Question

Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university: $$G P A=\beta_{0}+\beta_{1} P C+u$$ where \(P C\) is a binary variable indicating PC ownership. i. Why might PC ownership be correlated with \(u\) ? ii. Explain why \(P C\) is likely to be related to parents' annual income. Does this mean parental income is a good IV for \(P C\) ? Why or why not? iii. Suppose that, four years ago, the university gave grants to buy computers to roughly one-half of the incoming students, and the students who received grants were randomly chosen. Carefully explain how you would use this information to construct an instrumental variable for \(P C\).

Short answer

PC ownership correlates with unobserved factors affecting GPA. Parental income is related to PC ownership but may affect GPA directly, making it unsuitable as an IV. University grants can be a valid IV due to random assignment.

Step-by-step solution

Understanding Correlation with PC Ownership

PC ownership may be correlated with the error term \( u \) because there could be unobserved factors influencing both GPA and the likelihood of owning a PC. For example, students with higher motivation or better study habits might be more likely to own a PC, both contributing to higher GPA, inducing a correlation.

Assessing Relationship with Parental Income

PC ownership is likely related to parents' annual income because higher income families are more likely to afford PCs for their children. Although parental income is related to PC ownership, it does not necessarily serve as a good instrumental variable (IV) for PC because it may directly affect the GPA, violating the IV exogeneity requirement.

Constructing an Instrumental Variable

Given that the university randomly assigned computer grants, this random assignment can serve as an instrumental variable. The random nature of the grants means they are uncorrelated with any unobserved predictors of GPA \( u \). You can use grant receipt as an instrument for PC ownership, as it affects PC ownership but should not directly affect GPA.

Key concepts

PC Ownership

PC ownership might seem like a straightforward concept, but its implications can be quite telling. Imagine a student at a university. Whether or not they own a personal computer (PC) can significantly affect their college experience. From writing essays to conducting research and even watching online lectures, a PC is a crucial tool. However, PC ownership isn't just about access; it's often influenced by numerous factors.

For instance, students who own a PC might be more responsible or have better organizational skills because they need these traits to maintain and effectively use their technology. Ownership may also suggest a student's commitment to their studies.

In given studies, PC ownership is typically considered a binary variable meaning it is captured as 'yes' (1) or 'no' (0). It's important to consider how PC ownership might relate to other variables, like college GPA or even family background.

College GPA

College GPA is a measure of a student's academic performance. It is crucially important as it often reflects how well a student has absorbed their course material.

Several factors influence a student's GPA. These could range from study habits to availability of resources, like access to a PC, as mentioned above. Notably, having a higher GPA can open doors to more scholarships, better job prospects, and further educational opportunities.

When analyzing factors like PC ownership, it's essential to be aware that a student's GPA might also be influenced by those unseen motivators. Therefore, isolating factors like PC ownership when evaluating their true impact on GPA becomes difficult without proper statistical tools.

Parental Income

Parental income is often correlated with a student's access to resources, including PCs. Families with higher incomes are more likely to afford computers, influencing PC ownership.

However, when considering parental income as an instrumental variable (IV) for determining PC ownership, some challenges arise. While it relates to whether a student owns a PC, parental income can directly impact a student's opportunities and thus their GPA. Therefore, using parental income as an IV might violate the exogeneity requirement, which demands that the IV does not directly affect the dependent variable, in this case, the GPA.

That's why, despite the relationship between parental income and PC ownership, it might not be the best choice for an instrumental variable.

Random Assignment

Random assignment is a powerful tool to establish causality in studies. Consider when a university randomly gives out grants to buy computers for half of its students. In this scenario, the random grants serve as a natural experiment.

Since the assignment is random, it helps ensure that the decision of who receives a grant is not influenced by any unobserved characteristics. These characteristics are those omitted factors that typically mess with our ability to understand true causal effects—like that of PC ownership on GPA.

In this context, the grant receipt becomes an ideal instrumental variable. It primarily affects whether or not a student owns a PC. However, since the grants were assigned randomly, they do not directly affect the student's GPA, maintaining the essential condition of exogeneity for a valid instrumental variable. Thus, using such random assignment can significantly strengthen analysis and provide clearer insights into PC ownership's effect on GPA.