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chapter11_s - CHAPTER 11 SOLUTIONS TO PROBLEMS 11.1 Because...

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CHAPTER 11 SOLUTIONS TO PROBLEMS 11.1 Because of covariance stationarity, 0 γ = Var( x t ) does not depend on t , so sd( x t+h ) = 0 γ for any h 0. By definition, Corr( x t ,x t + h ) = Cov( x t ,x t+h )/[sd( x t ) sd( x t+h )] = 0 0 /( ) / . h h 0 γ γ γ γ γ = 11.3 (i) E( y t ) = E( z + e t ) = E( z ) + E( e t ) = 0. Var( y t ) = Var( z + e t ) = Var( z ) + Var( e t ) + 2Cov( z , e t ) = 2 z σ + 2 e σ + 2 0 = 2 z σ + 2 e σ . Neither of these depends on t . (ii) We assume h > 0; when h = 0 we obtain Var( y t ). Then Cov( y t , y t+h ) = E( y t y t+h ) = E[( z + e t )( z + e t+h )] = E( z 2 ) + E( ze t+h ) + E( e t z ) + E( e t e t+h ) = E( z 2 ) = 2 z σ because { e t } is an uncorrelated sequence (it is an independent sequence and z is uncorrelated with e t for all t . From part (i) we know that E( y t ) and Var( y t ) do not depend on t and we have shown that Cov( y t , y t+h ) depends on neither t nor h . Therefore, { y t } is covariance stationary. (iii) From Problem 11.1 and parts (i) and (ii), Corr( y t , y t+h ) = Cov( y t , y t+h )/Var( y t ) = 2 z σ /( 2 z σ + 2 e σ ) > 0. (iv) No. The correlation between y t and y t+h is the same positive value obtained in part (iii) now matter how large is h . In other words, no matter how far apart y t and y t+h are, their correlation is always the same. Of course, the persistent correlation across time is due to the presence of the time-constant variable, z . 11.5 (i) The following graph gives the estimated lag distribution: 58
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