Before moving to the logistic model, let us try to use our linear regression model to classify these two outcomes. We could for example fit a linear model to the default case if \( y_i > 0.5 \) and the no default case \( y_i \leq 0.5 \).
We would then have our weighted linear combination, namely
$$ \begin{equation} \boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} + \boldsymbol{\epsilon}, \tag{1} \end{equation} $$where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \boldsymbol{X} \) is our \( n\times p \) design matrix and \( \boldsymbol{\beta} \) represents our estimators/predictors.