is that these schools that were chosen to become Single-Shift may have intrinsic characteristics that could affect their results. If this is true, the results may reflect the effect of these intrinsic characteristics, rather than the impact of the Single-Shift policy. The third control group is formed by schools with a low proportion of students with more than two years of school delay. We showed, in Table 5, that there was a change in the student population in the treated schools, decreasing the age-grade distortion. If schools with a lower proportion of students with school-delay have on average higher increases, the increase in results of treated schools may be upwardly biased when compared to schools with a higher proportion of students with school delay. The main model is presented in equation (1) below:

- (1) Outcometi = β0 + β1 Schooli + β2 Yeart + β3 Single-Shift*Postit +

- β4 Schools of Tomorrowti + β5 More Educationti +

- β6 Mean # of items in the houseti + β7 % Mothers w/ higher edti +

- β8 Age-grade distortionti + β9 Participation rate in standardized testti + ε

Where _i_ refers to school and _t_ refers to year, from 2005 through 2015 (except for the analysis of schools that eventually became Single-Shift, when _t_ goes from 2005 through 2013). We use two outcomes in our models: the IDEB and the IDERio results. The more common DID model includes dummy variables for treated versus control groups, and for before and after the treatment was implemented. The model we are presenting is more restrictive than that. Instead of grouping the schools into four groups (treatment and control and before and after the intervention), when we include school fixed-effects ( _Schooli_ ) we control for all the individual characteristics that are fixed, not only whether they are in the treatment or control group. The same for the year fixed effect ( _Yeart_ ); instead of grouping the years in before and after the intervention, we are controlling for the average trend of the outcome.

As with the regular DID model, we are interested in analyzing the interaction of being a treated variable after the intervention. The variable _Single-Shift*Postit_ represents this interaction in equation (1). It is a dummy variable that varies by school _i_ over time _t_ . It equals one for the years after the intervention (after the school became Single-Shift), for the schools that became SingleShift, and is zero before or when the school never becomes Single-Shift. Therefore, coefficient _β3_ in equation (1) refers to the effect of being Single-Shift, measured by the outcome results. It represents the change in the outcome of the treated schools that is above the average trend of the outcome results, and above the level of the outcome for that school, compared to the trend when schools were not exposed to the treatment.

Not all Single-Shift elementary schools are CES and not all middle schools are CMS (but all CES and all CMS are Single-Shift). Therefore, we also estimate model (1) by including a specific variable for the _Ginasios_ ( _CMS*Postit_ ). The coefficients of the variables _Single-Shift*Postit_ and _CMS*Postit_ represent the change in outcome due specifically to the Single-Shift (non- _Ginasio_ ) and for the _Ginasio_ (certified Single-Shift middle school) treatments, respectively.<sup>17</sup> We do not do the same analysis for the CES because, by 2013, there were only 4 schools that were CES (and only 2 schools with complete information), and this sample only increases to 7 schools with complete information in 2014.

The model presented in equation (1) also includes controls that vary by school over time. The objective of including these variables is to control for school characteristics that are not fixed in

> 17 The triple-interaction ( _Ginasio Carioca*Post*Single-Shiftit_ ) means exactly the same as the interaction _Ginasio Carioca*Postit_ , since all _Ginasios Cariocas_ are Single-Shift schools.

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