Proof: Transformation matrices for simple linear regression
Index:
The Book of Statistical Proofs ▷
Statistical Models ▷
Univariate normal data ▷
Simple linear regression ▷
Transformation matrices
Metadata: ID: P285 | shortcut: slr-mat | author: JoramSoch | date: 2021-11-09, 15:19.
Theorem: Under ordinary least squares for simple linear regression, estimation, projection and residual-forming matrices are given by
\[\label{eq:slr-mat} \begin{split} E &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) \, 1_n^\mathrm{T} - \bar{x} \, x^\mathrm{T} \\ - \bar{x} \, 1_n^\mathrm{T} + x^\mathrm{T} \end{matrix} \right] \\ P &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \\ R &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_1 - n\bar{x}) - x_1^2 & \cdots & -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n \\ \vdots & \ddots & \vdots \\ -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n & \cdots & (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_n - n\bar{x}) - x_n^2 \end{matrix} \right] \end{split}\]where $1_n$ is an $n \times 1$ vector of ones, $x$ is the $n \times 1$ single predictor variable, $\bar{x}$ is the sample mean of $x$ and $s_x^2$ is the sample variance of $x$.
Proof: Simple linear regression is a special case of multiple linear regression with
\[\label{eq:slr-mlr} X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,\]such that the simple linear regression model can also be written as
\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n) \; .\]Moreover, we note the following equality:
\[\label{eq:b-est-cov-den} x^\mathrm{T} x - n\bar{x}^2 = (n-1) \, s_x^2 \; .\]
1) The estimation matrix is given by
which is a $2 \times n$ matrix and can be reformulated as follows:
\[\label{eq:E-qed} \begin{split} E &= (X^\mathrm{T} X)^{-1} X^\mathrm{T} \\ &= \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ \begin{matrix} 1_n & x \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\ &= \left( \left[ \begin{matrix} n & n\bar{x} \\ n\bar{x} & x^\mathrm{T} x \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\ &= \frac{1}{n x^\mathrm{T} x - (n\bar{x})^2} \left[ \begin{matrix} x^\mathrm{T} x & -n\bar{x} \\ -n\bar{x} & n \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\ &= \frac{1}{x^\mathrm{T} x - n\bar{x}^2} \left[ \begin{matrix} x^\mathrm{T} x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\ &\overset{\eqref{eq:b-est-cov-den}}{=} \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) \, 1_n^\mathrm{T} - \bar{x} \, x^\mathrm{T} \\ - \bar{x} \, 1_n^\mathrm{T} + x^\mathrm{T} \end{matrix} \right] \; . \end{split}\]
2) The projection matrix is given by
which is an $n \times n$ matrix and can be reformulated as follows:
\[\label{eq:P-qed} \begin{split} P &= X \, E = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \left[ \begin{matrix} e_1 \\ e_2 \end{matrix} \right] \\ &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \end{matrix} \right] \left[ \begin{matrix} (x^\mathrm{T} x/n) - \bar{x} x_1 & \cdots & (x^\mathrm{T} x/n) - \bar{x} x_n \\ -\bar{x} + x_1 & \cdots & -\bar{x} + x_n \end{matrix} \right] \\ &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \; . \end{split}\]
3) The residual-forming matrix is given by
which also is an $n \times n$ matrix and can be reformulated as follows:
\[\label{eq:R-qed} \begin{split} R &= I_n - P = \left[ \begin{matrix} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{matrix} \right] - \left[ \begin{matrix} p_{11} & \cdots & p_{1n} \\ \vdots & \ddots & \vdots \\ p_{n1} & \cdots & p_{nn} \end{matrix} \right] \\ &\overset{\eqref{eq:b-est-cov-den}}{=} \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} x^\mathrm{T} x - n\bar{x}^2 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & x^\mathrm{T} x - n\bar{x}^2 \end{matrix} \right] \\ &- \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \\ &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_1 - n\bar{x}) - x_1^2 & \cdots & -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n \\ \vdots & \ddots & \vdots \\ -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n & \cdots & (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_n - n\bar{x}) - x_n^2 \end{matrix} \right] \; . \end{split}\]∎
Sources: Metadata: ID: P285 | shortcut: slr-mat | author: JoramSoch | date: 2021-11-09, 15:19.