Consider the simple linear regression model Y = ?0 + ?1x +

Chapter , Problem 111MEE

(choose chapter or problem)

Consider the simple linear regression model $Y=\beta_{0}+\beta_{1} x+\epsilon$, with

$\mathrm{E}(\epsilon)=0, \mathrm{~V}(\epsilon)=\sigma^{2}$, and the errors $\epsilon$ uncorrelated.

a. $\text { Show that } E\left(\hat{\sigma}^{2}\right)=E\left(M S_{E}\right)=\sigma^{2}$

b. $\text { Show that } E\left(M S_{R}\right)=\sigma^{2}+\beta_{1}^{2} S_{x x}$

Equation Transcription:

$Y={\beta{}}_{0}+{\beta{}}_{1}x+\ ϵ$

$E(𝝐)=0,\ V(ϵ)={\sigma{}}^{2}$

$ϵ$

$show\ that\ E(\ {\widehat{\sigma{}}}^{2})=E(M{S}_{E})={\sigma{}}^{2}$

$E(M{S}_{R})={\sigma{}}^{2}+{{\beta{}}_{1}}^{2}{S}_{xx}$

Text Transcription:

Y=\beta_0+ \beta_1x+ \epsilon

E(\epsilon)=0, V(\epsilon)=\sigma 2

\epsilon

Show that E(\hat\sigma^2)=E(M S_E)=\sigma^2

Show that E(M S_R)=\sigma^2+\beta_1^2 S_x x

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