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Columns 1–6 in Table 4 present the estimated results for the dependent variable average Prova Brasil test scores in Portuguese for 5th grade students in basic education. Each column presents the results of an estimator (LS, FE and RE) with the matching and additional covariates. All the ATT measurements are statistically significant (at least at 5%) and oscillate between 5.47 and 7.49 points in the SAEB scale (values close to those presented in Table 2 using the simple difference in difference model). Proceeding with the presentation of Table 4, columns 7–12, present the estimated results for the dependent variable average test scores in mathematics. All the ATT measurements are statistically significant (at least at 5%) and oscillate between 3.94 and 5.60 points in the SAEB scale (also close to the values presented in Table 2). Furthermore, in terms of the estimates parameters for the covariates, one can also observe a positive and significant influence of the socioeconomic variables of the municipality and those that reflect the quality of school infrastructure. As for the other columns in Table 4, these present the estimated results for the dependent variables school passing rates and TMI. For these variables, as was the case in the differences in differences model presented in Table 2, there are no statistically significant ATT measurements. In short, the results for all estimators point to a significant and positive effect of the law change only on the students’ average proficiency. However, for each of those dependent variables, Table 4 presents six ATT estimations. Thus, aiming to define if one is more adequate than the other, the following test statistics are also presented for each triplet of estimators: (i) one test is the F′, related to the null eicosapentaenoic acid that the LS and FE results are the same (being the former a restricted case of the latter); and (ii) another test is the Hausman test, which has as its null hypothesis the fact that the FE and RE results do not significantly differ (so the RE model generates consistent and efficient estimates). As for the estimated results for the dependent variable average test scores in Portuguese, presented in columns 1–3, the F′=3.87 does not corroborate with the acceptance of the null hypothesis that the LS and FE estimates are equal. This indicates that the unrestricted model adjusts more adequately to the data – following the logic of a standard F test. Additionally, the Hausman test statistic points that the RE results are better adjusted to the data. Following the same logic for columns 4–12, both for Portuguese and mathematics test scores, for each triplet of estimators, there is indication that the RE results are better adjusted to the data. Therefore, what remains is testing which of the random effects estimations is more adequate. Note that the model in column 3 is a restricted form for the one in column 6 (for the case of the Portuguese test scores), and that the model in column 9 is a restricted form for the one in column 12 (for the case of the mathematics test scores). As such, another pair of F tests can be constructed using the sum of squared residuals – also presented in Table 4. Following that procedure, there is indication that both unrestricted estimators (column 6 and 12) are better adjusted to the data. In other words, there is indication that the best estimations for the ATT are: 6.09 for the Portuguese test scores; and 4.09 for the mathematics test scores. Regarding the first question, in relation to the distribution rule given by Eq. (1), it makes sense that competition between mayors for the quota-share is more present in the improvement of “5th grade test scores”, i.e., in the improvement of the quality of basic education, for the following reasons. The mayor can manipulate the “4th to 5th grade school passing rate” indicator, advising the schools not to retain students in a failing grade. However, this would lead to potentially unprepared 5th grade students, which would not have good test scores and would eventually penalize the quota. So manipulating passing rates cannot be a sustainable strategy.