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–1– Econ 220B, Winter 2010 Answers to midterm exam 1.) a.) E ( ˆ β 2 )= β 2 . b.) ε | X N ( 0 I T ) 2.) Let a ( µ )=1 and A ( µ )= ∂a ∂µ = 1 µ 2 . Then for ˆ µ T = T 1 P T t =1 y t by CLT T µ T µ ) L N (0 2 ) and from Hayashi Lemma 2.5 (the delta method) T [ a µ T ) a ( µ )] L N (0 , [ A ( µ )] 2 σ 2 ) . Hence v = σ 2 4 . But v 6 =l im T →∞ E ( q T µ 1 ) 2 because E ( q T ) does not exist. 3.) a.) b T = β + ³ X T t =1 x t x 0 t ´ 1 ³ X T t =1 x t ε t ´ T ( b T β )= ³ T 1 X T t =1 x t x 0 t ´ 1 ³ T 1 / 2 X T t =1 x t ε t ´ which converges in distribution to Q 1 times a N (0 2 Q ) variable, meaning T ( b T β ) L N ( 0 2 Q 1 ) . b.) Let R =(1 , 1 , 0 , 0 ,..., 0) . Then F =( b 1 b 2 ) s 2 R Ã T X t =1 x t x 0 t ! 1 R 0 1 ( b 1 b 2 ) c.) Note F = ( b 1 b 2 ) 2 s 2 R ¡P x t x 0 t ¢ 1 R 0 = h T R ( b β ) i 2 s 2 R ¡ T 1 P x t x 0 t ¢ 1 R 0 L h T R ( b β ) i 2 σ 2 RQ 1 R 0 .

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