Formulas Useful for Linear Regression Analysis and Related Matrix Theory

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.