Let 0 and l be the OLS intercept and slope estimators, respectively, and let u be the samp...

Let 0 and l be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!).

(i) Show that B1 can be written as&= B1 + V. . w.u. where w. = d./SST and d. = x. — x.

==1 w . = 0, to show that B1 and u are uncorrelated. [Hint: You are being asked to show that E[(fi1 — B1) . u] = 0.]

(iii) Show that B0 can be written as .wiui where wi=di/SSTX and di= xi -

(iv) Use parts (i) along with to show that 1 and are uncorrelated. [Hint: You are being asked to show that

(iii) show that 0 can be written as

(iv). Use parts (ii) and (iii) to show that

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