Solution-Final_Exam_2006 - PROF HONG FALL 2006 ECONOMICS...

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Unformatted text preview: PROF HONG FALL 2006 ECONOMICS 619 FINAL EXAM Notes: (1) This is a closed book/notes exam. There are 7 questions, with a total of 100 points; (2) you have 150 minutes; (3) suggestion: have a look at all problems and ...rst solve the problems you feel easiest; (4) good luck! 1. [15 pts] Suppose Y = + X + jXj"; where E(X) = 0; var(X) = 2 ; E(") = 0; var(") = 2 ; and " and X are independent. " X Both and are constants. (a) [5 pts] Find E(Y jX): (b) [5 pts] Find var(Y jX): (c) [5 pts] Show cov(Y; X) = 0 if and only if = 0: ANS: (a) E(Y jX) = E [ + X + jXj"jX] = + X + jXj E ("jX) = + X: (b) var(Y jX) = var( + X + jXj"jX) = var(jXj"jX) = X 2 2 : " (c)cov(Y; X) = cov( + X + jXj"; X) = cov(X; X) + cov(jXj"; X) = cov(X; X) + E[(jXj" E(jXj"))X] = cov(X; X) + E(XjXj") = 2 = 0 if and only if = 0: X Grading policy: 5 points each. Correct formula gets 1 point. 2. [10 pts] Suppose fXi gn is an i.i.d.N ( ; 2 ) random sample, where both and 2 i=1 are unknown parameters. Contruct an unbiased estimator for = 2 , and justify it is unbiased. ANS: A possible answer: Let 2 b = Xi2 Sn ; 2 where Sn = Pn i=1 (Xi X n )2 n 1 and X n = Pn i=1 Xi n . We could check the unbiasedness by 2 E(b) = E(Xi2 ) 2 E(Sn ) = + 2 2 = 2 1 (Remark: There are many solutions for this construction, say, we could also let Pn X2 2 b = i=1 i Sn ). n Grading policy: Knowing the concept of unbiasedness 2 points. Correct construction 8 points. 3. [12 pts] (a) [5 pts] Suppose X n = fXi gn is an i.i.d. random sample with i=1 n a population probability density f (x; ): Is X a su cient statistic for ? Give your reasoning. (b) [7 pts] Suppose X n = fXi gn is an i.i.d. random sample from a N ( ; ) i=1 population, where is unknown. Find a ONE DIMENSIONAL su cient statistic for . Give your reasoning. 2 [Note: The pdf of a N ( ; 2 ) is p21 2 exp (xi 2 ) :] 2 ANS: (a)Yes, since Yn f (xi ; ) = g(X n ; )h(X n ) f (X n ; ) = i=1 Q where g(X n ; ) = n f (xi ; ), h(X n ) = 1. i=1 Grading policy: Knowing factorization thm 1 point. (b) Yn (xi )2 f (xi ; ) = exp( ) i=1 i=1 2 Pn )2 1 n i=1 (xi ) (p ) exp( 2 2 Pn 2 Pn n 2 1 n i=1 xi + 2 i=1 xi ) exp( ) (p 2 2 Pn 2 n X n 2 1 n i=1 xi ) exp( ) exp( (p xi ) 2 2 i=1 p 1 2 i=1 f (x ; ) = = = = n Yn Pn 2 P 1 i=1 xi where g( n x2 ; ) = ( p2 )n exp( i 2 P i=1 Thus, n x2 is a su cent statistics for : i=1 i n X = g( x2 ; )h(xn ) i n 2 P ); and h(xn ) = exp( n xi ): i=1 2 Grading policy: Knowing factorization thm 1 point. Correct joint pdf 1 points. Correct separation 2 points. Correct construction 3 points. 4. [20 pts]: Suppose fXi gn are i.i.d.N (0; 2 ): There are two estimators for 2 : i=1 ^2 1 ^2 2 1X 2 = X ; n i=1 i n n P where X = n 1 n Xi : i=1 (a) [5 pts] Check whether ^ 2 and ^ 2 are unbiased for 2 . Give your reasoning. 2 1 (b) [15 pts] Which estimator, ^ 2 or ^ 2 ; is more e cient in terms of mean squared 2 1 error? Give your reasoning. ANS: (a) E(^ 2 ) 1 n n n 1 X 2 1X 1X 2 = E( X )= E(Xi ) = ( n i=1 i n i=1 n i=1 2 1X = (Xi n i=1 X)2 ; + 0) = 2 E(^ 2 ) = 2 n 1 P 2 2 where Sn = n 1 1 n (Xi X)2 and E (Sn ) = 2 . i=1 Thus, ^ 2 is unbiased but ^ 2 is not. 1 2 Grading policy: Knowing the concept of unbiasedness 1 points. Correct answer 2 points each. (b) 2 M SE(^ 2 ) = var(^ 2 ) + (E(^ 2 ) = var(^ 2 ) 1 1 1 1 n n n X X 1 1 X 1 2 2 = var( var(Xi ) = 2 [E(Xi4 ) X )= 2 n i=1 i n i=1 n i=1 n 1 X = [3 n2 i=1 4 4 2 n 2 E[Sn ] = n 1 2 n ; E(Xi2 )2 ] = 2 n 4 : 1 n2 M SE(^ 2 ) = var(^ 2 ) + (E(^ 2 ) 2 2 2 Since n^ 2 2 2 2 2 ) = var(^ 2 ) + 2 1); 4 = (n 2 1) Sn 2 X 2 (n 3 we have var Thus, 2(n 1) 4 1 + 2 n2 n 1 4 2 4 2 = < n n2 n 2n 1 n2 n^ 2 2 2 = 2(n 1) ) var(^ 2 ) = 2 2(n 1) n2 4 M SE(^ 2 ) = 2 4 = 4 4 = M SE(^ 2 ): 1 So, ^ 2 is more e cient in terms of MSE. 2 Grading policy: Right MSE formula 2 points. Getting MSE( 2 ) correctly 6 points, 1 2 and MSE( 2 ) correctly 7 points. 5. [10 pts] Suppose a sequence of random variables fZn g is de...ned as Zn PZn 1 1 n 1 n n 1 n (a) [4 pts] Does Zn converges in mean square to 0? Give your reasoning clearly. (b) [9 pts] Does Zn converges in probability to 0? Give your reasoning clearly. ANS: 1 1 1 1 1 2 (a) E(Zn 0)2 = E(Zn ) = n2 (1 n ) + n2 n = n + n2 (1 n ) ! 1. Thus, it does not coverge to zero in mean square. 1 (b) Take any " > 0; there exist N; when n > N; then n < "; thus lim Pr(jZn 0j > ") = lim Pr(Zn = n) = lim n!1 n!1 1 = 0: n!1 n Thus,it converge to zero in probability. Grading policy: Correct formula 1 points. 6. [13 pts] Let X n = fX1 ; X2 ; :::; Xn g be an independent but not identically distributed random sample with E(Xi ) = and V ar(Xi ) = 2 =i2 ; where i = 1; 2; :::; n: Both and 2 are unknown. De...ne a class of estimator for as ^= n X i=1 ci Xi : 4 P (a) [4 pts] Show that ^ is unbiased if and only if n ci = 1: i=1 (b) [9 pts] Find the most e cient unbiased estimator ^ from the class of ^ : P [Hint: n i2 = n(n + 1)(n + 2)=6:] i=1 ANS: (a) E(^ ) = n X i=1 E(ci Xi ) = n X i=1 ci E(Xi ) = n X i=1 ci = if and only if n X i=1 ci = 1: Grading policy: Correct formula 1 points. (b) var(^ ) = n X i=1 var(ci Xi ) = n X i=1 c2 var(Xi ) = i )2 = var(^ ) = 2 n X i=1 2 c2 =i2 i c2 =i2 i M SE(^ ) = var(^ ) + (E(^ ) Now, the most e cient unbiased ^ is found by solving n X i=1 n X i=1 fci gi=1 min M SE(^ ) = n s:t: n X i=1 2 (c2 =i2 ) i ci = 1: The Lagrange Function is L= F.O.C. 2 n X i=1 c2 =i2 i + (1 n X i=1 ci ) ci : 2ci : 1 So =i2 = 0 for all i n X ci = 0 i=1 2 5 and hence n X i=1 ci = n X i2 =1) 2 2 i=1 2 = Pn 2 i=1 i2 = 12 2 n(n + 1)(n + 2) ci = 6i2 : n(n + 1)(n + 2) SOC is satis...ed, since we have a quadratic objective function and a linear constraint. P 6i2 Xi Thus, the most e cient estimator is ^ = n n(n+1)(n+2) : i=1 Grading policy: Set up the problem correctly 3 points. FOC 2 point. SOC 1 point. Answer 3 points. N (ai ; 2 ); 7. [20 pts] Suppose fXi gn is an independent random sample and Xi i i=1 i = 1; :::; n; where ai and 2 are known constants that dier across dierent i' and is s, i an unknown parameter. Thus, the probability density of Xi is f (xi ; ) = p 1 2 2 i exp (xi 2 ai )2 2 i : (a) [10 pts] Find the MLE ^ for ; and check if it is a global maximizer. (b) [10 pts] Does ^ achieve the Cramer-Rao lower bound? Give your reasoning. ANS: (a) The log likelihood function is Yn Yn f (xi ; )] = log( l(x1 ; x2 ; :::; xn j ) = log[ i=1 n X (xi i=1 i=1 So, max l(x1 ; x2 ; :::; xn j ) , min F.O.C. n X (xi i=1 p 1 2 ) 2 i ai )2 2 2 i n X (xi i=1 ai )2 2 2 i : ai )ai 2 i =0 The solution is 6 To check whether it is globally maximum, we need to check the S.O.C: @ 2 l(x1 ; x2 ; :::; xn j ) = @ 2 n X a2 i i=1 ^ = Pn Pn xi a i 2 i 2 ai 2 i i=1 i=1 : 2 i < 0; which ensures the concavity. Thus, the log likelihood function is concave and ^ is the globally maximum. Grading policy: Set up the problem correctly 3 points. FOC 2 point. SOC 2 point. Answer 3 points. (b) Since ^ is unbiased for , the Cramer-Rao lower bound is h 1 @ 2 l(x1 ;x2 ;:::;xn j ) @ 2 E and 1 i=P n a2 i : i=1 2 i Thus, ^ achieves the Cramer-Rao lower bound. Pn xi ai V ar 2 i=1 i V ar(^ ) = 2 Pn a2 i i=1 2 i = Pn var(xi )a2 i 4 i i=1 i=1 Pn a2 i 2 2 i = Pn ( i=1 Pn i=1 2 a2 i i 4 i 2 ai 2 2 i 1 = Pn ) ( i=1 a2 i i : 2) Grading policy: Correctly calculating the bound 5 points. Variance 5 points. 7 ...
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This test prep was uploaded on 12/08/2007 for the course ECON 6190 taught by Professor Hong during the Fall '07 term at Cornell University (Engineering School).

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