Let 0 and 1 be the OLS intercept and slope estimators, respectively, and let u

Let β̂₀ and β̂₁ be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!).

(i) Show that β̂₂ can be written as β̂₁ = β₁ + Σⁿ_{i = 1wiui,} where w_i = d_i/SST_x and d_i = x_i – x̅.

(ii) Use part (i), along with Σⁿ_{i = 1wi} = 0, to show that β̂1 and u̅ are uncorrelated.

(iii) Show that β̂₀ can be written as β̂₀ = β₀ + u̅ – (β̂₁ = β₁) x̅.

(iv) Use parts (ii) and (iii) to show that Var (β̂₀) = s²/n + s²(x̅)²/SST_x.

(v) Do the algebra to simplify the expression in part (iv) to equation (2.58).