Question

Question 2: As in Question 1, the aim of this question is to review some of the basic concepts you have learned in the prerequisite courses. Consider the model Yi= Bo + B12; +ti i=1,...,n where €1,..., en are independent and identically distributed as normal with mean 0 and variance o?. Show that the least squares method and the maximum likelihood estimate method give the same estimate for Bo and B1, but they give different estimates for o?

Answer 1

Y = Bot Bisit Ei i=1,2,. .n. Let Firut we cetimate Bo B, of by leart square method. then by MLE. Za I ( 98-80-8,x) ? S= 3 své-Bo-B, 31) differenciated s with respect to Bo and Bi. as = -2 El Hi- B - Bi%) a po as = -25 yi-Bo-Boci) ori as zo. 23 7o Bo a ) 23. 70, -o . & (Yi - Bo B, ai) = 0. & y = nie + ß Exi R = 7 - 6 Iyi-Bo- By Di) si =o. È (Méxi) = Poßer + a 1 x 0 Ey = nRo+ 8 (2 2 - 0 In eo multiply Exi and in 6 n. Ayi xi = n R { x + nå ² oci Bei Vi = nBo & si + B (Eai) ?

8 = Eyixi - n [xi Eyi { Coc?) - (0)² 3= Sxy Now estimation of oth SSRes= (gi-Bo-Bixi) 2 SS Res with hos of X² dietribution freedom (n-3) degree SSRe is x² (n=25 EC SS Res) = (0-2) 0? 23 = ssRes (n=2) Ÿ is estimation of ² Bo, Blo²by Now we maximum . Y!= will estimate the likelihood thethed, Bot Bipit ci i=1,2-n. hly6, Po, po, 0) = 7 Loge epitome (yi Bo-Birre) 109Lz - Rulog pr - logos (46-60- 202 1

bure a log2 apo - 1x2 2 cui por Plat) Blog L = - 12 x 2 I (Yi-Bo-Bi xi) di 28 02 157 Both eq ④ and I gi = não + ß Exi ² yixi = 8o Zai + Ås Ex J I Same method as so we have Bo R seen in heart square will be same as LE 1 Pime = Sxy Now we delimate ² log L= n log PR an log or -1 & li-Bo-B, si ² söz = 0 mm to to t4i –ßo - Biz)? 2 El tog alog 30 or + sot ano = 22 = I (x-ß- Ô 33 ² yi-Bo-Bixi) ² È 19. - 8o Bixi) ?

we com les clearly see that a = by olsn SSReso -2 MLE = SSROS both are different,