Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Simple linear regression ▷ Maximum likelihood estimation

Theorem: Given a simple linear regression model with independent observations

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

the maximum likelihood estimates of $\beta_0$, $\beta_1$ and $\sigma^2$ are given by

$\label{eq:slr-mle} \begin{split} \hat{\beta}_0 &= \bar{y} - \hat{\beta}_1 \bar{x} \\ \hat{\beta}_1 &= \frac{s_{xy}}{s_x^2} \\ \hat{\sigma}^2 &= \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \end{split}$

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: With the probability density function of the normal distribution and probability under independence, the linear regression equation \eqref{eq:slr} implies the following likelihood function

$\label{eq:slr-lf} \begin{split} p(y|\beta_0,\beta_1,\sigma^2) &= \prod_{i=1}^n p(y_i|\beta_0,\beta_1,\sigma^2) \\ &= \prod_{i=1}^n \mathcal{N}(y_i; \beta_0 + \beta_1 x_i, \sigma^2) \\ &= \prod_{i=1}^n \frac{1}{\sqrt{2 \pi \sigma}} \cdot \exp \left[ -\frac{(y_i - \beta_0 - \beta_1 x_i)^2}{2 \sigma^2} \right] \\ &= \frac{1}{\sqrt{(2 \pi \sigma^2)^n}} \cdot \exp\left[ -\frac{1}{2 \sigma^2} \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i)^2 \right] \end{split}$

and the log-likelihood function

$\label{eq:slr-ll} \begin{split} \mathrm{LL}(\beta_0,\beta_1,\sigma^2) &= \log p(y|\beta_0,\beta_1,\sigma^2) \\ &= -\frac{n}{2} \log(2\pi) - \frac{n}{2} \log (\sigma^2) -\frac{1}{2 \sigma^2} \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i)^2 \; . \end{split}$

The derivative of the log-likelihood function \eqref{eq:slr-ll} with respect to $\beta_0$ is

$\label{eq:dLL-dbeta0} \frac{\mathrm{d}\mathrm{LL}(\beta_0,\beta_1,\sigma^2)}{\mathrm{d}\beta_0} = \frac{1}{\sigma^2} \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_i)$

and setting this derivative to zero gives the MLE for $\beta_0$:

$\label{eq:beta0-mle} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\hat{\sigma}^2)}{\mathrm{d}\beta_0} &= 0 \\ 0 &= \frac{1}{\hat{\sigma}^2} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i) \\ 0 &= \sum_{i=1}^n y_i - n \hat{\beta}_0 - \hat{\beta}_1 \sum_{i=1}^n x_i \\ \hat{\beta}_0 &= \frac{1}{n} \sum_{i=1}^n y_i - \hat{\beta}_1 \frac{1}{n} \sum_{i=1}^n x_i \\ \hat{\beta}_0 &= \bar{y} - \hat{\beta}_1 \bar{x} \; . \end{split}$

The derivative of the log-likelihood function \eqref{eq:slr-ll} at $\hat{\beta}_0$ with respect to $\beta_1$ is

$\label{eq:dLL-dbeta1} \frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\beta_1,\sigma^2)}{\mathrm{d}\beta_1} = \frac{1}{\sigma^2} \sum_{i=1}^n (x_i y_i - \hat{\beta}_0 x_i - \beta_1 x_i^2) \\$

and setting this derivative to zero gives the MLE for $\beta_1$:

$\label{eq:beta1-mle} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\hat{\sigma}^2)}{\mathrm{d}\beta_1} &= 0 \\ 0 &= \frac{1}{\hat{\sigma}^2} \sum_{i=1}^n (x_i y_i - \hat{\beta}_0 x_i - \hat{\beta}_1 x_i^2) \\ 0 &= \sum_{i=1}^n x_i y_i - \hat{\beta}_0 \sum_{i=1}^n x_i - \hat{\beta}_1 \sum_{i=1}^n x_i^2) \\ 0 &\overset{\eqref{eq:beta0-mle}}{=} \sum_{i=1}^n x_i y_i - (\bar{y} - \hat{\beta}_1 \bar{x}) \sum_{i=1}^n x_i - \hat{\beta}_1 \sum_{i=1}^n x_i^2 \\ 0 &= \sum_{i=1}^n x_i y_i - \bar{y} \sum_{i=1}^n x_i + \hat{\beta}_1 \bar{x} \sum_{i=1}^n x_i - \hat{\beta}_1 \sum_{i=1}^n x_i^2 \\ 0 &= \sum_{i=1}^n x_i y_i - n \bar{x} \bar{y} + \hat{\beta}_1 n \bar{x}^2 - \hat{\beta}_1 \sum_{i=1}^n x_i^2 \\ \hat{\beta}_1 &= \frac{\sum_{i=1}^n x_i y_i - \sum_{i=1}^n \bar{x} \bar{y}}{\sum_{i=1}^n x_i^2 - \sum_{i=1}^n \bar{x}^2} \\ \hat{\beta}_1 &= \frac{\sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} \\ \hat{\beta}_1 &= \frac{s_{xy}}{s_x^2} \; . \end{split}$

The derivative of the log-likelihood function \eqref{eq:slr-ll} at $(\hat{\beta}_0,\hat{\beta}_1)$ with respect to $\sigma^2$ is

$\label{eq:dLL-ds2} \frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\sigma^2)}{\mathrm{d}\sigma^2} = - \frac{n}{2\sigma^2} + \frac{1}{2(\sigma^2)^2} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2$

and setting this derivative to zero gives the MLE for $\sigma^2$:

$\label{eq:s2-mle} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\hat{\sigma}^2)}{\mathrm{d}\sigma^2} &= 0 \\ 0 &= - \frac{n}{2\hat{\sigma}^2} + \frac{1}{2(\hat{\sigma}^2)^2} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \\ \frac{n}{2\hat{\sigma}^2} &= \frac{1}{2(\hat{\sigma}^2)^2} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \\ \hat{\sigma}^2 &= \frac{1}{n} \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \; . \end{split}$

Together, \eqref{eq:beta0-mle}, \eqref{eq:beta1-mle} and \eqref{eq:s2-mle} constitute the MLE for simple linear regression.

Sources:

Metadata: ID: P287 | shortcut: slr-mle | author: JoramSoch | date: 2021-11-16, 08:34.