Question

The following is a simple model to measure the effect of a school choice program on standardized test performance [see Rouse (1998) for motivation and Computer Exercise \(\mathrm{C} 11\) for an analysis of a subset of Rouse's data]: $$s c o r e=\beta_{0}+\beta_{1} \text { choice }+\beta_{2} \text { faminc }+u_{1}$$ where score is the score on a statewide test, choice is a binary variable indicating whether a student attended a choice school in the last year, and faminc is family income. The IV for choice is grant, the dollar amount granted to students to use for tuition at choice schools. The grant amount differed by family income level, which is why we control for faminc in the equation. i. Even with faminc in the equation, why might choice be correlated with \(u_{1} ?\) ii. If within each income class, the grant amounts were assigned randomly, is grant uncorrelated with \(u_{1} ?\) iii. Write the reduced form equation for choice. What is needed for grant to be partially correlated with choice? iv. Write the reduced form equation for \(s c o r e .\) Explain why this is useful. (Hint: How do you interpret the coefficient on grant?)

Short answer

Choice might correlate with errors due to unobserved factors. Random grant assignments ensure it's uncorrelated with errors. Partial correlation with choice requires a non-zero coefficient on grant. The reduced form for score shows grant's impact.

Step-by-step solution

Understanding Correlation with Error Term

Choice might be correlated with the error term \(u_1\) because there might be unobserved factors influencing both the choice to attend a choice school and the student's test score. For example, parental involvement or motivation can simultaneously affect the decision to go to a choice school and the student's performance, causing a correlation between choice and \(u_1\) even after controlling for family income.

Assessing Random Allocation of Grants

If grant amounts within each income class were assigned randomly, then grant should be uncorrelated with the error term \(u_1\). Random assignment ensures that any unobserved factors affecting the outcomes do not systematically vary with the grant amounts, eliminating potential endogeneity concerns.

Writing Reduced Form for Choice

The reduced form equation for choice would be: \[ \text{choice} = \pi_0 + \pi_1 \text{grant} + \pi_2 \text{faminc} + v_1 \] For the grant to be partially correlated with choice, the coefficient \(\pi_1\) should be different from zero, indicating that the grant has a significant effect on the decision to attend a choice school.

Writing Reduced Form for Score

The reduced form equation for score is: \[ \text{score} = \gamma_0 + \gamma_1 \text{grant} + \gamma_2 \text{faminc} + v_2 \] This equation is useful because it shows the direct effect of the grant on the score. The coefficient \(\gamma_1\) provides an estimate of how changes in the grant amount, holding family income constant, impact test scores. If \(\gamma_1\) is significant, it suggests a causal relationship between grant amounts and the test score through the instrumenting process.

Key concepts

School Choice Programs

School choice programs have become increasingly popular in educational policy discussions. These programs give families the option to select schools outside of their designated public school zones. This choice can impact various student outcomes, including standardized test performance.
One main reason families opt for choice schools is the belief that these schools might offer better education or specialized programs that match their child's needs. However, measuring the actual effectiveness of these programs can be complex. There's a chance that students who choose these schools already have advantages such as higher parent involvement or motivation, which can influence both the decision to attend the school and a student's testing outcomes.
Understanding the impacts of school choice programs involves analyzing data to see how these factors play out in real-world settings. Education policy analysts often use statistical models to untangle the effects of attending a choice school from other influencing factors. By doing so, they aim to provide evidence on whether school choice programs ultimately benefit student academic performance or reinforce existing inequalities.

Standardized Test Performance

Standardized tests are a common tool used to measure student learning and school effectiveness. These tests provide a way to compare educational outcomes across different schools and districts. The scores from these tests can be influenced by various factors, including school choice.
When considering how school choice programs affect standardized test performance, it's essential to control for variables like family income. Family background can have a significant influence on educational outcomes, and failing to account for these factors can lead to misleading results.
In this context, educational researchers might set up a model where test scores are a function of attending a choice school (the choice variable), family income (faminc), and other unobserved factors (u1). This model helps ascertain whether attending a choice school genuinely impacts student performance, or if differences in scores are merely a result of underlying student advantages that weren't accounted for.

Instrumental Variables

Instrumental variables (IVs) are a crucial statistical technique used to address potential biases in regression analysis. In the case of school choice programs, an instrumental variable can help isolate the effect of attending a choice school on standardized test scores.
The IV must be correlated with the endogenous explanatory variable—"choice" in this scenario—but uncorrelated with the error term in the equation that predicts test score. The grant provided to students as tuition assistance is used as an IV here. Because the grants varied by family income, this variable can serve as a tool to distinguish the effect of attendance at a choice school from other factors.
A good instrumental variable helps to ensure that the relationship captured in the analysis is more causally robust, mitigating the risk of spurious correlations.

Reduced Form Equations

Reduced form equations in econometrics simplify complex relationships by depicting how an independent variable directly affects a dependent variable without considering intermediary steps or endogenous feedback.
For example, the reduced form equation for choice is set up as a linear equation that relates the choice of school to the variables of grants and family income. By analyzing this equation, researchers determine whether the grant influences the likelihood of attending a choice school by assessing if the coefficient is non-zero.
Understanding the reduced form equation for test scores is equally insightful. It shows the overall impact of grants on student performance, providing a clearer picture of the grant's effectiveness. Through coefficients like \(\gamma_1\) in the reduced form for score, it becomes evident how grant amounts influence scores directly, holding other factors constant.
These equations are vital for unbiased inferences in policy analysis, allowing researchers to evaluate interventions like grants without confounding effects distorting the results.