Problem Set 6-Solution.pdf - Introductory Econometrics...

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Introductory Econometrics - Problem Set VI (Suggested Solutions) 1 . Let n 0 be the total number of observations with Z i = 0 and n 1 be the total number of observations with Z i = 1. We have ˆ β 1, IV = n i = 1 ( Z i - ¯ Z )( Y i - ¯ Y ) n i = 1 ( Z i - ¯ Z )( X i - ¯ X ) = n i = 1 Z i Y i - n ¯ Z ¯ Y n i = 1 Z i X i - n ¯ Z ¯ X = n 1 ¯ Y 1 - n 1 n ( n 0 ¯ Y 0 + n 1 ¯ Y 1 ) n 1 ¯ X 1 - n 1 n ( n 0 ¯ X 0 + n 1 ¯ X 1 ) = n 0 n 1 n ¯ Y 1 - n 0 n 1 n ¯ Y 0 n 0 n 1 n ¯ X 1 - n 0 n 1 n ¯ X 0 = ¯ Y 1 - ¯ Y 0 ¯ X 1 - ¯ X 0 . 2 . (a) Plugging Y t = C t + I t into the expression for C t and re-organizing terms, we have C t = β 1 1 - β 2 + β 2 1 - β 2 I t + u t 1 - β 2 , which is the expression for C t . Plugging the above expression into Y t = C t + I t , we have Y t = β 1 1 - β 2 + 1 1 - β 2 I t + u t 1 - β 2 , which is the expression for Y t . (b) First, cov ( Y t , u t ) = cov ( β 1 1 - β 2 + 1 1 - β 2 I t + u t 1 - β 2 , u t ) = var ( u t ) 1 - β 2 > 0 . Now, cov ( I t , Y t ) = cov ( I t , β 1 1 - β 2 + 1 1 - β 2 I t + u t 1 - β 2 ) = var ( I t ) 1 - β 2 > 0 .

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