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Answer key for the midterm in 2012 1a.) x t ± y t " 1 , z t , z t " 1   U * ± C , ) , +   U C ² ) ² + ² ± ! t ± 1 T x t x t U " 1 ! t ± 1 T x t y t @ ² 2 ± T " 3   " 1 ! t ± 1 T y t " x t U * ±   2 b.) Let R ± 0,1,1   U and use ) ² ³ + ² s 2 R ! x t x t U   " 1 R U . Treat as Student t with T " 3 degrees of freedom. This is only an approximation since x t includes the lagged value of y . c.) SSR U is T " 3 times @ ² 2 above. To calculate SSR R estimate y t ± C y t " 1 ³ ) z t " z t " 1   ³ / t and treat SSR R " SSR U SSR U / T " 3   as F 1, T " 3   . The F statistic in (c) is exactly equal to the square of the t statistic in (b). 2a.) S ± E £ E / t 2 | x t   x t x t U ¤ ± E @ 2 x 1 t x t x t U   ± @ 2 " where row i column j element of " is given by 5 1 ij . b.) b ± ! x t x t U " 1 ! x t y t ± * ³ ! x t x t U " 1 ! x t / t T b " *   ± T " 1 ! x t x t U " 1 T " 1/2 ! x t / t L v Q " 1 z where z L N 0, S   . Thus T b " *   L v N 0 , Q " 1 SQ " 1   . c.) * ± GLS ± ! x ² t x ² t U " 1 ! x ² t y ´ t x ² t ± x 1 t " 1/2 x ² t y ´ t ± x 1 t " 1/2 y t

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3a.) ) ² + ² ± T " 1 ! 1 y 2 t 1 y 2 t " 1 T " 1 ! 1 y 2 t y 1 t ± 1 T " 1 ! y 2 t T " 1 ! y 2 t T " 1 ! y 2 t 2 " 1 T
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