Question

In a recent article, Evans and Schwab (1995) 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 Cath\(H S\) be a binary variable equal to one if the student attends a Catholic high school. A linear probability model is $$college =\beta_{0}+\beta_{1} Cath \quad H S+ other\quad factors +u$$ where the other factors include gender, race, family income, and parental education. i. Why might CathHS be correlated with \(u\) ? ii. Evans and Schwab 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?

Short answer

CathRel may serve as an IV for CathHS if it meets relevance and exogeneity conditions. The relevance can be tested; exogeneity cannot.

Step-by-step solution

Understanding the Nature of Correlation

In econometrics, the error term (\(u\)) represents unobservable factors affecting the dependent variable, which in this case is the probability of attending college. The variable \(CathHS\) could be correlated with \(u\) because students attending Catholic high schools might also have unobservable characteristics like higher family motivation towards education or religious influence, which also impact the probability of attending college.

Utilizing Standardized Test Scores

The standardized test score can be added as a control variable in the model to account for a student's academic ability, which might otherwise bias the estimate of attending a Catholic high school. By including this test score, the estimate of the effect of \(CathHS\) on attending college becomes more reliable, assuming all else is equal (ceteris paribus).

Requirements for a Valid Instrumental Variable

For \(CathRel\) to be a valid instrumental variable (IV) for \(CathHS\), two main conditions must be met. First, relevance: \(CathRel\) must be correlated with \(CathHS\), which we know is already met if being Catholic increases the likelihood of attending a Catholic high school. Second, exogeneity: \(CathRel\) should not be correlated with the error term \(u\), meaning it should not directly affect the probability of attending college outside its correlation with \(CathHS\). The relevance condition can be tested statistically, typically using a test for instrument strength.

Evaluating CathRel as an Instrument

As being Catholic significantly increases the likelihood of attending a Catholic high school and assuming it does not affect the probability of attending college directly through other pathways, \(CathRel\) could be a convincing instrument. However, if being Catholic affects unobservable factors captured by \(u\), like a preference for education, it might not be a valid IV. It's essential to analyze whether \(CathRel\) fulfills the exogeneity condition rigorously to prevent endogeneity bias.

Key concepts

Linear Probability Model

The Linear Probability Model (LPM) is a straightforward approach to estimating the probability of a binary outcome. In this method, we use a linear regression framework to predict the probability of an event occurring, which in this context is attending college.
While the model's setup is simple, it involves using continuous predictors to model a binary outcome. Here's how it works:

• The dependent variable is binary (0 or 1) — 1 if the student attends college, 0 otherwise.
• Independent variables, like attending a Catholic high school ( CathHS) and other factors, such as gender, race, and family income, influence the probability of the dependent variable.

One key concern with LPM is that its predictions can fall outside the [0,1] range, which does not realistically represent probabilities. However, it remains popular because it's easy to implement and interpret. It gives a sense of which factors positively or negatively affect the likelihood of an outcome.

Instrumental Variables

Instrumental Variables (IV) technique is crucial in econometric analysis when the independent variable is suspected to be correlated with the error term ( u). This correlation can lead to biased estimates, obstructing a clear understanding of the causal relationship between variables.
Here's how IV works:

• We introduce another variable, called an instrument, to capture the variation in the independent variable that is not correlated with the error term.
• In our context, CathRel is proposed as an IV for CathHS, requiring two key conditions: relevance (correlation with CathHS) and exogeneity (no correlation with the error term u).

Testing for relevance involves checking if being Catholic increases the likelihood of attending a Catholic high school, while determining exogeneity requires careful analysis to ensure no direct effect on college attendance beyond its impact on CathHS.

Ceteris Paribus

Ceteris Paribus is a Latin phrase meaning "other things being equal." In econometrics, this concept ensures that the effect of a change in one variable is measured while keeping all other variables constant. It's akin to isolating the impact of one factor amidst the complexity of real-world influences.
Using ceteris paribus helps:

• Provide a clearer understanding of the economic relationship being studied by isolating particular effects.
• Allow researchers to make more accurate predictions or assertions about causality.

In our exercise, controlling for a student's standard test score ensures that the estimate of attending a Catholic high school reflects its unique effect on college attendance, keeping other variables unchanged.

Econometric Analysis

Econometric Analysis is a scientific approach to understanding economic relationships using statistical techniques. It involves formulating models, estimating coefficients, and interpreting results to gain insights into economic phenomena.
Key components include:

• Developing a theoretical model that reflects the real-world relationship we want to study.
• Specifying an econometric model aligning with data structures, like the LPM in our case.
• Using statistical methods, such as regression analysis, to estimate the model parameters.

Econometric analysis empowers researchers to scrutinize hypotheses with empirical data, improving decisions and policy formations by transforming complex data into actionable insights.