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

Chapter , Problem 110SE

(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. $\text { Show that } \operatorname{cov}\left(\hat{\beta}_{0}, \hat{\beta}_{1}\right)=-\bar{x} \sigma^{2} / S_{x x}$

b. $\text { show that } \operatorname{cov}\left(\bar{Y}, \widehat{\beta}_{1}\right)=0$

Equation Transcription:

$Y={\beta{}}_{0}+{\beta{}}_{1}x+\ ϵ$

$E(𝝐)=0,\ V(ϵ)={\sigma{}}^{2}$

$ϵ$

$show\ that\ cov\ (\ {\widehat{\beta{}}}_{0},{\widehat{\beta{}}}_{1})=-\overline{x}{\sigma{}}^{2}/{S}_{xx}$

$show\ that\ cov\ (\ \overline{Y,}\ {\widehat{\beta{}}}_{1})=0$

Text Transcription:

Y=\beta_0+ \beta_1x+ \epsilon

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

\epsilon

show that cov ( \hat \beta_0,\hat \beta_1)=-x \sigma 2/Sxx

show that cov ( \bar Y, \widehat \beta 1)=0

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