Covariance term in simple linear regression

Question

I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps the covariance term $\text{Cov}(\bar{y}, \hat{\beta_1})$ comes up and I don't know how to deal with it. Any help would be appreciated!

Answer 1

I was trying to prove the same and struggled a bit but finally got to a solution. I know this was asked long back but there might be other people interested.

$$Cov (\bar{Y}, \hat{\beta_1}) = E[(\bar{Y} - E(\bar{Y}))(\hat{\beta}_1 - \beta_1)]\\ = E[(\beta_0 + \bar{X}\beta_1 + \bar{\epsilon} - \beta_0 - \bar{X} \beta_1) (\hat{\beta}_1 - \beta_1)]\\ = E(\bar{\epsilon} (\hat{\beta}_1 - \beta_1)]\\ = E(\bar{\epsilon} \hat{\beta}_1) - \beta_1 E(\bar{\epsilon})\\ = E(\bar{\epsilon} \hat{\beta}_1) $$ Now we know that $$ \hat{\beta}_1 = \sum_{i=1}^n \dfrac{(X_i - \bar{X})Y_i}{\sum(X_i - \bar{X})^2}$$ which we can write as $$ \hat{\beta}_1 = \sum_{i=1}^n c_i Y_i$$ for convenience. Now consider, $$ Cov(\bar{\epsilon}, \hat{\beta}_1) = E(\bar{\epsilon} \hat{\beta}_1)\\ = Cov(\sum c_i \epsilon_i , \bar{\epsilon})\\ = \sum_{i=1}^n \sum_{j=1}^n c_i\times \dfrac{1}{n} Cov(\epsilon_i, \epsilon_j) \\ \text{by independence we get,}\\ = \sum_{i=1}^n \dfrac{c_i}{n} \sigma^2\\ = \dfrac{\sigma^2}{n}\sum_i c_i = 0$$ This is because $$ c_i = \dfrac{(X_i - \bar{X})}{\sum_{j=1}^n (X_j - \bar{X})^2}\\ \sum_{i=1}^n c_i = \dfrac{1}{\sum_{j=1}^n (X_j - \bar{X})^2} \sum_{i=1}^n (X_i - \bar{X})\\ =\dfrac{1}{\sum_{j=1}^n (X_j - \bar{X})^2} (n\bar{X} - n\bar{X}) = 0 $$

Answer 2

$$ {\rm Cov}\bigg[\frac 1 n \sum y_i,\ \frac{\sum(x_i - \bar x)y_i}{\sum(x_i-\bar x)^2}\bigg] = \frac 1 n \times \frac{\sum(x_i-\bar x)\sigma^2}{\sum(x_i - \bar x)^2} = 1$$ it does not equal zero this way I believe