Question

Correlation and Regression a) Determine the value of a (as a function of n and y), which minimizes S function given below. Prove your answer with all the details. s=30,-a)? b) Consider the following two models: Modell:Y, = B. +B,X, + u Model2:Y,= 2, + QX: +e; where Ñ =X; -X (i) Are the OLS estimators of constant terms for both models identical? Are their variances the same? Prove your answer. Are the OLS estimators of slope terms for both models identical? Are their variances the same? Prove your answer. c) Show that the R2 from the regression of X on Y is the same as the R2 from the regression of Y on X.

Answer 1

a) S. (yi -a The value of x for which la determined ley - is minh can. de 20 3) 5 2(%; -a) (-- =) { = na. 2) a= 25.4 Ą Iyi , Sis minm. Model 1. Y = Bot Boxi thi lé = Yi-Bo-Bixi The as of OLs estimator follows: Bo & B. can lee der benined component +) (309-6-8, )" ]

22 (y: - 80 - 8, xi) (-x) = 0 I - {xiy = ßo {xi +8, Ixi? (SUP) -- where, - 2 (y; -ß - ß. 2.) (-1) 20 Eyie mß. + . Sa - (nxj. "= i n (x) - ({xi) (x) gives. nExiti- {xi & Ii = (n Boxi + miß, 22:2) - (mẾo Izi + $ 162.)). ..nexiy; - {xi Edi = niß Exi² & ß. (2x) - Ex: Yi - 1 - {xi Iyi Bil {x² (2x) 1) 323 - mas. 3) 5(2-50) (9:-5) = (x - 2) [22 - 05] . Verslo [va (1) rae (f) Say Ssan

EN JY WZ gosti - stein sa Boj prix b) For , Y = XB + ų. oli 10 1 xn . (4,,.... ny u = xn Ex)) +(2x)] vol dir (?) vac (6 (

Model 2: Yido to, xi tle is Xi = x:- - 2 E =). Ery i li Yi - do- dzi of do 8 & can lee The OLS estim ator determined as follows: 9.36 2 [ 16: 7. 23 (* -do -6, Fi) (-) . 5) * 3 ye não + a goza 8. Now, - {xi {(x; -7) (1) o for, . where, 3y - mdo -) Zo syg

- 22 (: - 26- 2,2i) (-x) = o. =). Eñy - 20 Zñi +2 Exe - Eñiya & añr. =) . { (rif) Ji - â {(x - 2)² I can lee = E Xi Yi - n Iyi 2 (-5) -** ý . I ai Yi – m ł Slaina) :) 2 = Say Sxx Sxy= {(2; -2) (y. -7) {(2, - x) For y = x B x + a. wher, 86-si x' Y. (Y ) e - (linen) = ).

Gisp (3): -($'$)" bi) from (16 the oLs e E' f IT- 1X, X₂-7 li xi- . Xn-7 Ill. identical XI XI from (le) - ll xnx of consta n = (x; -7) 7 in) from slope Cla for 13(-5) 263-8) from (id) & In slope lim c) for the se - FE(x: +372 (58)". wzbuat o m 01 R Ř Sya O Seegressier R? Å Say U O & u vore (20) = 2 . Say= Syx الماید we (). (24)

bi) from (16) & (2a) he found that, for both models is not the OLS estimators of constant term identical. from (1e) & oLs estimators (2e). the variances of for both the models terem is also not of constant identicas 3 ) trom (la) & (26), the OLS estimator of slope both the models is Colentical. of variance from (id) & (22), the OLs estimator of the slope lion for looth the models is identical. c) for the regression of x ont. for regression of Y on X., RE: ß Say Some so you Sayy some sony Say= Syx R² for regression et Youx " & x ont au equad