I was going to start a thread for the same question, but came across this one, unanswered. I've gotten some of the solution finished, but I'm stuck on Cov(y_i, y_i hat) (see below).
Prove that Var(e_i) = σ^2 [1 -1/n - (X_i-X bar)^2 / ∑(X_i-X bar)^2].
I have thus far:
Var(e_i) = Var(y_i - y_i hat) = Var(y_i) - Var(y_i hat) - 2*Cov(y_i, y_i hat)
Where Var(y_i) = σ^2 (by assumption of the linear model)
Var(y_i hat) = (σ^2)/n + σ^2*(X_i-X bar)^2 / ∑(X_i-X bar)^2](solved with some some manipulation)
Can anyone pick it up from there and solve Cov(y_i, y_i hat) ?? I've tried over and over to no avail. Thanks.
Edit: I should state that I know what the answer to Cov() should be in order for the final proof answer to be correct. What I need help with is the steps to solve the Cov().