Question

4. Consider the simple linear regression model: Vi=Ay+βίζί +Ej, for i=1, . . . , n. Write out the expression for y, β,e, and X such that the model can be written in matrix orim

Answer 1

Let the simple linear regession model has data points. The regression model is

$y_i=\beta _0+\beta _1x_i+\epsilon_i;i=1,2,3,...,n$

In matrix notation, the model is $\overrightarrow{y}=\textbf{X}\overrightarrow{\beta }+\overrightarrow{\epsilon}$ . Here

$\overrightarrow{y}$ is the column vector,

$\begin{pmatrix} y_1\\ y_2\\ y_3\\ ....\\ y_n \end{pmatrix}$

$\textbf{X}$ is the matrix,

$\begin{pmatrix} 1&x_1\\ 1&x_2\\ 1&x_3\\ ....&....\\ 1&x_n \end{pmatrix}$

The regresion coefficients are

0.1

random error $\overrightarrow{\epsilon}$ is the column vector,

$\begin{pmatrix} \epsilon_1\\ \epsilon_2\\ \epsilon_3\\ ....\\ \epsilon_n \end{pmatrix}$

Thus the matrix model for the SLR is

123 123 xx:x 1 123