Question

Use GPA3 for this exercise. The data set is for 366 student-athletes from a large university for fall and spring semesters. [A similar analysis is in Maloney and McCormick $(1993),$ but here we use a true panel data set. Because you have two terms of data for each student, an unobserved effects model is appropriate. The primary question of interest is this: Do athletes perform more poorly in school during the semester their sport is in season? (i) Use pooled OLS to estimate a model with term GA (trmgpa) as the dependent variable. The explanatory variables are spring, sat, hsperc, female, black, white, frstsem, tothrs, crsgpa, and season. Interpret the coefficient on season. Is it statistically significant? (ii) Most of the athletes who play their sport only in the fall are football players. Suppose the ability levels of football players differ systematically from those of other athletes. If ability is not adequately captured by SAT score and high school percentile, explain why the pooled OL.S estimators will be biased. (iii) Now, use the data differenced across the two terms. Which variables drop out? Now, test for an in-season effect. (iv) Can you think of one or more potentially important, time-varying variables that have been omitted from the analysis?

Short answer

Athletes may perform worse during their sport season, but pooled OLS may be biased due to omitted variables like athletic ability. First differences remove some biases, showing 'season' effects.

Step-by-step solution

Run Pooled OLS Regression

In this step, perform a pooled ordinary least squares (OLS) regression using the term GPA (trmgpa) as the dependent variable. Include the following explanatory variables: spring, sat, hsperc, female, black, white, frstsem, tothrs, crsgpa, and season. The coefficient on the variable 'season' will indicate whether the term GPA is affected during the season of the sport. Interpret this coefficient to see if athletes perform differently when their sport is in season. Analyze its statistical significance using the p-value. A significant p-value (typically less than 0.05) would mean that the season had a significant effect on GPA.

Discuss Potential Bias in Pooled OLS

If the athletic ability affects term GPA but is not fully captured by SAT scores and high school percentile, the pooled OLS estimator may be biased. This is because the omitted variable (athletic ability) may correlate with the included variables (like SAT score or hsperc) and could be influencing GPA. This bias occurs particularly if the omitted ability level is systematic across a group, such as football players.

Perform First Differences Analysis

In order to eliminate unobserved, time-invariant factors (like ability), take the first difference across the two terms for each variable. In this differenced data, variables like 'hsperc' and 'female' will drop out because they do not change over time. Re-run the regression on the differenced data to isolate the effect of 'season'. In this setup, check if 'season' still significantly affects GPA, indicating an in-season effect.

Identify Omitted Time-Varying Variables

Consider variables that could change between terms and affect a student's GPA, but were not included in the regression. These might include variables such as 'number of study hours per week', 'participation in study groups', or 'changes in health or family circumstances'. These omissions could lead to omitted variable bias if they correlate with the variables already in the model.

Key concepts

Pooled OLS Regression

Pooled Ordinary Least Squares (OLS) regression is a method used when you have panel data, which means repeated observations of the same entities, in this case, students over two terms. This technique involves stacking all observations into one data set and ignoring any potential differences between the entities over time. In the context of studying student-athletes' GPA, the model includes variables such as `spring`, `sat`, `hsperc`, `female`, `black`, `white`, `frstsem`, `tothrs`, `crsgpa`, and `season`.
The "season" variable is particularly interesting, as it measures whether being in-season affects athletes' GPA. You determine the effect by examining the coefficient of "season" in the regression. If the coefficient is positive or negative, it suggests that being in-season helps or hinders academic performance, respectively. Evaluating whether this coefficient is statistically significant involves checking the p-value. A p-value under 0.05 would tell you the season's impact is statistically significant and not likely due to chance.
This method provides insights, but it may overlook important details about differences between student-athletes that don't change over the terms.

Unobserved Effects Model

An unobserved effects model is more sophisticated than a simple pooled OLS, as it accounts for unseen factors that do not change over the time. These might include intrinsic attributes such as a student's motivation or personal ability, which remain constant over both observed terms. The traditional pooled OLS ignores these factors, possibly leading to inaccurate estimates of the effect of season on GPA.
By acknowledging these unobserved effects, you adjust your analysis to better isolate the impact of the variables of interest, such as the term in which an athlete's sport is in season. The model uses individual-specific effects to represent unobservable differences between students. These effects allow you to control for them and make the estimated relationship between GPA and seasonal factors more precise.
This approach generally provides a more robust understanding of the factors influencing GPA, beyond those that have been directly measured and included as variables in your model.

First Differences Analysis

First differences analysis is a technique used to address the issue of unobserved, time-invariant factors. This method involves computing the difference of each variable's value between two time periods for each entity in the panel data. When you perform first differences, variables that do not change, like `hsperc` and `female`, naturally disappear because their change over time is zero.
Applying this analysis to the student-athlete GPA data lets you focus on how changes in variables like `season` across terms impact GPA. Importantly, it removes the bias from unobserved characteristics that are consistent over time, such as a student's innate ability or motivation—those influences that could not be directly measured.
By examining whether the differenced data still indicates a significant effect of being in-season, you can confirm if the time-specific components, such as seasonality, are crucial in affecting the athletes' academic performance, independent of unchanging personal qualities.

Omitted Variable Bias

Omitted variable bias occurs in regression analysis when a relevant variable is left out of the model, causing the estimated effects of the included variables to be distorted. In the context of analyzing student-athletes' performance, suppose a variable like `athletic ability` is not included, and it correlates with variables like `SAT` scores. If this happens, the effect of being in-season could be confounded with the influence of missing variables, leading to inaccurate conclusions.
Moreover, the omission of important time-varying variables, such as changes in study habits or participation in study groups, can also affect the findings. Since these factors might vary between semesters, their absence could skew the results, suggesting an impact from being in-season that might actually arise from omitted influences.
To minimize omitted variable bias, it is vital to critically assess which factors should be included based on their potential to affect the outcome and the explanatory variables to ensure the most accurate model possible.