Vector Matrix Notation

Vector matrix notation with binary variables

Now let’s write this same model in vector-matrix notation. We need to specify vectors $\mathbf{Y}$, $\boldsymbol{\beta}$ and $\boldsymbol{\epsilon}$ and design matrix $\mathbf{X}$.
If we let $i=1, \ldots, n$, we can see that we have $n$ equations: $$\begin{aligned} y_1 & =\alpha+\beta I_{x_1}+\epsilon_1 \\ y_2 & =\alpha+\beta I_{x_2}+\epsilon_2 \\ &\vdots \\ y_n & = \alpha+\beta I_{x_n}+\epsilon_n \end{aligned}$$ where, for $i=1,\ldots,n$, $$\beta I_{x_i}= \begin{cases} \beta \hspace{0.5cm} \mbox{if } x_i=1 \\ 0 \hspace{0.5cm} \mbox{if } x_i=0. \end{cases}$$ Now suppose that $m$ out of the $n$ observations have $x_i=1$ and the remaining $n-m$ observations have $x_i=0$. We may group these data such that $$\begin{aligned} y_1 & =\alpha+\beta +\epsilon_1 \\ y_2 & =\alpha+\beta +\epsilon_2 \\ &\vdots \\ y_m & =\alpha+\beta +\epsilon_m \\ y_{m+1} & =\alpha +\epsilon_{m+1} \\ &\vdots \\ y_n & = \alpha+\epsilon_n \end{aligned}$$ We group all the observations, $y_i$, into an $n$ dimensional vector $Y$, and the errors $\epsilon_i$ into another column vector $\boldsymbol{\epsilon}$: $$\begin{aligned} \mathbf{Y} =&\left( \begin{array}{c} y_{1} \\ \vdots \\ y_{m} \\ y_{m+1} \\ \vdots \\ y_{n} \\ \end{array} \right), \quad \quad \quad \quad \boldsymbol{\epsilon} &= \left( \begin{array}{c} \epsilon_{1} \\ \vdots \\ \epsilon_{m} \\ \epsilon_{m+1} \\ \vdots \\ \epsilon_{n} \\ \end{array} \right).\\ \end{aligned}$$ Similarly, we stack the two parameters, $\alpha$ and $\beta$, into another column vector: $$\begin{aligned} \boldsymbol{\beta} &=&\left( \begin{array}{c} \alpha \\ \beta \end{array} \right)\end{aligned}$$ Finally, we append a vector of ones with the single predictor for each $i$, and create a matrix with two columns of the following form: $$\begin{aligned}\mathbf{X}&=&\left( \begin{array}{cc} 1 & 1 \\ \vdots & \vdots \\ 1 & 1\\ 1 & 0 \\ \vdots & \vdots \\ 1 & 0 \end{array} \right). \end{aligned}$$ such that when we multiply $\mathbf{X}$ by $\boldsymbol{\beta}$

$$\mathbf{X}\boldsymbol{\beta} = \left( \begin{array}{c} \alpha +\beta \\ \vdots \\ \alpha +\beta \\ \alpha \\ \vdots \\ \alpha \end{array} \right).\\ $$