I am trying to compute $Var(e_i)$.
So far I have
$Var(e_i)=Var(y_i-\hat y_i)=Var(y_i)+Var(\hat y_i)-2cov(y_i,\hat y_i)$
Now, I know that
$Cov(y_i,\hat y_i)=var(\hat y_i)$
but how do I prove this? (without using matrices)
But anyway, from there I have $Var(e_i)=var(y_i)-var(\hat y_i)= \sigma^2 -var(\overline y+\hat \beta_1 (x_i-\overline x))$
$=\sigma^2-var(\overline y)-(x_i-\overline x)^2var(\hat\beta_1) -2cov(\overline y,\hat \beta_1 )(x_i-\overline x)$
$\sigma^2-var(\sum y_i /n) - (x_i-\overline x)^2\sigma^2\sum(x_i-\overline x)^2$
$var(\sum y_i /n)=\sum(var(y_i))n^2 = \sigma ^2 / n$
So I end up with $Var(e_i)=\sigma^2(1-(1/n)-(x_i-\bar x)^2\sum (x_i-\overline x)^2)$
Is this correct?
