This exercise uses data from the STAR experiment introduced to illustrate fixed and random effects for grouped data. In the STAR experiment, children were randomly assigned within schools into three types of classes: small classes with 13-17 students, regular-sized classes with 22-25 students, and regular-sized classes with a full-time teacher aide to assist the teacher. Student scores on achievement tests were recorded as well as some information about the students, teachers, and schools. Data for the kindergarten classes are contained in the data file star.

a. Estimate a regression equation (with no fixed or random effects) where READSCORE is related to SMALL, AIDE, TCHEXPER, BOY, WHITE_ASIAN, and FREELUNCH. Discuss the results. Do students perform better in reading when they are in small classes? Does a teacher's aide improve scores? Do the students of more experienced teachers score higher on reading tests? Does the student's sex or race make a difference?

b. Re-estimate the model in part (a) with school fixed effects. Compare the results with those in part (a). Have any of your conclusions changed? [Hint: specify SCHID as the cross-section identifier and ID as the "time" identifier.]

c. Test for the significance of the school fixed effects. Under what conditions would we expect the inclusion of significant fixed effects to have little influence on the coefficient estimates of the remaining variables?

d. Re-estimate the model in part (a) with school random effects. Compare the results with those from parts (a) and (b). Are there any variables in the equation that might be correlated with the school effects? Use the LM test for the presence of random effects.

e. Using the \(t\)-test statistic in equation (15.36) and a 5\% significance level, test whether there are any significant differences between the fixed effects and random effects estimates of the coefficients on SMALL, AIDE, TCHEXPER, WHITE_ASIAN, and FREELUNCH. What are the implications of the test outcomes? What happens if we apply the test to the fixed and random effects estimates of the coefficient on \(B O Y\) ?

f. Create school-averages of the variables and carry out the Mundlak test for correlation between them and the unobserved heterogeneity.