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pq2_2ak - 1 ∆ X β 3 ∆ X = β 1 β 3 ∆ X The expected...

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Econ 410 - Fall 2006 Practice Questions for the Second Midterm - Part II - Answer Key 1. (a) E ( u i | X i ) = E ³ u i X i | X i ´ ; this step follows from the fact that X i carries the same information as X i (indeed, X i is just 1 X i , so if we know X i , we know X i as well). Then: E ³ u i X i | X i ´ = 1 X i E ( u i | X i ) = 0 , because the conditional mean assumption holds in the original model. It follows that the conditional mean assumption holds in the new model as well. (b) V ar ( u i | X i ) = V ar ³ u i X i | X i ´ = 1 X 2 i V ar ( u i | X i ) = ( αX i ) 2 X 2 i = α 2 So, the error term is homoskedastic. (c) V ar ( u i | X i ) = V ar ³ u i X i | X i ´ = 1 X 2 i V ar ( u i | X i ) = α 2 X 2 i So, in this case the error term u i would be heteroskedastic. 2. (a) The expected change in Y when X changes by X and D = 1 is: Y = β 0 + β 1 ( X + X )+ β 2 + β 3 ( X + X ) β 0 β 1 X β 2 β 3 X = β
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Unformatted text preview: 1 ∆ X + β 3 ∆ X = ( β 1 + β 3 ) ∆ X The expected change in Y when X changes by ∆ X and D = 0 is: ∆ Y = β + β 1 ( X + ∆ X ) − β − β 1 X = β 1 ∆ X (b) From ( a ) , we can say that this happens when β 3 = 0 (c) When D = 1 , the population regression function is: E ( Y i | X i , D i = 1) = β + β 1 X i + β 2 + β 3 X i = ( β + β 2 )+( β 1 + β 3 ) X i When D = 0 , the population regression function is: E ( Y i | X i , D i = 0) = β + β 1 X i (d) From ( c ) , we can say that this happens when β 2 = 0 and β 3 = 0 . 1...
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