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 \(\boldsymbol{\beta}\)

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