Solution-Final_Exam_2006

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

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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 + j X j "; where E ( X ) = 0 ; var ( X ) = ² 2 X ; E ( " ) = 0 ; var ( " ) = ² 2 " ; and " and X are independent. Both ° and ± are constants. (a) [5 pts] Find E ( Y j X ) : (b) [5 pts] Find var ( Y j X ) : (c) [5 pts] Show cov ( Y; X ) = 0 if and only if ± = 0 : ANS: (a) E ( Y j X ) = E [ ° + ±X + j X j " j X ] = ° + ±X + j X j E ( " j X ) = ° + ±X: (b) var ( Y j X ) = var ( ° + ±X + j X j " j X ) = var ( j X j " j X ) = X 2 ² 2 " : (c)cov ( Y; X ) = cov ( ° + ±X + j X j "; X ) = ±cov ( X; X ) + cov ( j X j "; X ) = ±cov ( X; X ) + E [( j X j " ° E ( j X j " )) X ] = ±cov ( X; X ) + E ( X j X j " ) = ±² 2 X = 0 if and only if ± = 0 : Grading policy: 5 points each. Correct formula gets 1 point. 2. [10 pts] Suppose f X i g n i =1 is an i.i.d. N ( ³; ² 2 ) random sample, where both ³ and ² 2 are unknown parameters. Contruct an unbiased estimator for ´ = ³ 2 , and justify it is unbiased. ANS: A possible answer: Let b ´ = X 2 i ° S 2 n ; where S 2 n = P n i =1 ( X i ° X n ) 2 n ° 1 and X n = P n i =1 X i n . We could check the unbiasedness by E ( b ´ ) = E ( X 2 i ) ° E ( S 2 n ) = ³ 2 + ² 2 ° ² 2 = ³ 2 1
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(Remark: There are many solutions for this construction, say, we could also let b ´ = P n i =1 X 2 i n ° S 2 n ). Grading policy: Knowing the concept of unbiasedness 2 points. Correct construction 8 points. 3. [12 pts] (a) [5 pts] Suppose X n = f X i g n i =1 is an i.i.d. random sample with a population probability density f ( x; µ ) : Is X n a su¢ cient statistic for µ ? Give your reasoning. (b) [7 pts] Suppose X n = f X i g n i =1 is an i.i.d. random sample from a N ( µ; µ ) population, where µ is unknown. Find a ONE DIMENSIONAL su¢ cient statistic for µ . Give your reasoning.
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