伍德里奇计量经济学第六版答案Chapter-15

CHAPTER 15
TEACHING NOTES
When I wrote the first edition, I took the novel approach of introducing instrumental variables as a way of solving the omitted variable (or unobserved heterogeneity) problem. Traditionally, a
neous equations models. Occasionally, IV is first seen as a method to solve the measurement error problem. I have even seen texts where the first appearance of IV methods is to obtain a consistent estimator in an
AR(1) model with AR(1) serial correlation.
The omitted variable problem is conceptually much easier than simultaneity, and stating the conditions needed for an IV to be valid in an omitted variable context is straightforward. Besides, most modern applications of IV have more of an unobserved heterogeneity motivation.
A leading example is estimating the return to education when unobserved ability is in the error term. We are not thinking that education and wages are jointly determined; for the vast majority of people, education is completed before we begin collecting information on wages or salaries. Similarly, in studying the effects of attending a certain type of school on student performance, the choice of school is made and then we observe performance on a test. Again, we are primarily concerned with unobserved factors that affect performance and may be correlated with school choice; it is not an issue of simultaneity.
The asymptotics underlying the simple IV estimator are no more difficult than for the OLS estimator in the bivariate regression model. Certainly consistency can be derived in class. It is also easy to demonstrate how, even just in terms of inconsistency, IV can be worse than OLS if the IV is not completely exogenous.
At a minimum, it is important to always estimate the reduced form equation and test whether the IV is partially correlated with endogenous explanatory variable. The material on multicollinearity and 2SLS estimation is a direct extension of the OLS case. Using equation (15.43), it is easy to explain why multicollinearity is generally more of a problem with 2SLS estimation.
Another conceptually straightforward application of IV is to solve the measurement error problem, although, because it requires two measures, it can be hard to implement in practice.
Testing for endogeneity and testing any overidentification restrictions is something that should be covered in second semester courses. The tests are fairly easy to motivate and are very easy to implement.
While I provide a treatment for time series applications in Section 15.7, I admit to having trouble finding compelling time series applications. These are likely to be found at a less aggregated level, where exogenous IVs have a chance of existing. (See also Chapter 16 for examples.)
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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.
15.2 (i) It seems reasonable to assume that dist and u are uncorrelated because classrooms are not usually assigned with convenience for particular students in mind.
(ii) The variable dist must be partially correlated with atndrte. More precisely, in the reduced form
atndrte = 0 + 1priGPA + 2ACT + 3dist + v,
we must have 3 0. Given a sample of data we can test H0: 3 = 0 against H1: 3 0 using a t test.
(iii) We now need instrumental variables for atndrte and the interaction term,
priGPA atndrte. (Even though priGPA is exogenous, atndrte is not, and so priGPA atndrte is generally correlated with u.) Under the exogeneity assumption that E(u|priGPA,ACT,dist) = 0, any function of priGPA, ACT, and dist is uncorrelated with u. In particular, the interaction priGPA dist is uncorrelated with u. If dist is partially correlated with atndrte then priGPA dist is partially correlated with priGPA atndrte. So, we can estimate the equation
stndfnl = 0 + 1atndrte + 2priGPA + 3ACT + 4priGPA atndrte + u
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by 2SLS using IVs dist, priGPA, ACT, and priGPA dist. It turns out this is not generally optimal. It may be better to add priGPA2 and priGPA ACT to the instrument list. This would give us overidentifying restrictions to test. See Wooldridge (2002, Chapters 5 and 9) for further discussion.
15.3 It is easiest to use (15.10) but where we drop. Remember, this is allowed because
= and similarly when we replace x with y. So the numerator in the formula for is
where n1 = is the number of observations with z i = 1, and we have used the fact that /n1 = , the average of the y i over the i with z i = 1. So far, we have shown that the
numerator in is n1(). Next, write as a weighted average of the averages over the two subgroups:
= (n0/n) + (n1/n),
where n0 = n n1. Therefore,
= [(n n1)/n] (n0/n) = (n0/n) ( - ).
Therefore, the numerator of can be written as
(n0n1/n)().
By simply replacing y with x, the denominator in can be expressed as (n0n1/n)(). When we take the ratio of these, the terms involving n0, n1, and n, cancel, leaving
= ()/().
15.4 (i) The state may set the level of its minimum wage at least partly based on past or expected current economic activity, and this could certainly be part of u t. Then gMIN t and u t are correlated, which causes OLS to be biased and inconsistent.
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