chapter15_s

chapter15_s - CHAPTER 15 SOLUTIONS TO PROBLEMS 15.1(i It...

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CHAPTER 15 SOLUTIONS TO PROBLEMS 15.1 (i) It has been fairly well established that socioeconomic status affects student performance. The error term u contains, among other things, family income, which has a positive effect on GPA and is also very likely to be correlated with PC ownership. (ii) Families with higher incomes can afford to buy computers for their children. Therefore, family income certainly satisfies the second requirement for an instrumental variable: it is correlated with the endogenous explanatory variable [see (15.5) with x = PC and z = faminc ]. But as we suggested in part (i), faminc has a positive affect on GPA , so the first requirement for a good IV, (15.4), fails for faminc . If we had faminc we would include it as an explanatory variable in the equation; if it is the only important omitted variable correlated with PC , we could then estimate the expanded equation by OLS. (iii) This is a natural experiment that affects whether or not some students own computers. Some students who buy computers when given the grant would not have without the grant. (Students who did not receive the grants might still own computers.) Define a dummy variable, grant , equal to one if the student received a grant, and zero otherwise. Then, if grant was randomly assigned, it is uncorrelated with u . In particular, it is uncorrelated with family income and other socioeconomic factors in u . Further, grant should be correlated with PC : the probability of owning a PC should be significantly higher for student receiving grants. Incidentally, if the university gave grant priority to low-income students, grant would be negatively correlated with u , and IV would be inconsistent. 15.3 It is easiest to use (15.10) but where we drop z . Remember, this is allowed because 1 () n i i zz = i x x = 1 ( n ii i zx x = ) and similarly when we replace x with y . So the numerator in the formula for 1 ˆ β is 11 1 1 nn n i i i i zy y zy ny ny == = ⎛⎞ −= = ⎜⎟ ⎝⎠ ∑∑ , where n 1 = is the number of observations with z i = 1, and we have used the fact that / n 1 = 1 n i i z = 1 n i = 1 y , the average of the y i over the i with z i = 1. So far, we have shown that the numerator in 1 ˆ is n 1 ( 1 y y ). Next, write y as a weighted average of the averages over the two subgroups: y = ( n 0 / n ) 0 y + ( n 1 / n ) 1 y , 85
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where n 0 = n n 1 . Therefore, 1 y y = [( n n 1 )/ n ] 1 y – ( n 0 / n ) 0 y = ( n 0 / n ) ( 1 y - 0 y ). Therefore, the numerator of 1 ˆ β can be written as ( n 0 n 1 / n )( 1 y 0 y ). By simply replacing y with x , the denominator in 1 ˆ can be expressed as ( n 0 n 1 / n )( 1 x 0 x ). When we take the ratio of these, the terms involving n 0 , n 1 , and n , cancel, leaving 1 ˆ = ( 1 y 0 y )/( 1 x 0 x ). 15.5 (i) From equation (15.19) with σ u = x , plim 1 ˆ = 1 + (.1/.2) = 1 + .5, where 1 ˆ is the IV estimator. So the asymptotic bias is .5.
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chapter15_s - CHAPTER 15 SOLUTIONS TO PROBLEMS 15.1(i It...

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