a.)

Write an equation for the CEF that is consistent with the ATEs presented in Table V, Column (1) in Krueger (1999)

\[ \mathbb{E}\!\left[Y_{ig} \mid D^{S}_{ig}, D^{RA}_{ig}\right] = \alpha_g + \beta_g D^{S}_{ig} + \delta_g D^{RA}_{ig} \]

Part B: Empirical Analysis

Using the Project Star data provided in the AER package, estimate the effect of class size on student achievement. Substantively, you will be replicating the Column (1) results in Table 5 of Krueger (1999), except with testing data reported in raw scores, not percentiles. Thus, the scale of your estimated effects will be different.

  1. Estimate the class-size effects for each grade, K through 3, using OLS and report your results in a table using modelsummary analogously to Column (1) in the OLS panels of Table V in Krueger (1999).

  2. Write a sentence that describes the statistical significance of small-class effect for kindergartners. Write a sentence explaining whether adding an aide to a regular class has a statistically significant effect on kindergartner test scores.

  3. Repeat the sentences in (b) for the first, second and third-grade results.

  4. Use the scale function to create standardized (\(z\)-score) version of the test-score variables. Repeat the exercise in (a).

  5. Write a sentence that interprets the results for kindergartners. Write a sentence that compares the results for first, second and third-graders with those for kindergartners.

Results and answers

a.)

OLS Estimates of Class Size Effects by Grade (Raw Scores)
K 1st 2nd 3rd
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
Standard errors in parentheses.
small 13.899*** 29.781*** 19.394*** 15.587***
(2.409) (2.807) (2.710) (2.395)
regular+aide 0.314 11.959*** 3.479 -0.291
(2.310) (2.686) (2.566) (2.302)
Constant 918.043*** 1039.393*** 1157.807*** 1228.506***
(1.641) (1.836) (1.849) (1.715)
$N$ 5786 6379 6049 5967
$R^2$ 0.01 0.02 0.01 0.01

b.)

The small-class effect for kindergartners is positive and highly statistically significant at the 1% level. By contrast, adding an aide to a regular class does not have a statistically significant effect on kindergartner test scores.

c.)

For first graders, small classes raise test scores by about 29.8 points and are highly significant, while adding an aide increases scores by roughly 12 points and is also significant. In second grade, small classes add about 19.4 points (significant), but the aide effect of 3.5 points is not significant. By third grade, small classes add about 15.6 points (significant), while the aide effect is essentially zero and not significant.

d.)

Estimated ATEs for Small Class Sizes (Standardized Test Scores, SD Units)
K 1st 2nd 3rd
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
Standard errors in parentheses. Outcomes are standardized (mean 0, SD 1).
small 0.188*** 0.326*** 0.231*** 0.213***
(0.033) (0.031) (0.032) (0.033)
regular+aide 0.004 0.131*** 0.041 -0.004
(0.031) (0.029) (0.031) (0.031)
Constant -0.058** -0.137*** -0.083*** -0.066**
(0.022) (0.020) (0.022) (0.023)
$N$ 5786 6379 6049 5967
$R^2$ 0.01 0.02 0.01 0.01

e.)

For kindergartners, being in a small class raises test scores by about 0.19 standard deviations, a meaningful and statistically significant improvement, while adding an aide has no effect. Compared with kindergartners, the small-class benefits are larger for first graders (0.33 SD) and remain strong but slightly smaller for second and third graders (about 0.23 SD each).