Econometric Analysis of Cross Section and Panel Data - Free

Additional Single-Equation Topics 135
independence across clusters. An example is studying teenage peer e¤ects using a
large sample of neighborhoods (the clusters) with relatively few teenagers per neighborhood.
Or, using siblings in a large sample of families. The asymptotic analysis is
with fixed cluster sizes with the number of clusters getting large. As we will see in
Section 11.5, handling within-cluster correlation in this context is relatively straightforward.
In fact, when the explanatory variables are exogenous, OLS is consistent
and asymptotically normal, but the asymptotic variance matrix needs to be adjusted.
The same holds for 2SLS.
Problems
6.1. a. In Problem 5.4d, test the null hypothesis that educ is exogenous.
b. Test the the single overidentifying restriction in this example.
6.2. In Problem 5.8b, test the null hypothesis that educ and IQ are exogenous in the
equation estimated by 2SLS.
6.3. Consider a model for individual data to test whether nutrition a¤ects productivity
(in a developing country):
logðproducÞ ¼d 0 þ d 1 exper þ d 2 exper 2 þ d 3 educ þ a 1 calories þ a 2 protein þ u 1
ð6:35Þ
where produc is some measure of worker productivity, calories is caloric intake per
day, and protein is a measure of protein intake per day. Assume here that exper,
exper 2 , and educ are all exogenous. The variables calories and protein are possibly
correlated with u 1 (see Strauss and Thomas, 1995, for discussion). Possible instrumental
variables for calories and protein are regional prices of various goods such as
grains, meats, breads, dairy products, and so on.
a. Under what circumstances do prices make good IVs for calories and proteins
What if prices reflect quality of food
b. How many prices are needed to identify equation (6.35)
c. Suppose we have M prices, p 1 ; ...; p M . Explain how to test the null hypothesis
that calories and protein are exogenous in equation (6.35).
6.4. Consider a structural linear model with unobserved variable q:
y ¼ xb þ q þ v;
Eðv j x; qÞ ¼0