Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Simple linear regression ▷ Distribution of estimates

Theorem: Assume a simple linear regression model with independent observations

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

and consider estimation using ordinary least squares. Then, the estimated parameters are normally distributed as

$\label{eq:slr-olsdist} \left[ \begin{matrix} \hat{\beta}_0 \\ \hat{\beta}_1 \end{matrix} \right] \sim \mathcal{N}\left( \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right], \, \frac{\sigma^2}{(n-1) \, s_x^2} \cdot \left[ \begin{matrix} x^\mathrm{T}x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \right)$

where $\bar{x}$ is the sample mean and $s_x^2$ is the sample variance of $x$.

$\label{eq:slr-mlr} X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,$

such that \eqref{eq:slr} can also be written as

$\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n)$

and ordinary least sqaures estimates are given by

$\label{eq:mlr-ols} \hat{\beta} = (X^\mathrm{T} X)^{-1} X^\mathrm{T} y \; .$

From \eqref{eq:mlr} and the linear transformation theorem for the multivariate normal distribution, it follows that

$\label{eq:y-dist} y \sim \mathcal{N}\left( X\beta, \, \sigma^2 I_n \right) \; .$

From \eqref{eq:mlr-ols}, in combination with \eqref{eq:y-dist} and the transformation theorem, it follows that

$\label{eq:b-est-dist} \begin{split} \hat{\beta} &\sim \mathcal{N}\left( (X^\mathrm{T} X)^{-1} X^\mathrm{T} X\beta, \, \sigma^2 (X^\mathrm{T} X)^{-1} X^\mathrm{T} I_n X (X^\mathrm{T} X)^{-1} \right) \\ &\sim \mathcal{N}\left( \beta, \, \sigma^2 (X^\mathrm{T} X)^{-1} \right) \; . \end{split}$

Applying \eqref{eq:slr-mlr}, the covariance matrix can be further developed as follows:

$\label{eq:b-est-cov} \begin{split} \sigma^2 (X^\mathrm{T} X)^{-1} &= \sigma^2 \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ \begin{matrix} 1_n & x \end{matrix} \right] \right)^{-1} \\ &= \sigma^2 \left( \left[ \begin{matrix} n & n\bar{x} \\ n\bar{x} & x^\mathrm{T} x \end{matrix} \right] \right)^{-1} \\ &= \frac{\sigma^2}{n x^\mathrm{T} x - (n\bar{x})^2} \left[ \begin{matrix} x^\mathrm{T} x & -n\bar{x} \\ -n\bar{x} & n \end{matrix} \right] \\ &= \frac{\sigma^2}{x^\mathrm{T} x - n\bar{x}^2} \left[ \begin{matrix} x^\mathrm{T} x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \; . \end{split}$

Note that the denominator in the first factor is equal to

$\label{eq:b-est-cov-den} \begin{split} x^\mathrm{T} x - n\bar{x}^2 &= x^\mathrm{T} x - 2 n\bar{x}^2 + n\bar{x}^2 \\ &= \sum_{i=1}^{n} x_i^2 - 2 n \bar{x} \frac{1}{n} \sum_{i=1}^{n} x_i + \sum_{i=1}^{n} \bar{x}^2 \\ &= \sum_{i=1}^{n} x_i^2 - 2 \sum_{i=1}^{n} x_i \bar{x} + \sum_{i=1}^{n} \bar{x}^2 \\ &= \sum_{i=1}^{n} \left( x_i^2 - 2 x_i \bar{x} + \bar{x}^2 \right) \\ &= \sum_{i=1}^{n} \left( x_i^2 - \bar{x} \right)^2 \\ &= (n-1) \, s_x^2 \; . \end{split}$

Thus, combining \eqref{eq:b-est-dist}, \eqref{eq:b-est-cov} and \eqref{eq:b-est-cov-den}, we have

$\label{eq:slr-olsdist-qed} \hat{\beta} \sim \mathcal{N}\left( \beta, \, \frac{\sigma^2}{(n-1) \, s_x^2} \cdot \left[ \begin{matrix} x^\mathrm{T}x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \right)$

which is equivalent to equation \eqref{eq:slr-olsdist}.

Sources:

Metadata: ID: P282 | shortcut: slr-olsdist | author: JoramSoch | date: 2021-11-09, 09:09.