Proof: Multiple linear regression is a special case of the general linear model
Theorem: Multiple linear regression is a special case of the general linear model with number of measurements $v = 1$, such that data matrix $Y$, regression coefficients $B$, noise matrix $E$ and noise covariance $\Sigma$ equate as
\[\label{eq:mlr-glm} Y = y, \quad B = \beta, \quad E = \varepsilon \quad \text{and} \quad \Sigma = \sigma^2\]where $y$, $\beta$, $\varepsilon$ and $\sigma^2$ are the data vector, regression coefficients, noise vector and noise variance from multiple linear regression.
Proof: The linear regression model with correlated errors is given by:
\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; .\]Because $\varepsilon$ is an $n \times 1$ vector and $\sigma^2$ is scalar, we have the following identities:
\[\begin{split} \mathrm{vec}(\varepsilon) &= \varepsilon \\ \sigma^2 \otimes V &= \sigma^2 V \; . \end{split}\]Thus, using the relationship between multivariate normal and matrix normal distribution, equation \eqref{eq:mlr} can also be written as
\[\label{eq:mlr-dev} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{MN}(0, V, \sigma^2) \; .\]Comparing with the general linear model with correlated observations
\[\label{eq:glm} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; ,\]we finally note the equivalences given in equation \eqref{eq:mlr-glm}.
- Wikipedia (2022): "General linear model"; in: Wikipedia, the free encyclopedia, retrieved on 2022-07-21; URL: https://en.wikipedia.org/wiki/General_linear_model#Comparison_to_multiple_linear_regression.
Metadata: ID: P329 | shortcut: mlr-glm | author: JoramSoch | date: 2022-07-21, 08:28.