STAC67H3 Final: Exercise 1
Document Summary
Stac67 week 1: exercise solutions: exercise 1. It su ces to show that pn i=1(cid:0)xi x(cid:1)(yi y(cid:1) = pn. Xi=1 (cid:0)xi x(cid:1)2 = n n i=1 xiyi nxy. 1 = pn i=1(cid:0)xi x(cid:1)(yi y(cid:1) i=1(cid:0)xi x(cid:1)2. 2(cid:1) i 2xix + x i=1 x 2 i i 2x n. 2 i nx i 2x(cid:0)nx(cid:1) + nx. Xi=1 (cid:0)xi x(cid:1)(yi y(cid:1) = n. Xi=1 (cid:0)xiyi xiy yix + nxy(cid:1) Prove that the least squares estimators are unbiased, i. e. prove that e(cid:0) 1(cid:1) = 1 and e(cid:0) 0(cid:1) = Xi nx = i=1 ai = 0 since, n. Xi x i=1 aixi = 1, j=1(xj x(cid:1)2(cid:19)xi (cid:18) Xi=1 i=1 a2 a2 i = i = = pn j=1(xj x(cid:1)2(cid:1)2 (cid:0)pn i=1(xi x(cid:1)2. Using r we nd the following for sat data set, n. X 2 i = 1021. 487 n = 105: compute the least squares estimates of 0 and 1. 1 i n pn i=1 xipn i=1 xi(cid:1)2 n(cid:0)pn i=1 xiyi 1.
