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The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$

  1. What is ${\rm cov}(e_i, \hat y_i)$?
  2. What is ${\rm cov}(\epsilon_i, \hat \beta_1)$?
  3. What is ${\rm cov}(e_i, \epsilon_i)$?

For 1, I am writing ${\rm cov}(e_i, \hat y_i)$ as \begin{align} {\rm cov}(e_i, \hat \beta_0+\hat\beta_1x_i) &= {\rm cov}(e_i, \hat\beta_0) + x_i {\rm var}(e_i, \hat\beta_1) \\ &= {\rm cov}(\bar y - \hat \beta_1 \bar x, y_i-\bar y - \hat \beta_1(x_i-\bar x)) \\ &= {\rm cov}(\bar y, y_i) - {\rm var}(\bar y) - \bar x {\rm cov}(\hat \beta_1, y_i) + (x_i -\bar x)\bar x{\rm var}(\hat \beta_1) \\ &= \frac{\sigma^2}{n}-\frac{\sigma^2}{n}-\frac{(x_i - \bar x)\bar x}{\sum (x_i - \bar x)^2}\sigma^2 + \frac{(x_i - \bar x)\bar x}{\sum (x_i - \bar x)^2}\sigma^2 \\ &= 0 \end{align}

For 2, ${\rm cov}(\epsilon_i, \hat \beta_1) = {\rm cov}(\epsilon_i, \frac{\sum (x_i - \bar x)y_i}{\sum (x_i - \bar x)^2} = \frac{(x_i - \bar x)}{\sum (x_i - \bar x)^2}\sigma^2$

For 3, \begin{align}{\rm cov}(e_i, \epsilon_i) &= {\rm cov}(\epsilon_i, y_i-\bar y - \hat \beta_1(x_i-\bar x) ) \\ &= {\rm cov}(\epsilon_i, y_i) - {\rm cov}(\epsilon_i, \bar y) - {\rm cov}(\epsilon_i, \hat \beta_1(x_i - \bar x)) \\ &= \sigma^2 - \frac{\sigma^2}{n} - \frac{(x_i - \bar x)^2}{\sum (x_i - \bar x)^2}\sigma^2 \end{align}

Can someone look at my derivation and tell me if there is any mistake?

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