Question

2. In the regression model Y-Χβ+ E, Xis a fixed n x k matrix of rank k S11, E(c)-0 and E(es')-σ2Ω where Ω is a known non-singular matrix. The GiLS estimator of B is given by the formula Consider the following data 16 31 2 3 51 4 10 Assuming that Ay a) b) Calculate the GLS estimate of β in the model Y,Xß + ε Calculate the OLS estimate c) Compare it the two estimates and comment on efficiency.

Answer 1

$a.~\Omega=\begin{pmatrix} 1 & 0 & 0 &0 &0 \\ 0&1 & 0&0 & 0\\ 0&0 &1 & 0 &0 \\ 0&0 &0 &1 &0 \\ 0&0 &0 & 0 & 10 \end{pmatrix},~\Omega^{-1}=\begin{pmatrix} 1 & 0 & 0 &0 &0 \\ 0&1 & 0&0 & 0\\ 0&0 &1 & 0 &0 \\ 0&0 &0 &1 &0 \\ 0&0 &0 & 0 & 0.1 \end{pmatrix},~X^\prime=(-6,-3,0,3,6)\\ X^\prime \Omega^{-1}X=\begin{pmatrix} -6, &-3, & 0,& 3, & 0.6 \end{pmatrix}\begin{pmatrix} -6\\ -3\\ 0\\ 3\\ 6\\ \end{pmatrix}=6^2+3^2+3^2+(0.6)*6=57.6\\ X^\prime \Omega^{-1}Y=\begin{pmatrix} -6, &-3, & 0,& 3, & 0.6 \end{pmatrix}\begin{pmatrix} 3\\ 5\\ 5\\ 6\\ 2\\ \end{pmatrix}=-6*3+(-3)*5+3*6+0.6*2\\ =-13.8\\ \hat{\beta}_{GLS}=(X^\prime \Omega^{-1}X)^{-1}X^\prime \Omega^{-1}Y=(-13.8)/57.6=-0.2396\\ b.~\hat{\beta}_{OLS}=(X^\prime X)^{-1}X^\prime Y=(-6*3+(-3)*5+3*6+6*2)/((-6)^2+(-3)^2+3^2+6^2)=-0.0333\\$

$c.~Var(\hat{\beta}_{GLS})=(X^\prime\Omega^{-1}X)^{-1}X^\prime \Omega^{-1}Disp(Y)\Omega^{-1}X(X^\prime\Omega^{-1}X)^{-1}\sigma^2=(X^\prime\Omega^{-1}X)^{-1}\sigma^2\\ since~Disp(Y)=\sigma^2\Omega\\ Nar(\hat{\beta}_{OLS})=(X^\prime X)^{-1}\sigma^2\\ \\ Since~ X^\prime \Omega^{-1}X=57.6~and~(X^\prime X)=(-6)^2+(-3)^2+3^2+6^2=90\\ MSE_{GLM}=\sum_{i=1}^5(Y_i-\hat{\beta}_{GLM}X_i)^2=(3+0.2396*(-6))^2+(5+0.2396*(-3))^2+(5+0.2396*0)^2+(6+0.2396*(2))^2+(2+0.2396*(6))^2=99.5669\\ MSE_{OLS}=\sum_{i=1}^5(Y_i-\hat{\beta}_{OLS}X_i)^2=(3+0.0333*(-6))^2+(5+0.0333*(-3))^2+(5+0.0333*0)^2+(6+0.0333*(2))^2+(2+0.0333*(6))^2=98.4949\\ Standard~error~of~\hat{\beta}_{GLM}=\sqrt{99.5669/57.6}=1.3148\\ Standard~error~of~\hat{\beta}_{OLS}=\sqrt{98.4949/90}=1.0461\\ Hence~\hat{\beta}_{OLS}~is~more~efficient~than~\hat{\beta}_{GLS}.$