## 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:

$\textbf{y} = \textbf{X} \mathbf{\beta} + \mathbf{\epsilon}$