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Answer key for the 2006 f nal exam (econ 220b). 1. See the textbook. 2. (a) b =( X 0 X ) 1 X 0 Y . The OLS estimator is unbiased because the errors are strictly exogenous, and it is consistent because both the bias and variance decrease to zero in the limit. (b) ˆ β GLS =( X 0 V 1 X ) 1 X 0 V 1 y . The GLS estimator is also unbiased and consistent for the same reasons as the OLS estimator. (c) If we could estimate α consistently, the feasible GLS estimator would still be consistent (unless the function V were very poorly behaved). The feasible GLS estimator will probably be biased, though. Trans- forming the errors (i.e. premultiplying by ˆ V 1 / 2 ) will remove their strict exogeneity. 3. (a) ˆ β IV =( X 0 Z ) 1 X 0 ( + u ) = β + ³ T 1 X x t z 0 t ´ 1 ³ T 1 X x 0 t u t ´ β (b) T ( ˆ β IV β )= ³ T 1 X x t z t ´³ T 1 / 2 X x t u t ´ The IV estimator is asymptotically normal because the second term is the sum of a martingale di

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## This note was uploaded on 03/02/2012 for the course ECON 220b taught by Professor Hamilton,j during the Spring '08 term at UCSD.

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