5. In a simple linear regression for (li, yi) where i = 1,2,...,n, Y = Bo + Biai + € where {e}_1 are i.i.d. N(0,0%). (a)

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5 In A Simple Linear Regression For Li Yi Where I 1 2 N Y Bo Biai Where E 1 Are I I D N 0 0 A 1 (193.1 KiB) Viewed 29 times

5. In a simple linear regression for (li, yi) where i = 1,2,...,n, Y = Bo + Biai + € where {e}_1 are i.i.d. N(0,0%). (a) Derive the normal equations that the LS estimators, Bo and ßı, satisfy. (b) Show that the LS estimators of Bo and B, and g2 are given by ei = Szy Ss and Bo = Ý - Br. 72 02 n Sex where Sce = L1=(21 - )2 and Sxy= 21-11i - 7)(Y; - Y). (c) Show that Bo ~ N(B0,0% + _-)) and Ŝi ~ N(BL, (d) Show that o? cov(Bo, ßi) Sex (e) Let êi = Y; – Bo - Bili. Show that coulêi, Bo) = 0, and ê, and ßo are independent. Let 1.Σ. - - : (Y; – Bo – Bux:)? €2. TE 72 1 a 2 n n i=1 i=1 Show that ô2 and Bo are independent. (f) Let Y; = Bo + B12;. Show that ΣΥ. - Y)? - ΣΥ - Y)2 +ΣΥ. - Yή)?. - i=1 i=1 i=1 (Hint: the normal equations in (a) are helpful.) (g) Show that the 100(1 - a)% prediction interval of Yo taken at r = ro is given by 1 (20 - )2 Bo + 110 Etn-2,0/21/1+ + n SIE where tn 2,0/2 is the upper a/2 percentile of a t distribution with (n-2) degree of freedom.