8.2 Mathematical Model

\[ y_{i} = \beta_{0} + \beta_{1}x_{1i} + \ldots + \beta_{p}x_{pi} + \epsilon_{i}\]

The assumptions on the residuals \(\epsilon_{i}\) still hold:

• They have mean zero
  
• They are homoscedastic, that is all have the same finite variance: \(Var(\epsilon_{i})=\sigma^{2}<\infty\)
  
• Distinct error terms are uncorrelated: (Independent) \(\text{Cov}(\epsilon_{i},\epsilon_{j})=0,\forall i\neq j.\)

The regression model relates \(y\) to a function of \(\textbf{X}\) and \(\mathbf{\beta}\), where \(\textbf{X}\) is a \(nxp\) matrix of \(p\) covariates on \(n\) observations and \(\mathbf{\beta}\) is a length \(p\) vector of regression coefficients.

In matrix notation this looks like:

\[ \textbf{y} = \textbf{X} \mathbf{\beta} + \mathbf{\epsilon} \]