This preview shows page 1. Sign up to view the full content.
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: 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. 3. [15 pts] (a) Suppose X n = fXi gn is an i.i.d. random sample with a population i=1 probability density f (x; ): Is X n a su cient statistic for ? Give your reasoning. (b) Suppose X n = fXi gn is an i.i.d. random sample from a N ( ; ) population, i=1 where is unknown. Find a su cient statistic for . Give your reasoning. 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 1 2 1 1X = (Xi n i=1 X)2 ; 2 . Give your reasoning. (b) [15 pts] Which estimator, ^ 2 or ^ 2 ; is more e cient in terms of mean squared 2 1 error? Give your reasoning. 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) [6 pts] Does Zn converges in probability to 0? Give your reasoning clearly. 6. [10 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 : N (ai ; 7. [20 pts] Suppose fXi gn is an independent random sample and Xi i=1 2 s, i = 1; :::; n; where ai and i are known constants that dier across dierent i' and an unknown parameter. Thus, the probability density of Xi is f (xi ; ) = p 1 2
2 i P (a) [4 pts] Show that ^ is unbiased if and only if n ci = 1: i=1 (b) [6 pts] Find the most e cient unbiased estimator ^ from the class of ^ : P [Hint: n i2 = n(n + 1)(n + 2)=6:] i=1 2 i ); is 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 CramerRao lower bound? Give your reasoning. 2 ...
View
Full
Document
This note 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).
 Fall '07
 HONG
 Economics, Econometrics

Click to edit the document details