Question

Question 1 (10 marks) Suppose the data consist of repeated observations (Yit, xT), t = 1, ... ,T, for each individual i = 1,..., n. Here Yit € {0, 1, 2,... } is a count response and Xit is a k x 1 covariate vector. Consider a log-linear random intercept model Yit|bi ~ Poisson(1it), with lit E(Yit|bi) = elit and Nit = zB+bi. Here Zit is a p x 1 design vector built from Xit, and {bi} are i.i.d. N(0,02) random variables. Note Yit and Yjs are independent if i 7 j. Also Yit and Yis are conditionally independent given bi. Hint : The moment generating function (mgf) for b; is M(t) = Elebit) = e3o24. (a) Compute E(Yit). [3] (b) Compute Var(Yit). [3] (c) Compute Cov(Yit, Yis) where t# s. [4]

Answer 1

ANSWER: Let the data consist of repeat observationg (Lite xeito ),t= 1,-- iti for each individual este...n. Here yet € {0,1,2,...} is a count response and Xit is a kxl Covariate vector. Q ElYit) - FE18+|bi) ECOME Ele (ZAB+ +be) - Elezz'Bebe) C eZX'B elebi) . ettre tot Itul Et Be 1202 ete!8 +0% (6 Var(yit). VE ( Hitlbi) + Ev (Yitlbi) V lehet) + Ele nat) Ele 2nit )- El Cent) + Elent) Ele 2 ( 2t'p+b)) - ( e!!8+0%a)37 € 34/3 4042 Zeits bi (2) - eZ & LB 70% te Zieß +0%/ H it's+202 - ez2+B+02 + eZit!B+r%

(C) cov (hit ng is) E (Wie 918) - El 914) Etyis) EE ( Yiz Yislba) -&78+ 042 e Zr's +042 S E [ 141+'/bi) E ( 4s [b7)] -e 24'B +215/B702 = [enc enis) - 21t'p+2sB + o2 E Cemitthis) 24 18+ ZislB toe Efetit If+bf+ zis's+bi) - 2+*+218p-p que e zit's+2is's Ele2bi) 216'5 +Zigletre ZAB +215'B Hbi ezit'B +2islB+02 (29 - eZit's e Pit's +2isip le 21.204) е You Thank enring give like it helps to motivate me to do more work friend. please