Question

We are interested in studying the effects of attending a private high school on the probability of attending college. For concreteness, let college be a binary variable equal to one if a student attends college, and zero otherwise. Let PrivateHS be a binary variable equal to one if the student attends a private high school. A linear probability model is:

college=B₀+B₁PrivateHS+other factors+u

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

(v) Propose an alternative instrument for PrivateHS and discuss whether the two requirements needed are valid.

Answer 1

(v) The regression model is given as College=0+1PrivateHS+other factors+u where the dependent variable or College represents the probability of attending college of any student chosen for the study or research and PrivateHS denotes the student attendance in private school which is the independent variable in the given model and u represents the error term in the regression model. The other factors constituting the other independent variables in the model include race, family income, and parental education of the students in the study. Now, in this case, an appropriate and statistically valid instrumental variable or IV for PrivateHS could be overall family size and structure of the students chosen for the research study as the overall family size can practically influence both the probability or decision of private schooling of the students as well as usually it can be evidently inferred or assumed that larger the overall family size and composition, the lower the chance of attending private school as well as college for the students due to various practical reasons such as financial constraints( even for families with substantially high overall income levels), diversified parental attitude and inclination towards the children in the household, and other socio-economic determinants or factors. Hence, all these practical factors or attributes can also affect the decision to attend college and pursue higher education for any student as well. Therefore, it could essentially imply a strong and convincing correlation between the independent variable PivateHS and the overall family size and composition of the sample students chosen for the concerned research study thereby signifying the statistical validity of the overall family size and composition as a possible IV. Furthermore, the proposed IV or the overall family size would also be uncorrelated with the error term or u in the regression model given in the question.