Linear classifier

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.

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