A long standing question of interest to economists who study education is how outcomes vary for students who attend private schools as compared to public schools. Some early work on this question focused on test scores and compared students in Catholic high schools to students in public high schools, finding students at Catholic schools had significantly higher test scores. In a journal article, Even and Schwab (1995) studied the effects of attending a Catholic high school on the probability of attending college.
Let collegei be a binary variable equal to 1 if a student goes on to college and 0 if not; Let CathHSi be 1 if they attended a Catholic High School and 0 if not(assume for the sake of this question there are only Catholic and public High School)
Let the following LPM describe the relationship of interest.
collegei = beta0 + beta1 * CathHSi + beta2*W1i + beta3*W2i +....+ beta r+1* W ri+ ui
note that W1,W2,....WR are the included exogenous variables.
a. why might CathHSi be correlated with ui?
b. The authors of this article have data on a standardized test taken when each student was a sophomore. how might they use this data to improve their estimates of the effects of attending a Catholic school?
c. let CathReli be equal to1 if the student is Catholic and 0 otherwise. Discuss the validity of the variable as possible intrument for attending Catholic School.
d. being Catholic has a significant effect on probability of attending a Catholic high and the t for th null that the coefficient on being Cathloic is 0 against a two sided null is 5. Do I have a weak instrument?
e. Suppose another instrument is the availability of privately funded scholarships to attend Catholic school, Describe how I can test whether both instruments are exogenous, given I assume at least one is.
f. Do you think CathRel is a convincing instrument for CathHS? What about availability of scholarships? why or why not?