The residuals \( \hat{\epsilon} \) are in turn given by $$ \hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, $$ and with $$ \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, $$ we have $$ \hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, $$ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.