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ECON4004 - Econometrics 2

Tutorial 3

Question 1 (Exercise 15.3, Wooldridge). Consider the simple regression model         ⃞  ⃞

and let z be a binary instrumental variable (IV) for x. We know that in the population and using an IV we have ⃞         , ሻ/,  . Hence, the estimated coefficient ⃞  in a sample of size n is equal to

     ⃞ୀ⃞(⃞)       ̅

Use (1) to show that the IV estimator ⃞  in (1) can be written as

⃞      ,

ሺ1ሻ

ሺ2ሻ

where ⃞  and ⃞  are the sample averages of ⃞  and ⃞  over the part of the sample with ⃞         0, and where  and ⃞  are the sample averages of ⃞  and ⃞  over the part of the sample with ⃞   1. This estimator, known as a grouping estimator, was first suggested by Wald (1940).


Question 2 (Exercise 15.8, Wooldridge) Suppose you want to test whether girls who attend a girls’ high school do better in math than girls who attend coed schools. You have a random sample of senior high school girls from a state in the United States, and score is the score on a standardized math test. Let girlhs be a dummy variable indicating whether a student attends a girls’ high school.

(i) What other factors would you control for in the equation? (You should be able to reasonably collect data on these factors.)

(ii) Write an equation relating score to girlhs and the other factors you listed in part (i).

(iii) Suppose that parental support and motivation are unmeasured factors in the error term in part (ii). Are these likely to be correlated with girlhs? Explain.

(iv) Discuss the assumptions needed for the number of girls’ high schools within a 20-mile radius of a girl’s home to be a valid IV for girlhs.

(v) Suppose that, when you estimate the reduced form for girlhs, you find that the coefficient on  numghs  (the number  of girls’ high  schools within  a  20-mile radius)  is negative  and statistically significant. Would you feel comfortable proceeding with IV estimation where numghs is used as an IV for girlhs? Explain.


Question 3 (Exercise 15. 10, Wooldridge) In an article, Evans and Schwab (1995, EV) studied the effects of attending a Catholic high school on the probability of attending college. For concreteness, let college be a binary variable equal to unity if a student attends college, and zero otherwise. Let CathHS be a binary variable equal to one if the student attends a Catholic high school. On would like to estimate the equation

 ൌ   ⃞  ⃞   ⃞ ,

where the other factors include gender, race, family income, and parental education.

(i) Why might CathHS be correlated with u?

(ii) EV have data on a standardized test score taken when each student was a sophomore. What can be done with this variable to improve the ceteris paribus estimate of attending a Catholic high school?

(iii) Let CathRel be a binary variable equal to one if the student is Catholic. Discuss the two requirements needed for this to be a valid IV for CathHS in the preceding equation. Which of these can be tested?

(iv) Not surprisingly, being Catholic has a significant positive effect on attending a Catholic high school. Do you think CathRel is a convincing instrument for CathHS?

Reference: W. N. Evans and R. M. Schwab (1995), “Finishing High School and Starting College: Do Catholic Schools Make a Difference?” The Quarterly Journal of Economics, Vol. 110, No. 4 (Nov. 1995), pp. 941-974.