Formulas Useful for Linear Regression Analysis and Related Matrix Theory

Abstract

Linear model. By $\fancyscript{M}= \{\varvec{\mathbf{y }}, \varvec{\mathbf{X }}{\varvec{\beta }}, \sigma ^2\varvec{\mathbf{V }}\}$ we mean that we have the model $\varvec{\mathbf{y }}= {\varvec{\mathbf{X }}}{\varvec{\beta }}+ {\varvec{\varepsilon }}$, where $\mathrm{E }(\varvec{\mathbf{y }}) = \varvec{\mathbf{X }}{\varvec{\beta }}\in \mathbb {R} ^n$ and $\text{cov}(\varvec{\mathbf{y }}) = \sigma ^2\varvec{\mathbf{V }}$, i.e., $\mathrm{E }({\varvec{\varepsilon }}) = \varvec{\mathbf{0 }}$ and $\text{cov}({\varvec{\varepsilon }}) = \sigma ^2\varvec{\mathbf{V }}$; $\fancyscript{M}$ is often called the Gauss–Markov model.