Covariance and Correlation for Regression Model

#1
I've been looking over some regression models lately and I came across one which, although similar, differs from the "standard" simple linear model. I was hoping somebody could provide some assistance with some properties that I'm confused with.

Assuming the regression form:

\(y_{i} = \beta_{0} + \beta_{1}(x_{i}-\bar{x}) + \epsilon_{i}\)

with expected value:

\({\bf E}[y_{i}] = \hat{\beta}_{0} + \hat{\beta}_{1}(x_{i}-\bar{x})\)

where \(\hat{\beta}_{0} = \bar{y}\) and \(\hat{\beta}_{1} = \frac{S_{XY}}{S_{XX}}\)

and, from what I've worked out:

\({\bf E}[\hat{\beta}_{0}] = \beta_{0}\), \({\bf E}[\hat{\beta}_{1}] = \beta_{1}\)

and:
\(\text{Var}(y_{i}) = \sigma^{2}\), \(\text{Var}(\hat{\beta}_{0}) = \frac{\sigma^{2}}{n^{2}}\)**, \(\text{Var}(\hat{\beta}_{1}) = \frac{\sigma^{2}}{S_{XX}}\)**

**These could be wrong.

How can it be shown that:

(a)

\(\text{Cov}(y_{i}, \hat{\beta}_{1}) = \frac{\sigma^{2}(x_{i}-\bar{x})}{\sum (x_{i}-\bar{x})^{2}}\)

I know that the covariance formula is given by:

\(\text{Cov}(y_{i}, \hat{\beta_{1}}) = {\bf E}[(y_{i} - {\bf E}[y_{i}])(\hat{\beta_{1}} - {\bf E}[\hat{\beta_{1}}])]\)

However, I'm confused about how to manipulate this formula to yield the desired result.

(b)

\(\text{Corr}(\hat{\beta}_{0}, \hat{\beta}_{1}) = 0\)

Here, I know that if it can be shown that:

\(\text{Cov}(\hat{\beta_{0}}, \hat{\beta}_{1}) = 0\)

it follows that:

\(\text{Corr}(\hat{\beta}_{0}, \hat{\beta}_{1}) = 0\)

However, as in part (a), I'm confused about how to develop the covariance formula accordingly.

Thanks!
 
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