hws03 - Economics 140A Answers to Exercise 3 1. (Fall 1995...

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Economics 140A Answers to Exercise 3 1. (Fall 1995 Final Examination) a) The OLS estimator of is B = P n t =1 X t Y t P n t =1 X 2 t = + P n t =1 X t U t P n t =1 X 2 t : Because X t is exogenous and EU t = 0 , EB = + P n t =1 X t EU t P n t =1 X 2 t = The variance of the OLS estimator is E ( B EB ) 2 = E n t =1 X t U t P n t =1 X 2 t ± 2 = P n s =1 P n t =1 X s X t E ( U s U t ) ( P n t =1 X 2 t ) 2 = P n t =1 X 2 t ± 2 ( P n t =1 X 2 t ) 2 = ± 2 P n t =1 X 2 t ; where the second line follows because the regressor is exogenous and can be conditioned upon and the third line follows because the error is homoskedastic and uncorrelated. Because U t is a Gaussian random variable, if we condition on the regressor then P n t =1 X t U t P n t =1 X 2 t ± N & 0 ; ± 2 P n t =1 X 2 t ± ; Gaussian random variable. Because B = + P n t =1 X t U t P n t =1 X 2 t , B ± N ± 2 P n t =1 X 2 t ± ; and ²P n t =1 X 2 t ³ 1 2 ( B ) ± ± N (0 ; 1) : Because a standard Gaussian density is symmetric around zero, the shortest interval is one that sets the slope of the density function to be equal and of 1
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opposite sign at each end point, and is thus symmetric. The corresponding probability interval is P " 1
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hws03 - Economics 140A Answers to Exercise 3 1. (Fall 1995...

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