Exploring residuals for classical linear-regression models

• Residuals should be normally distributed with mean zero
• The leverage values from the diagonal of hat matrix \(\mathbf{H} = \mathbf{X}(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T\).

\[ \mathbf{\hat{y}} = \mathbf{X}\hat{\beta} = \mathbf{X}[(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}] = \mathbf{H}\mathbf{y} \]

• Expected variance given by: \(\text{Var}(e_i) = \sigma^2 (1 - h_{ii})\)
• For independent explanatory variables, it should lead to a constant variance of residuals.