40  LM 6. GLS

40.1 Linear model

Let \(y_i=\beta x_i+\epsilon_i\), \(i=1,2\), where \(\epsilon_1 \sim N(0,\sigma^2)\) and \(\epsilon_2 \sim N(0,2\sigma^2)\), and \(\epsilon_1\) and \(\epsilon_2\) are statistically independent.

If \(x_1=1\) and \(x_2=-1\), obtain \(\hat \beta\) and its variance.

Linear model. Solutions

\[\hat{\beta}=\left( X^T V^{-1} X \right)^{-1} X^TV^{-1} y\]

\[V=\begin{pmatrix} 1 & 0 \\ 0 & 2 \\ \end{pmatrix},\ V^{-1}=\begin{pmatrix} 1 & 0 \\ 0 & 1/2 \\ \end{pmatrix}\]

\[X^T V^{-1}=(1, -1) \begin{pmatrix} 1 & 0 \\ 0 & 1/2 \\ \end{pmatrix} =( 1, -1/2)\\ \]

\[ \left(X^T V^{-1}X \right)^{-1}= \left(( 1, -1/2) ( 1, -1)' \right)^{-1} = \left(1+1/2 \right)^{-1} = 2/3\\ \]

\[X^TV^{-1}y=(1, -1/2)\begin{pmatrix} y_1 \\ y_2 \\ \end{pmatrix}=y_1-1/2 y_2\]

\[\Rightarrow \hat{\beta}=\left( X^T V^{-1} X \right)^{-1} X^TV^{-1} y=\frac{2}{3} \left( y_1- \frac{1}{2}y_2 \right) \]

\[Var(\hat{\beta})= \frac{4}{9} \left(Var(y_1)+\frac{1}{4} Var(y_2) \right)= \frac{4}{9} \left(\sigma^2+\frac{1}{4} 2 \sigma^2 \right)= \frac{2}{3}\sigma^2 \]