Consider the general linear model

Chapter 11, Problem 71E

(choose chapter or problem)

Consider the general linear model

                  \(Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\ldots+\beta_{k} x_{z}+\varepsilon\)

where \(E(\varepsilon)=0\) and \(V(\varepsilon)=o^{2}\). Notice that \(\widehat{\beta_{i}}=a \widehat{\beta}\) , where the vector a is defined by

\(a_{j}=\left\{\begin{array}{l}

1, i f j=i \\

0, \text { if } j \neq i

\end{array}\right.

\)

Use this to verify that \(E\left(\widehat{\beta_{i}}\right)=\beta_{i}\) and \(V\left(\widehat{\beta_{i}}\right)=c_{0} o^{2}\), where cii is the element in row i and column i of (XX)-1.

Equation transcription:

Text transcription:

Y=\beta{0}+beta{1} x{1}+\beta{2} x{2}+\ldots+\beta_{k} x{k}+varepsilon

E(\varepsilon)=0

V(varepsilon)=o^{2}

\widehat{{i}}=a{\beta}

a{j}{array}{l}

1, i f j=i \\

0, { if } j neq i

\end{array}

E(\widehat{\beta{i}})=\beta{i}

V(widehat{beta{i}})=c{0} o^{2}