Bettinger et al. (2017)

The problem is related to the study by Bettinger et al. (2017) that investigates the effects of taking college courses online versus in-person on student achievement. Let's break down the questions based on the given regression model:

### (a) Describe all the variables on the right-hand side of the model, including those denoted by Greek letters.

The regression model provided is:

\[

y_{ict} = \delta \cdot Online_{ict} + y_{i,\tau < t} \cdot \alpha + X_{it} \cdot \beta + \pi_c + \phi_t + \psi_{b(it)} + \rho_{p(it)} + \epsilon_{ict}

\]

- **\( Online_{ict} \)**: A binary variable indicating whether student \( i \) took course \( c \) online in period \( t \).

- **\( y_{i,\tau < t} \)**: Previous academic performance or course grades of student \( i \) before time \( t \), included to control for past performance.

- **\( X_{it} \)**: A vector of student-level characteristics, such as demographics (age, gender, ethnicity) or academic background.

- **\( \pi_c \)**: Course fixed effects, controlling for differences between various courses \( c \).

- **\( \phi_t \)**: Time fixed effects, controlling for changes over time that could affect all students.

- **\( \psi_{b(it)} \)**: Fixed effects for student's high school (or another baseline variable \( b(it) \)), controlling for unobserved factors related to the high school that the student attended.

- **\( \rho_{p(it)} \)**: Instructor or professor fixed effects, controlling for variations in the quality of instruction between different professors.

- **\( \epsilon_{ict} \)**: The error term, capturing unobserved factors that affect student grades in course \( c \) at time \( t \).

### (b) Which of the right-hand side variables is considered endogenous by Bettinger et al. and why?

The variable **\( Online_{ict} \)** is likely considered endogenous. This is because the decision to take an online course may be influenced by unobserved factors that also affect student performance, such as motivation, convenience, or other personal circumstances. These unobserved factors could lead to biased estimates if not properly addressed.

### (c) Bettinger et al. employ an instrumental variables strategy based on the variables Offered\(_{b(i)ct}\) and Distance\(_{ict}\). Describe these variables and discuss the sources of exogenous variation for each of them.

- **Offered\(_{b(i)ct}\)**: This variable likely represents whether an online version of the course is available to students at their high school \( b(i) \) in period \( t \). This is an instrument because the availability of online courses is likely exogenous to the individual student's characteristics or performance.

- **Distance\(_{ict}\)**: This variable measures the physical distance between the student's location and the nearest location offering the in-person version of the course. This distance introduces exogenous variation because students living farther from the in-person location may be more likely to opt for the online course due to convenience, but this distance is unrelated to their academic abilities or outcomes.

The exogenous variation comes from the fact that both the offering of online courses and the distance to an in-person course are external to the individual student's academic abilities and thus can be used as valid instruments to control for the endogenous nature of the **\( Online_{ict} \)** variable.

### (d) How do Bettinger et al. specify the instrumental variables regression? What are the advantages of this approach?

The instrumental variables (IV) approach is likely specified in two stages:

1. **First stage**: Regress \( Online_{ict} \) on the instruments (Offered\(_{b(i)ct}\) and Distance\(_{ict}\)) and other control variables.

   \[

   Online_{ict} = \pi \cdot Offered_{b(i)ct} + \rho \cdot Distance_{ict} + Z_{ict}

   \]

   where \( Z_{ict} \) represents other covariates.

   

2. **Second stage**: Use the predicted values of \( Online_{ict} \) from the first stage to estimate the impact of taking online courses on student outcomes (course grades, etc.).

The advantage of this IV approach is that it helps address the endogeneity issue by using exogenous sources of variation (availability and distance) to isolate the causal effect of taking an online course, as these instruments are unlikely to be correlated with unobserved factors affecting student performance.

### (e) Summarize the key empirical findings and their policy implications.

While the exact findings are not provided in the image, a typical study like this might find that online courses have either a neutral or negative impact on student performance compared to in-person courses. This could have important policy implications regarding the expansion of online education, suggesting that while it may increase access, careful attention must be paid to its effects on learning outcomes. Policies may need to focus on improving the quality of online courses or supporting students who struggle in these environments.