Wooldridge IE AISE SSM ch11 - CHAPTER 11 SOLUTIONS TO...

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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 58 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 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 )] = 00 0 /( ) / . hh γγ ⋅= 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 + 2 0 = 2 z + 2 e . 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 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 /( 2 z + 2 e ) > 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:
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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 59 lag 01 2 3 4 5 6 7 8 9 10 11 12 coefficient 0 .04 .08 .12 .16 By some margin, the largest effect is at the ninth lag, which says that a temporary increase in wage inflation has its largest effect on price inflation nine months later. The smallest effect is at the twelfth lag, which hopefully indicates (but does not guarantee) that we have accounted for enough lags of gwage in the FLD model. (ii) Lags two, three, and twelve have t statistics less than two. The other lags are statistically significant at the 5% level against a two-sided alternative. (Assuming either that the CLM assumptions hold for exact tests or Assumptions TS.1 through TS.5 hold for asymptotic tests.) (iii) The estimated LRP is just the sum of the lag coefficients from zero through twelve: 1.172. While this is greater than one, it is not much greater, and the difference from unity could be due to sampling error. (iv) The model underlying and the estimated equation can be written with intercept α 0 and lag coefficients δ 0 , 1 , , 12 . Denote the LRP by θ 0 = 0 + 1 + + 12 . Now, we can write 0 = 0 1 2 12 . If we plug this into the FDL model we obtain (with y t = gprice t and z t = gwage t ) y t = α 0 + ( 0 1 2 12 ) z t + 1 z t -1 + 2 z t -2 + + 12 z t -12 + u t = 0 + 0 z t + 1 ( z t -1 z t ) + 2 ( z t -2 z t ) + + 12 ( z t -12 z t ) + u t .
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This note was uploaded on 09/11/2011 for the course ECONOMICS eco375 taught by Professor Suzuki during the Spring '11 term at University of Toronto- Toronto.

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Wooldridge IE AISE SSM ch11 - CHAPTER 11 SOLUTIONS TO...

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