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Unformatted text preview: CHAPTER 14 SOLUTIONS TO PROBLEMS 14.1 First, for each t > 1, Var( u it ) = Var( u it u i,t 1 ) = Var( u it ) + Var( u i,t 1 ) = 2 2 u , where we use the assumptions of no serial correlation in { u t } and constant variance. Next, we find the covariance between u it and u i,t+ 1 . Because these each have a zero mean, the covariance is E( u it u i,t+ 1 ) = E[( u it u i,t1 )( u i,t+ 1 u it )] = E( u it u i,t+ 1 ) E( ) E( u i,t 1 u i,t+ 1 ) + E( u i,t 1 u it ) = E( ) = 2 it u 2 it u 2 u because of the no serial correlation assumption. Because the variance is constant across t , by Problem 11.1, Corr( u it , u i,t+ 1 ) = Cov( u it , u i,t+ 1 )/Var( u it ) = 2 /(2 ) u 2 u = .5. 14.3 (i) E( e it ) = E( v it i v ) = E( v it ) E( i v ) = 0 because E( v it ) = 0 for all t . (ii) Var( v it i v ) = Var( v it ) + 2 Var( i v ) 2 Cov( v it , i v ) = 2 v + 2 E( 2 i v ) 2 E( v it i v ). Now, 2 2 2 E( ) v i t a v 2 u = = + and E( v it i v ) = = 1 1 ( ) T it is s v v = T E 1 T [ 2 a + 2 a + + ( 2 a + 2 u ) + + 2 a ] = 2 a + 2 u / T . Therefore, E( 2 i v ) = 1 T t = 1 it i T E v ( ) v = 2 a + 2 u / T . Now, we can collect terms: Var( v it i v ) = . 2 2 2 2 2 2 2 ( ) ( / ) 2 ( / a u a u a u T T + + + + ) Now, it is convenient to write = 1 / , where 2 u / T and 2 a + 2 u / T . Then Var( v it i v ) = ( 2 a + 2 u ) 2 ( 2 a + 2 u / T ) + 2 ( 2 a + 2 u / T ) = ( 2 a + 2 u ) 2(1 / ) + (1 / ) 2 = ( 2 a + 2 u ) 2 + 2 + (1 2 / + / ) = ( 2 a + 2 u ) 2 + 2 + (1 2 / + / ) = ( 2 a + 2 u ) 2 + 2 + 2 + = ( 2 a + 2 u ) + = 2 u . This is what we wanted to show. 78 (iii) We must show that E( e it e is ) = 0 for t s . Now E( e it e is ) = E[( v it i v )( v is i v )] = E( v it v is ) E( i v v is ) E( v it i v ) + 2 E( 2 i v ) = 2 a 2 ( 2 a + 2 u / T ) + 2 E( 2 i v ) = 2 a 2 ( 2 a + 2 u / T ) + 2 ( 2 a + 2 u / T ). The rest of the proof is very similar to part (ii): E( e it e is ) = 2 a 2 ( 2 a + 2 u / T ) + 2 ( 2 a + 2 u / T ) = 2 a 2(1 / ) + (1 / ) 2 = 2 a 2 + 2 + (1 2 / + / ) = 2 a 2 + 2 + (1 2 / + / ) = 2 a 2 + 2 + 2 + = 2 a + = 0....
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 Spring '08
 BREMAN
 Econometrics

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