In table 2, we present regression discontinuity estimates for student-level baseline covariates. As we mentioned in the empirical strategy subsection, we provide effect estimates under various bandwidth choices. As can be expected from the densities of these covariates, discontinuity estimates at the cutoff point are statistically and practically insignificant, regardless of the bandwidth choice. We also tested whether baseline test performance is significantly different, and table 3 presents the results (“Pretreatment” column). All the discontinuity estimates are statistically and practically insignificant. Table 4 presents discontinuity estimates for baseline school characteristics (“Pretreatment” column). As can be inferred from online appendix figure A3 (panel E), estimated discontinuities are significant only for the share of schools providing customized instructional materials. Though the discontinuities are statistically significant for this variable, we argue that this does not invalidate our identifying assumption because the estimated discontinuities are very small (approximately 0.14). While we cannot test for the differences in other school-level covariates due to data limitations, we further argue that school characteristics are less likely to be significantly different between the two groups given that elementary education is mandatory in Korea and that the Ministry of Education makes a significant effort to homogenize school-level characteristics across the schools. All in all, we proceed with the assumption that school funding is as good as random at the cutoff point, given the results presented in figures 3 and 4, online appendix figures A1 to A3, and tables 2 to 4.
Regression Discontinuity Estimates for Baseline Student-level Covariates
| . | Bandwidth (h) . | |||
|---|---|---|---|---|
| Baseline outcomes . | h = 0.009 . | h = 0.012 . | h = 0.015 . | h = 0.018 . |
| Share of female students | 0.007 | 0.004 | 0.007 | 0.008 |
| (0.017) | (0.015) | (0.013) | (0.012) | |
| [25,297] | [36,164] | [50,954] | [69,739] | |
| Share of students preparing schoolwork | −0.027 | −0.025 | −0.026 | −0.022 |
| (0.024) | (0.019) | (0.016) | (0.014) | |
| [25,186] | [35,996] | [50,712] | [69,408] | |
| Share of students reviewing schoolwork | −0.027 | −0.026 | −0.023 | −0.017 |
| (0.027) | (0.021) | (0.018) | (0.016) | |
| [25,190] | [35,985] | [50,726] | [69,423] | |
| Share of students taking online lectures | −0.004 | −0.001 | −0.003 | 0.001 |
| (0.024) | (0.021) | (0.018) | (0.016) | |
| [25,131] | [35,918] | [50,621] | [69,299] | |
| Number of family members | 0.074 | 0.066 | 0.046 | 0.034 |
| (0.051) | (0.042) | (0.036) | (0.032) | |
| [25,311] | [36,181] | [50,973] | [69,765] | |
| . | Bandwidth (h) . | |||
|---|---|---|---|---|
| Baseline outcomes . | h = 0.009 . | h = 0.012 . | h = 0.015 . | h = 0.018 . |
| Share of female students | 0.007 | 0.004 | 0.007 | 0.008 |
| (0.017) | (0.015) | (0.013) | (0.012) | |
| [25,297] | [36,164] | [50,954] | [69,739] | |
| Share of students preparing schoolwork | −0.027 | −0.025 | −0.026 | −0.022 |
| (0.024) | (0.019) | (0.016) | (0.014) | |
| [25,186] | [35,996] | [50,712] | [69,408] | |
| Share of students reviewing schoolwork | −0.027 | −0.026 | −0.023 | −0.017 |
| (0.027) | (0.021) | (0.018) | (0.016) | |
| [25,190] | [35,985] | [50,726] | [69,423] | |
| Share of students taking online lectures | −0.004 | −0.001 | −0.003 | 0.001 |
| (0.024) | (0.021) | (0.018) | (0.016) | |
| [25,131] | [35,918] | [50,621] | [69,299] | |
| Number of family members | 0.074 | 0.066 | 0.046 | 0.034 |
| (0.051) | (0.042) | (0.036) | (0.032) | |
| [25,311] | [36,181] | [50,973] | [69,765] | |
Notes: Standard errors clustered at the school level are in parentheses. The number of observations is in brackets. Regression discontinuity estimates are derived using local linear specification (i.e., ) and the triangular kernel function.
Regression Discontinuity Estimates for Pre- and Post-treatment Achievement Level
| . | Effect Estimates . | |
|---|---|---|
| Outcome variables . | Pretreatment . | Post-treatment . |
| Reading | −0.021 | −0.023 |
| (0.014) | (0.022) | |
| [69,678] | [63,794] | |
| Mathematics | 0.007 | −0.065** |
| (0.013) | (0.026) | |
| [69,683] | [63,779] | |
| English | −0.013 | −0.046* |
| (0.017) | (0.024) | |
| [69,684] | [63,794] | |
| Social studies | −0.003 | −0.053** |
| (0.015) | (0.025) | |
| [69,664] | [63,808] | |
| Science | −0.001 | −0.038** |
| (0.010) | (0.019) | |
| [69,667] | [63,806] | |
| . | Effect Estimates . | |
|---|---|---|
| Outcome variables . | Pretreatment . | Post-treatment . |
| Reading | −0.021 | −0.023 |
| (0.014) | (0.022) | |
| [69,678] | [63,794] | |
| Mathematics | 0.007 | −0.065** |
| (0.013) | (0.026) | |
| [69,683] | [63,779] | |
| English | −0.013 | −0.046* |
| (0.017) | (0.024) | |
| [69,684] | [63,794] | |
| Social studies | −0.003 | −0.053** |
| (0.015) | (0.025) | |
| [69,664] | [63,808] | |
| Science | −0.001 | −0.038** |
| (0.010) | (0.019) | |
| [69,667] | [63,806] | |
Notes: An outcome variable is a dummy variable that indicates whether a student received the “below average” grade. The effect estimate is derived from a regression of this indicator variable on the treatment variable that varies at the school level. The effect estimate indicates the average difference in the proportion of students scoring “below average” between treated and untreated schools. Regression discontinuity estimates are derived under the bandwidth choice of 0.018. Standard errors clustered at the school level are in parentheses. The number of observations is in brackets. Regression discontinuity estimates are derived using local linear specification (i.e., ) and the triangular kernel function. ** and * indicate statistical significance at the 5% and 10% levels, respectively.
Regression Discontinuity Estimates for Baseline and Post-treatment School Characteristics
| . | Effect Estimates . | |
|---|---|---|
| . | Pretreatment . | Post-treatment . |
| Share of teachers with master's degree or higher | −0.028 | 0.043 |
| (0.030) | (0.031) | |
| [679] | [674] | |
| Share of newly hired teachers | −0.006 | 0.014 |
| (0.026) | (0.025) | |
| [676] | [672] | |
| Share of schools operating after school | −0.040 | 0.047 |
| (0.060) | (0.040) | |
| [680] | [675] | |
| Share of schools operating summer school | −0.040 | 0.114** |
| (0.054) | (0.054) | |
| [680] | [675] | |
| Share of schools utilizing outside resources | −0.104 | 0.085 |
| (0.086) | (0.082) | |
| [680] | [675] | |
| Share of schools providing customized materials | −0.143* | 0.048 |
| (0.081) | (0.066) | |
| [680] | [675] | |
| . | Effect Estimates . | |
|---|---|---|
| . | Pretreatment . | Post-treatment . |
| Share of teachers with master's degree or higher | −0.028 | 0.043 |
| (0.030) | (0.031) | |
| [679] | [674] | |
| Share of newly hired teachers | −0.006 | 0.014 |
| (0.026) | (0.025) | |
| [676] | [672] | |
| Share of schools operating after school | −0.040 | 0.047 |
| (0.060) | (0.040) | |
| [680] | [675] | |
| Share of schools operating summer school | −0.040 | 0.114** |
| (0.054) | (0.054) | |
| [680] | [675] | |
| Share of schools utilizing outside resources | −0.104 | 0.085 |
| (0.086) | (0.082) | |
| [680] | [675] | |
| Share of schools providing customized materials | −0.143* | 0.048 |
| (0.081) | (0.066) | |
| [680] | [675] | |
Notes: Regression discontinuity estimates under the bandwidth choice of 0.018. Standard errors are in parentheses. The number of observations is in brackets. Regression discontinuity estimates are derived using local linear specification (i.e., ) and the triangular kernel function. ** and * indicate statistical significance at the 5% and 10% levels, respectively.