Regressand and regressor data

Summary

• Given: A sample of size \(n\) .
• Each sample point consists of \(\left( y_i,x_{i,1}, \ldots ,x_{i,k} \right)\) ( \(x_{i,1}\) is typically 1 for all \(i\) )
• The \(y\) -observations are called the regressand data
• The \(x\) -observations are called regressor data.
• Definition of \(x_i\) for \(i=1, \ldots ,n\) :

\[x_i=\begin{bmatrix}x_{i,1} \\ ⋮ \\ x_{i,k}\end{bmatrix}\]

• \(x_i\) is \(k×1\) .
• Definition of \(y\) :

\[y=\begin{bmatrix}y_1 \\ ⋮ \\ y_n\end{bmatrix}\]

• \(y\) is \(n×1\) .
• Definition of \(X\) :

\[X=\begin{bmatrix}x_{1,1} & \ldots & x_{1,k} \\ ⋮ & ⋱ & ⋮ \\ x_{n,1} & ⋯ & x_{n,k}\end{bmatrix}\]

• \(X\) is \(n×k\) .
• Since \(x'_i\) is row \(i\) in \(X\) for \(i=1, \ldots ,n\) , we can write

\[X=\begin{bmatrix}x'_1 \\ ⋮ \\ x'_n\end{bmatrix}\]

• If \(x_{i,1}=1\) for all \(i\) then the first column of \(X\) is a column of 1s.