \hat{\beta}_1 \bar{x} \; . \end{split}$

From the second equation, we can derive the estimate for the slope:

$\label{eq:slr-ols-sl} \begin{split} \hat{\beta}_1 \sum_{i=1}^n x_i^2 + \hat{\beta}_0 \sum_{i=1}^n x_i &= \sum_{i=1}^n x_i y_i \\ \hat{\beta}_1 \sum_{i=1}^n x_i^2 + \left( \bar{y} - \hat{\beta}_1 \bar{x} \right) \sum_{i=1}^n x_i &\overset{\eqref{eq:slr-ols-int}}{=} \sum_{i=1}^n x_i y_i \\ \hat{\beta}_1 \left( \sum_{i=1}^n x_i^2 - \bar{x} \sum_{i=1}^n x_i \right) &= \sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i \\ \hat{\beta}_1 &= \frac{\sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i}{\sum_{i=1}^n x_i^2 - \bar{x} \sum_{i=1}^n x_i} \; . \end{split}$

Note that the numerator can be rewritten as

$\label{eq:slr-ols-sl-num} \begin{split} \sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i &= \sum_{i=1}^n x_i y_i - n \bar{x} \bar{y} \\ &= \sum_{i=1}^n x_i y_i - n \bar{x} \bar{y} - n \bar{x} \bar{y} + n \bar{x} \bar{y} \\ &= \sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i - \bar{x} \sum_{i=1}^n y_i + \sum_{i=1}^n \bar{x} \bar{y} \\ &= \sum_{i=1}^n \left( x_i y_i - x_i \bar{y} - \bar{x} y_i + \bar{x} \bar{y} \right) \\ &= \sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y}) \end{split}$

and that the denominator can be rewritten as

$\label{eq:slr-ols-sl-den} \begin{split} \sum_{i=1}^n x_i^2 - \bar{x} \sum_{i=1}^n x_i &= \sum_{i=1}^n x_i^2 - n \bar{x}^2 \\ &= \sum_{i=1}^n x_i^2 - 2 n \bar{x} \bar{x} + n \bar{x}^2 \\ &= \sum_{i=1}^n x_i^2 - 2 \bar{x} \sum_{i=1}^n x_i - \sum_{i=1}^n \bar{x}^2 \\ &= \sum_{i=1}^n \left( x_i^2 - 2 \bar{x} x_i + \bar{x}^2 \right) \\ &= \sum_{i=1}^n (x_i - \bar{x})^2 \; . \end{split}$

With \eqref{eq:slr-ols-sl-num} and \eqref{eq:slr-ols-sl-den}, the estimate from \eqref{eq:slr-ols-sl} can be simplified as follows:

$\label{eq:slr-ols-sl-qed} \begin{split} \hat{\beta}_1 &= \frac{\sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i}{\sum_{i=1}^n x_i^2 - \bar{x} \sum_{i=1}^n x_i} \\ &= \frac{\sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} \\ &= \frac{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y})}{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2} \\ &= \frac{s_{xy}}{s_x^2} \; . \end{split}$

Together, \eqref{eq:slr-ols-int} and \eqref{eq:slr-ols-sl-qed} constitute the ordinary least squares parameter estimates for simple linear regression.

Sources:

Metadata: ID: P271 | shortcut: slr-ols | author: JoramSoch | date: 2021-10-27, 08:56.