Econ220B_final_09

Econ220B_final_09 - 1 Econ 220B Final Exam Winter 2009...

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–1– Econ 220B Final Exam Winter 2009 DIRECTIONS: No books or notes of any kind are allowed. Answer all questions on separate paper. 250 points are possible on this exam. 1.) (60 points total) The least-squares regression model y = X β + ε makes a lot of assumptions, including: (i) X 0 X is of full rank k (ii) E ( ε 2 t )= σ 2 (iii) plim( X 0 X /T )= Q of full rank k (iv) T 1 / 2 P T t =1 x t ε t L N ( 0 2 Q ) . In addition, we often make one or more of the following assumptions: (v) E ( ε | X )= 0 ; (vi) E ( εε 0 | X )= σ 2 I T ; (vii) ε | X N ( 0 2 I T ) . (viii) x t is strictly deterministic Questions (a)-(f) below involve certain statements about properties of the least-squares regression coe cient b =( X 0 X ) 1 X 0 y and variance estimate s 2 =( T k ) 1 e 0 e for e = y Xb . In each of the questions, you can assume that (i)-(iv) are satis f ed. The question is, which further assumptions besides (i)-(iv) would you need in order to prove the stated result? For example, if you think assumptions (i), (ii), and (v) alone are su cient to prove that E ( b )= β , then your answer to part (a) should be “v”. If you think that (i), (iii), (v) and (vi) are needed to prove that E ( b )= β , then your answer to part (a) should be “v and vi”. Note that you should not give any proofs or arguments— your answer in each case should be some combination of the numerals “v”, “vi”, “vii”, or “viii”. a.) E ( b )= β b.) b is the best linear unbiased estimator of β c.) E [( b β )( b β ) 0 | X ]= σ 2 ( X 0 X ) 1 d.) m 1 ( Rb r ) 0 [ s 2 R ( X 0 X ) 1 R 0 ] 1 ( Rb r ) F ( m, T k ) when R β = r for R an ( m × k ) matrix e.) b p β f.) ( Rb r ) 0 [ s 2 R ( X 0 X ) 1 R 0 ] 1 ( Rb r ) L χ 2 m when R β = r for R an ( m × k ) matrix
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–2– 2.) (50 points total) Consider a regression of a scalar
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Econ220B_final_09 - 1 Econ 220B Final Exam Winter 2009...

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