chapter18_s - CHAPTER 18 SOLUTIONS TO PROBLEMS 18.1 With...

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CHAPTER 18 SOLUTIONS TO PROBLEMS 18.1 With z t 1 and z t 2 now in the model, we should use one lag each as instrumental variables, z t- 1,1 and z t- 1,2 . This gives one overidentifying restriction that can be tested. 18.3 For δ β , y t z t = y t z t + ( ) z t , which is an I(0) sequence ( y t z t ) plus an I(1) sequence. Since an I(1) sequence has a growing variance, it dominates the I(0) part, and the resulting sum is an I(1) sequence. 18.5 Following the hint, we have y t y t -1 = x t x t -1 + x t -1 y t -1 + u t or Δ y t = Δ x t – ( y t -1 x t -1 ) + u t . Next, we plug in Δ x t = γ Δ x t -1 + v t to get Δ y t = ( Δ x t -1 + v t ) – ( y t -1 x t -1 ) + u t = βγ Δ x t -1 – ( y t -1 x t -1 ) + u t + v t 1 Δ x t -1 + ( y t -1 x t -1 ) + e t , where 1 = , = –1, and e t = u t + v t . 18.7 If unem t follows a stable AR(1) process, then this is the null model used to test for Granger causality: under the null that gM t does not Granger cause unem t , we can write unem t = β 0 + β 1 unem t- 1 + u t E( u t | unem t -1 , gM t -1 , unem t -2 , gM t -2 , ) = 0 and | 1 | < 1. Now, it is up to us to choose how many lags of gM to add to this equation. The simplest approach is to add gM t -1 and to do a t test. But we could add a second or third lag (and probably not beyond this with annual data), and compute an F test for joint significance of all lags of gM t . 18.9 Let be the forecast error for forecasting y n +1 , and let 1 ˆ n e + 1 ˆ n a + be the forecast error for forecasting Δ y n +1 . By definition, = y n +1 1 ˆ n e + ˆ n f = y n +1 – ( + y n ) = ( y n +1 y n ) = Δ y n +1 = , where the last equality follows by definition of the forecasting error for Δ y n +1 . ˆ n g ˆ n g ˆ n g 1 + ˆ n a 110
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SOLUTIONS TO COMPUTER EXERCISES C18.1 (i) The estimated GDL model is l gprice = .0013 + .081 gwage + .640 gprice -1 (.0003) (.031) (.045) n = 284, R 2 = .454. The estimated impact propensity is .081 while the estimated LRP is .081/(1 – .640) = .225. The estimated lag distribution is graphed below. lag 01 2 3 4 5 6 7 8 9 10 11 12 0 .02 .04 .06 .08 coefficient .1 (ii) The IP for the FDL model estimated in Problem 11.5 was .119, which is substantially above the estimated IP for the GDL model. Further, the estimated LRP from GDL model is much lower than that for the FDL model, which we estimated as 1.172. Clearly we cannot think of the GDL model as a good approximation to the FDL model.
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This note was uploaded on 05/03/2009 for the course ECON 418 taught by Professor Breman during the Spring '08 term at University of Arizona- Tucson.

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chapter18_s - CHAPTER 18 SOLUTIONS TO PROBLEMS 18.1 With...

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