A simple linear regression model [tex]y=\beta_{0}+\beta_{1} x+\varepsilon [/tex] with [tex]\varepsilon \sim N I D\left(0, \sigma^{2}\right)[/tex].

[FOWKFOWKIE]

Consider the simple linear regression model \(\displaystyle y=\beta_{0}+\beta_{1} x+\varepsilon \) with \(\displaystyle \varepsilon \sim N I D\left(0, \sigma^{2}\right)\).. Then give a concise but clear proof of the following.

(1) The least squares estimators $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$ are uncorrelated.
(2) The covariance between $\bar{y}$ and $\hat{\beta}_{1}$ is zero.
(3) Show that $\hat{y}_{0}$ is an unbiased predictor of $y_{0}$.
(4) Use $b$ to show that $P V\left(\hat{y}_{0}\right)=\sigma^{2}\left(1+\frac{1}{n}+\frac{\left(x-\bar{x}_{0}\right)^{2}}{S_{x x}}\right)$
(5) the maximum value of $r^{2}$ is less than 1 if the data contain repeated (different) observations on $y$ at the same value of $x$.

 

[Deleted member 4993]

Consider the simple linear regression model \(\displaystyle y=\beta_{0}+\beta_{1} x+\varepsilon \) with \(\displaystyle \varepsilon \sim N I D\left(0, \sigma^{2}\right)\).. Then give a concise but clear proof of the following.

(1) The least squares estimators $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$ are uncorrelated.
(2) The covariance between $\bar{y}$ and $\hat{\beta}_{1}$ is zero.
(3) Show that $\hat{y}_{0}$ is an unbiased predictor of $y_{0}$.
(4) Use $b$ to show that $P V\left(\hat{y}_{0}\right)=\sigma^{2}\left(1+\frac{1}{n}+\frac{\left(x-\bar{x}_{0}\right)^{2}}{S_{x x}}\right)$
(5) the maximum value of $r^{2}$ is less than 1 if the data contain repeated (different) observations on $y$ at the same value of $x$.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.