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PARENTS KNOW THEM BETTER: THE EFFECT OF OPTIONAL EARLY ENTRY ON PUPILS' SCHOOLING ATTAINMENT.

I. INTRODUCTION

An impressive body of literature has investigated the effect that age at school entry has on pupils' educational attainment. (1) This is because it is widely acknowledged that understanding the right moment for children to start formal learning may generate more effective schooling in terms of skills' acquisition and human capital accumulation that may eventually lead to better labor market opportunities. (2) Almost all existing empirical studies point toward the existence of a positive relationship between age at school entry and educational attainment and--despite some recent works that have questioned methodological aspects suggesting more sophisticated approaches--this result appears to be fairly robust. (3) In the light of this evidence, governments are very cautious about lowering the age for compulsory education, even if this may expedite labor market entry of new generations, reduce public expenditure for child care, not to mention increasing parental participation in the labor market.

Indeed, although entry age seems to be positively related to attainment, many education systems still offer an age-range wherein parents can decide when to enroll their children. In the United Kingdom, for example, children have the option of starting school in the September after their fourth birthday. In Finland, Ireland, Italy, Netherlands, and Sweden regulations give the opportunity to anticipate school entry by about 1 year depending on different rules and limitations. In the United States, optional early entry is a common practice in some States and, in several cases, it is conditional upon the evaluation of the child's schooling aptitude. These normative approaches are based on the common belief that the process of intellectual development is related to the inherent nature of each child, so that fixing an age threshold that peremptorily applies uniformly to the entire population could, on one hand, hamper the development of young pupils who are particularly gifted and already ready to start learning and, on the other, it may prejudice those children who need additional out-of-school experience. Although this type of normative setting is widespread in developed countries, to the best of our knowledge, there are no empirical studies providing evidence on its effect on schooling achievements, hence, the target of this paper is to provide a first empirical evaluation of the impact of optional early entry on educational achievement.

We investigate this issue using Italian data collected by the National Institute for the Evaluation of the Education and Training System (INVALSI) containing information on the universe of students who attended the second and the fifth grade in primary school during academic year 2011/2012 and 2013/2014, respectively. (4) Our empirical approach exploits the presence of exogenous age threshold imposed by the Italian law separating pupils who have the option to enroll in advance from those who do not. In particular, since we observe schooling outcomes of pupils who have this option, regardless of whether they actually take it up or not, we can investigate if the opportunity to decide whether children are ready for school may generate a better match between entry age and pupils' characteristics with positive effects on educational performance.

Our empirical design is as follows. In each classroom, pupils in the middle of the age distribution had no option to enroll in advance. Conversely, children at both ends of the age distribution are those who had the option of early entry. In particular, those at the beginning of the age scale (the younger) are those who decided to enroll earlier while those at the end (the older) are pupils who had the option to enroll in advance in the previous year but did not take it up. By evaluating if any discontinuity arises in test scores at the two cutoff points encompassing students with no option of early enrollment, it is possible to assess if children who have the option to enroll in advance perform differently from those who do not.

Our results suggest that, ceteris paribus, pupils with optional school entry perform better than their peers. This effect may be the consequence of two mechanisms. On one hand, early enrollees are actually high-ability pupils who gain from early enrollment. Indeed, it is possible that young pupils who are particularly smart benefit from interaction with more mature peers and these positive externalities affect their schooling outcomes. On the other, students who decide not to take advantage of the option are those who profit more than others from out-of-school experience, hence, their performance improves when they start school later. This evidence implies that, although lowering cutoff birth date for first enrollment for the entire population could have negative effects, giving parents the option to anticipate school entry for their offspring could be a valuable measure.

The paper proceeds as follows. The next section highlights data sources and variables used in the analysis. Section III describes our identification strategy. Section IV discusses results, while Section V contains tests targeted to assess the internal validity of our design. Discussion and concluding remarks are addressed in Sections VI and VII, respectively.

II. DATA

A. Source

Data used in this work have been collected by the National Institute for the Educational Evaluation of Instruction and Training (INVALSI) which yearly assesses students' ability in Italian (indicated as reading) and mathematics. (5) Tests are administered in the primary school (Grades 2 and 5), in the lower secondary school (Grades 6 and 8), and in the upper secondary school (Grade 10). (6) For the purpose of the present study, we use data on primary education and in particular the second grade of the school year 2011/2012 and the fifth grade of the school year 2013/2014 since norms regulating enrollment for these specific cohorts enable us to design appropriate econometric strategies. (7) We use information on about 460,000 pupils for each grade. (8)

The dataset of INVALSI contains standardized tests scores in reading and mathematics and provides relevant information on pupils (gender, date and country of birth, region of residence, type of preschool care), parents (country of birth, occupational status, educational level), schools (size, hours per week), and classrooms (size). Moreover, since INVALSI identifies each year a number of schools where the test is undertaken in the presence of an external supervisor, in our specifications, it is possible to add a control for problems related to cheating. (9) All variables used in the analysis are described in Appendix Table Al, while summary statistics are presented in Tables A2-A9.

B. Normative Frame

Our identification strategy exploits norms regulating first enrollment and compulsory education in Italy in force since the academic year 2009/2010, hence, we consider the cohorts of students who were in the second grade during year 2011/2012 and in the fifth grade during year 2013/2014. The institutional background is as follows.

The Italian primary school year starts in September. In any given year (say year r), all pupils who are 6 years old and those who will be 6 by December 31 must start school in September. The law also permits enrollment of pupils who will turn 6 by April 30 in year t +1. Crucially, this is only an option and it is apparent that for these pupils self-selection into school is almost certainly related to their (potentially unobserved) characteristics. This implies that in each class, along with regular students, there are early enrollees as well as pupils who became 6 between January 1 and April 30 in t whose parents decided not to enroll them in advance at i-1. As summarized in Box 1, this institutional framework gives rise to classrooms wherein students' age at school entry ranges from 65 months (those turning 6 in April of t+ 1) to 80 months (those turning 6 in January of t).
BOX 1 : PUPILS ENTRY AGE AND EARLY
ENTRY OPTION FOR PUPILS IN SECOND
AND FIFTH GRADE

Age in Months        Month at
at School          Which Pupils
Entry                 Turno          Groups

65              [April.sub.t+l]
66              [March.sub.t+l]      Group (i): Early
67              [February.sub.t+l]   enrollees: option
68              [January.sub.t+l]    taken

69              [December.sub.t]
70              [November.sub.t]
71              [October.sub.t]      Group (ii): Regulars:
72              [September.sub.t]    no option
73              [August.sub.t]
74              [July.sub.t]
75              [June.sub.t]
76              [May.sub.t]

77              [April.sub.t]
78              [March.sub.t]        Group (iii): Not early
79              [February.sub.t]     enrollees: option
80              [January.sub.t]      not taken


Therefore, in each classroom, we have three groups of pupils selected according to the date they turned 6: group (i) January-April of year i+1; group (ii) May-December of year t; and group (iii) January-April of year t. Group (i) had the option of early enrollment and used it, group (ii) did not have the option, and group (iii) had the option but did not use it. We can manage this setting to evaluate if, ceteris paribus, test scores of groups (i) and (iii) are higher than those of group (ii). The exogenous thresholds ensure that by merging group (i) and group (iii), we obtain a pool of students similar to those of group (ii). In particular, since we have all pupils involved in primary education, when merging group (i) and group (iii) we obtain the entire population of pupils born in the first 4 months of the year regardless of their decision concerning early entry. These children are fully comparable with those of group (ii) in the sense that--apart from age--the only difference between them is that the former had the option of early entry.

At this stage, it is important to note that the aim of the strategy described above is to implement same-grade comparisons, that is, to evaluate the relative performance of pupils enrolled in the same grade. Indeed, an alternative strategy could be based on same-age comparisons, that is, whenever data on two contiguous grades are available, it could be possible to consider all pupils born in a given calendar year by merging those enrolled in the lower grade with early enrollees in the higher one. In this case, pupils with the option of early entry--irrespective of whether they took it up or not--could be compared with others. (10) However, for cohorts that had first enrollment after 2009/2010, INVALSI decided not to evaluate contiguous grades. For this reason, our strategy is mainly based on a same-grade analysis designed to control for age differences. Notwithstanding this, a robustness check based on same-age comparisons will be provided considering different cohorts for a single Italian region characterized by a specific institutional context.

C. Some Descriptive Statistics

We can now turn our attention to the statistics presented in Tables A2-A5 where pupils' characteristics are evaluated according to their age at school entry. From these tables, it appears that regulars have a parental background (parents' education and occupational status) that is homogeneously distributed across entry age. Instead, early enrollees are atypical since their parents tend to be more educated and employed in high-profile occupations. Moreover, the percentage of females in this group is higher than for regulars. When looking at the group of pupils who decided not to take the option up, we can see they also differ from regulars but in an opposite way with respect to early enrollees since the share of males is higher and they come from less educated families. From Tables A6-A9, it appears that when merging group (i) and group (iii), we obtain pupils whose observable characteristics are similar to those of group (ii). Even so, in order to support the idea that these two pools of students are comparable some statistical tests are required and we will specifically address this issue in Section V.

III. THE IDENTIFICATION STRATEGY

As discussed above, norms for first enrollment give rise to a particular classroom composition consisting of pupils who cannot opt for early entry and those who can. However, these two groups of children are different in terms of age since pupils having the option have no peers with the same age in the group of regulars. This has a significant implication for our study: the effect that age may have on test scores (through both different age at test and different age at school entry) cannot be completely untangled when comparing the two groups in a standard ordinary least squares regression framework. To overcome this problem, we apply a two-side regression discontinuity design (RDD) engineered as follows.

A. Step 1: Evaluating Two Discontinuities

Consider a RDD framework where test scores are the dependent variable, while age at school entry (in months) is used as the running variable. (11) According to Box 1, in each grade, there are two exogenous cutoffs. The first separates pupils entering school at the 68th and 69th month of age, that is, those turning 6 in January t + 1 and December t, respectively. These students have 1month difference in age (the former are younger at test and at school entry) and they also differ since the former had the option to enroll in advance and decided to take it while the latter did not have the option. We can indicate the effect (if any) of these two elements on test scores as [[beta].sub.Age] and [[beta].sub.EOEE] respectively, where EOEE stands for exploited optional early entry. In addition, these two groups differ from each other because early enrollees have been selected for early enrollment possibly because of some of their unobserved characteristics whose effect on test scores can be indicated as [[beta].sub.Selection]. By indicating with [[beta].sup.JD.sub.RDD], the parameter estimated at this cutoff point using a RDD, we have:

(1) [mathematical expression not reproducible]

In Equation (1), we decompose the parameter [[beta].sup.JD.sub.RDD] according to the three elements highlighted above. It follows that, by evaluating this discontinuity, it is clearly not possible to establish any causal relation between optional early entry and test scores, since the regression discontinuity (RD) setup would not be internally valid.

Now, let us turn our attention to the second cutoff point separating children turning 6 in April and May of year f, that is, those entering school at the 77th and 76th month, respectively. By indicating with [[beta].sup.AM.sub.RDD], the RDD parameter estimated at this cutoff point, we have:

(2) [mathematical expression not reproducible]

In Equation (2), captures three elements as well: [[beta].sub.Age] which has a negative sign since pupils born in April are older than those born in May; PNE0EE which indicates the effect on scores of having the option of early enrollment but deciding not to use it (NEOEE stands for not exploited optional early entry); [[beta].sub.Unselection] which captures the effect on scores of unobserved components that also influence the decision not to enroll in advance. It follows that, even in this case, parameter [[beta].sup.AM.sub.RDD] cannot be used to establish any relation between optional early entry and test scores. However, as formally explained in the following section, by taking together parameters prod [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] we can move toward the identification of this effect.

B. Step 2: Summing Up Discontinuities

Once [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] have been evaluated, it is possible to sum them up. Formally, using Equations (1) and (2), we have:

(3) [mathematical expression not reproducible]

The first result we gather from Equation (3) is that the effect of age on test scores is sidelined. Indeed, whatever the effects of age at the time of test and of age at school entry on scores, when summing up [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD], they would be differentiated out. This implies that our strategy fully addresses concerns related to these variables in achieving identification. Second, let us assume that by merging early enrollees and those not exploiting the option, we obtain a pool of pupils whose composition is identical to that of regulars. If this assumption holds, we have:

(4) [[beta].sub.Unselection] = - [[beta].sub.Selection]

and

(5) [mathematical expression not reproducible]

In this case, Equation (4) states that the compositional difference between regulars and early enrollees, and that between regulars and those who did not use the option, offset each other, so that the sum [[beta].sup.JD.sub.RDD] + [[beta].sup.AM.sub.RDD] captures the effect of optional early entry on test scores (Equation (5)).

IV. RESULTS

A. Graphical Investigation

We start our analysis with some graphical inspections. In Figures 1-4, the dependent variable (standardized test scores) is plotted on the vertical axis, and the corresponding rating variable (age at school entry) is plotted on the horizontal axis. In Figure 1, we consider reading test scores for pupils in the second grade, while Figure 2 refers to mathematics tests. Figures 3 and 4 consider the fifth grade and refer to reading and mathematics, respectively. Each of these figures is constructed by using more than 460,000 observations and they should help in answering the question of whether there are discontinuities in the outcome at the cutoff points. Some crucial issues, which apply to all grades and subjects, are worth noting. First, when looking at pupils with no option for early entry, a clear relationship between age and scores arises. As discussed in the Introduction, this evidence is consistent with that reported in other works. Although we cannot establish if the positive link between age and scores is due to a direct effect of entry age or, instead, it is caused by age at test, this evidence is instructive about the presence of compositional differences arising between regulars and pupils located at the ends of the age distribution. Second, all figures show a fairly marked discontinuity at the cutoff separating pupils aged 68 months and those aged 69 at school entry. This means that early enrollees present an advantage in scores that ranges between 1 and 4 points. Furthermore, for this group, the link between age and score is negative, probably because the younger the pupil selected for early enrollment, the higher his/her schooling attitude. Turning our attention to the cutoff point separating regulars from later enrollees (76-77 months), we detect a situation that mirrors that observed on the left-hand side of the previous cutoff: the relation between age and score is negative since older unselected pupils are probably those with very low schooling attitude. Furthermore, the jumps appear to be much smaller than those detected at the first cutoff. All in all, these pictures indicate a scenario where the two discontinuities are not likely to sum up to zero, pointing toward a positive effect of optional early entry on scores. However, the different sign of the link between age and scores for regulars with respect to other groups highlights the fact that heterogeneous selection processes are driving pupils into the pool of early enrollees depending on their month of birth. Consequently, the evaluation of potential compositional differences at the two cutoffs needs to be carefully considered.

B. Evaluating Discontinuities

In Tables 1 and 2, we report the RDD estimates for the second grade for reading and mathematics test scores, respectively, while Tables 3 and 4 refer to reading and mathematics scores for the fifth grade. In columns 1 and 2 of these tables, we show estimates of cutoff points [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] respectively. Each is obtained by implementing a local linear regression without covariates on the two bins adjacent to the cutoff point allowing the slope and intercept to differ on either side of it (Hahn, Todd, and van der Klaauw 2001). Because of the sample size and the large number of observations available in the neighborhood of the cutoff points, the statistical power of the nonparametric/local strategy is less of an issue in this context, implying this method is more appropriate than the parametric/global one.

Since we are modeling the case of a discrete running variable, particular attention should be devoted to the assessment of standard errors and confidence intervals (CIs). Indeed, a common practice to determine CI is to use standard errors that are clustered by the running variable (CRV standard errors) as suggested by Lee and Card (2008). However, as recently pointed out by Kolesar and Rothe (2017), this procedure is likely to generate CIs that are not actually reliable. In particular, discreteness of the running variable causes problems if the number of support points close to the cutoff is small and, in this case, using a small bandwidth is not sufficient to make the bias of the estimator negligible vis-avis its standard deviation. In the light of this, we evaluate "honest" CIs according to the methods presented by Armstrong and Kolesar (2016) and Kolesar and Rothe (2017). In practice, we fix the bandwidth around the cutoff point according to the "length-optimal-bandwidth-method" whose procedure is explained in Armstrong and Kolesar (2016) and evaluate CIs by means of the bounded second derivative (BSD) procedure which requires fixing a bound K to the second derivative of the expectation function. In this case, we set K according to the heuristics described in Section 5.1 of Kolesar and Rothe (2017). (12) In addition, as a robustness check, we also report bounded misspecification error (BME) honest CIs. These are based on the assumption that the specification bias at zero is no worse at the cutoff than away from it (see Section 5.2 of Kolesar and Rothe 2017). Although this type of CI is specifically designed for discrete running variables, it tends to be quite conservative and, in practice, uninformative if there is substantia] uncertainty about the magnitude of the specification errors. For this reason, our comments will be mainly focused on BSD CIs.

To ease reading of Tables 1-4, aside from showing honest CIs, we report the average normalized standard errors for both BDS and BME methods as well as CRV standard errors that, as expected, are always lower than the average normalized BSD ones. (13) Finally in column 3 of these tables, we report the sum of the two parameters [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] test'n8 f[degrees]r the null hypothesis that [[beta].sup.JD.sub.RDD] + [[beta].sup.AM.sub.RDD] = 0 using both the BSD and the BME average normalized standard errors.

The numbers reported in these tables are fairly consistent with Figures 1 - 4. Pupils turning 6 in January of year t+ 1 perform better than those turning 6 in December of t. At the second grade, their advantage is about 2.3 points in reading and 4.4 points in mathematics. These differences are still present at the fifth grade (1.5 points in both reading and mathematics) and their statistical significance is supported by both the BSD and the BME procedures. As discussed above, these differences are also related to unobserved characteristics driving pupils toward early enrollment and are probably mitigated by age differences between the two groups. Turning our attention to the cutoff point separating pupils turning 6 in May and April of year t, we find that the latter perform significantly worse than the former according to the BSD procedure. Their penalty is estimated to be of about 0.5 points for both reading and mathematics and, even in this case, compositional issues may influence the results.

Consider now the sum of the two parameters [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] reported in column 3 of Tables 1 -4. The sum of the two discontinuities should switch off any age effect as well as any compositional effect shaping difference in test scores. If Equation (4) holds, therefore this sum provides an estimate of the effect of the option of early entry on students' outcomes. Numbers reported in Tables 1-4 show that, in all cases, the sum + [[beta].sup.JD.sub.RDD] + [[beta].sup.AM.sub.RDD] is significantly larger than zero and the results are robust with both BSD and BME methods. The reported parameters indicate that optional early entry positively affects test scores generating an improvement of about 1.8 points in reading and 3.8 points in mathematics at the second grade and of about 1.0 point in both subjects at the fifth grade, highlighting that this positive effect lasts for the entire period of primary education.

V. THE IMPACT OF COMPOSITIONAL DIFFERENCES ON SCORES

Numbers reported in the previous section capture the effect of optional early entry on test scores if and only if Equation (4) is satisfied, that is, the selection mechanisms driving pupils into early enrollment are fully compensated by the "inverse" selection process inducing pupils not to enroll in advance. As a consequence, understanding if there are different selection/unselection processes at the two cutoffs and, most importantly, assessing how they affect scores is something that must be considered carefully.

Indeed, in the specific case under analysis, it is plausible that the "selection/unselection" processes may not offset each other. This could happen since the two discontinuities do not separate regulars from pupils born in the same month. Instead, the first cutoff divides regulars from early enrollees born in January, while the second cutoff divides regulars from pupils not enrolled in advance who were born in April. In the presence of different selection mechanisms driving students born in January and April into early enrollment, our identification strategy would be undermined since Equation (4) would not be valid and the sum [[beta].sup.JD.sub.RDD] + [[beta].sup.AM.sub.RDD] would capture differences of the two selection processes too. In the following, we present some tests to evaluate if this might influence our results.

A. Strategy 1: Evaluating Discontinuity in Observables at the Two Cutoffs

A first step is based on the use of information about pupil's background and parental characteristics. Indeed, our data provide a very wide set of useful information such as gender, country of birth, type of preschool care, class size, school size, school hours per week, father's country of birth, mother's country of birth, 8 dummies for mother's employment status, 8 dummies for father's employment status, 4 dummies for mother's educational level, 4 dummies for father's educational level, 20 dummies for geographical regions, and a dummy variable for pupils whose examination has been undertaken in the presence of an external supervisor. All these variables can be used to investigate if compositional differences are affecting our results by evaluating if they present discontinuities at the two cutoffs that do not sum to zero. Indeed, since these control variables are all recorded as dummy or multiple dummy variables, we cannot adopt a standard RDD wherein each single covariate is used as an outcome variable, thus the following strategy is adopted. (14) We run a regression where standardized test scores are explained by all available covariates but age. Then, from this regression, we derive fitted values of standardized test scores and use these as the outcome variable in the RD setup where age at school entry is used as running variable. The main idea is that, despite the fact that significant discontinuities may arise at each cutoff, their sum should contain the average difference of observable characteristics at the two cutoffs with respect to regular pupils. In this way, it is possible to test if they actually sum to zero, as required by Equation (4).

In Figures 5-8, we graphically show the implied discontinuities. In all figures, jumps appear to be statistically significant and, crucially, their size is very similar. In Table 5, we report the estimates for the two discontinuities (for all grades and subjects) as well as BSD normalized standard errors and statistical tests for the null hypothesis that their sum is equal to zero. In all cases, we cannot reject the null. This implies that, as far as observable characteristics are concerned, Equation (4) holds.

B. Strategy 2: Adding Covariates to the RDD Specification

As a second step, we use observable characteristics and include them as control variables in our RDD. In Tables 6 and 7, we show the RD estimates of parameters [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] for the second and the fifth grade, respectively. In this case, estimates are obtained after adding all control variables listed in the previous section including also school and class fixed effects.

A picture emerges that is actually unchanged with respect to the case where no covariates were included: early entrants do better than regulars, late entrants do worse than regulars but the size of the latter difference falls short of the former. We are aware that this type of comparison between estimates with and without covariates is not a perfect test for measuring selection; however, the fact that observables do not matter at all in explaining parameters [[beta].sup.JD.sub.RDD] and [[beta].sup.AM.sub.RDD] would imply that our results are not entirely explained by compositional issues.

C. Strategy 3: Implementing Same-Age Comparisons

So far, we have provided evidence that Equation (4) holds with respect to several observable characteristics. However, concern may arise about unobserved pupils' features that may differ at the two cutoffs. In this part of the paper, we tackle this issue by adopting a strategy based on same-age comparisons. The idea--borrowed from the grade retention literature--requires the use of two contiguous grades in order to reconstruct the entire cohort of children born in a given year. To this end, regulars and those who have not taken the option--who are enrolled in the lower grade--are merged with early enrollees attending the higher grade. In this way, it is possible to rank the entire population of pupils born in a given calendar year according to their month of birth. The effect of optional early enrollment can then be evaluated by means of a standard RDD inspecting if test scores are statistically different at the threshold separating pupils having the option of early entry from those who do not. Thereby, compositional issues are completely bypassed since early enrollees and pupils born in the same month who did not take the option are compared with regulars.

In our case, same-age comparisons cannot be applied straightforwardly since INVALSI data provide information on two contiguous grades only for the cohorts of students who had first enrollment in years 2007/2008 and 2006/2007 and were attending the fifth and the sixth grade, respectively, during the school year 2011/2012. For these cohorts, norms regulating first enrollment were slightly different from those discussed above. In particular, the main difference was that for pupils in the sixth grade, at the time of their first enrollment, it was possible to start primary school directly from the second grade "skipping" the first one. This specific option was available to all pupils conditional upon the evaluation of individual aptitude and skills. This norm, which has not been applied since the school year 2007/2008, imposes a severe caveat on our strategy, that is, in the sixth grade, we cannot disentangle pupils who have used the early entry option from "skippers." It is important to remark that the separation of these two different "types" of early enrollees is necessary since otherwise we would pool pupils of the same age with different years of education.

In order to handle this caveat, we use information provided by an official report of the Italian Ministry of Education (2007) regarding the incidence of pupils who enrolled directly in the second grade in 2006/2007 throughout Italy. Indeed, the report shows that the use of this practice varied widely across regions, Southern ones being located at the top of the ranking with 18% of students in the second grade who did not attend the first grade against the 7% in the North. Interestingly and crucially for our purposes, the Veneto region that year recorded only 0.3% of skippers. For this reason, we implement our same-age comparisons using only pupils in the sixth and in the fifth grades in 2011/2012 in schools located in Veneto, thereby avoiding any bias that may arise by pooling early enrollees with less educated children. (15) We carry on our analysis using more than 37,000 observations.

Our strategy works as follows. Consider children born in 2001 who had the option of early enrollment only if they were born between January and April. Consequently, we obtain the entire cohort of students born between January and April 2001 by merging those who have exploited the option and are attending the sixth grade with those enrolled in the fifth grade who have not. This group can be compared with pupils with no option enrolled in the fifth grade, born between May and December 2001. Our main focus then is on the presence of discontinuity in test scores arising at the threshold separating pupils with and without the option for early entry.

In Figures 9 and 10, we present the graphical investigation of the effect of optional early entry on reading and mathematics scores, respectively. Note that in both figures, the running variable is ranked from January to December, hence unlike previously, from older to younger pupils. Also, on the vertical axis, we report within-grade normalized scores that have been obtained from raw scores available in our dataset. (16) From these graphs, the positive relation between age and score clearly emerges. Furthermore, if we turn our attention to the cutoff separating pupils born in April from those born in May, we detect a significant jump in educational performance. Point estimates are reported in Table 8. The advantage for pupils who can opt for early enrollment is of about 1.1% points in reading and 0.9% points in mathematics. In this case, we are confident that the selection/unselection processes fully offset each other since we are considering the entire regional population of pupils born in April 2001 regardless of their choice on early entry. Furthermore, in this table, we also present results of the McCrary's test (McCrary 2008) implemented to test for homogeneous density across month of birth. According to this procedure, no statistical difference arises in terms of density at the cutoff that provides further support for this econometric exercise.

VI. DISCUSSION

Existing empirical evidence emphasizes the negative impact on scores that may arise when reducing the age for first enrollment. This evidence is mostly derived by comparing the average performance of classmates born in different quarters so that the conclusion of education economists as well as that of educational psychologists is that out-of-school experience is required in order to achieve the adequate maturity before starting formal learning. This view has been recently questioned by authors (Barua and Lang 2016; Black, Devereux, and Sal vanes 2011; Buckles and Hungerman 2013) who criticize certain methodological aspects on which this evidence is grounded and the debate is still ongoing. Our work considers the issue of early entry in primary school from a different perspective, since it evaluates the effect on scores of optional early entry. Our results point toward a single direction: pupils who have this option perform better than those who do not.

Our findings are consistent with improvements in the performance of pupils whose parents decided to delay school entry. In principle, delayed entrance may positively affect student outcomes because slightly older children are more developmentally aligned with the demands and opportunities of formal schooling. As this idea is widespread, in the United States there is a tendency known as "redshirting" (named after the sports term indicating a player kept out of the game for a year to mature) which stresses the benefit of longer experience in comparatively playful environments. At the same time, our study highlights the fact that pupils who have actually been selected for early enrollment register better test scores. Some mechanisms shaping this evidence can be figured out. First, early enrollees could be particularly gifted pupils who benefit from the interaction with more mature classmates, in other words, there can be some positive externalities from mature children toward those who are younger and particularly receptive. Positive externalities may also arise because these smart students may engage in competition with older classmates and hence perform better than they would have done in a classroom with same age peers. This is in line with findings recently reported in Cascio and Schanzenbach (2016) who have highlighted the benefit for young pupils of having more mature classmates.

In conclusion, we argue that the debate opposing "school readiness" to "free play" should be reassessed focusing on the individual's inherent learning disposition, and recognition that parents are better placed to know their children's intellectual maturity.

VII. CONCLUDING REMARKS

The issue concerning the age at which a child should start school is a controversial one in educational policy. Despite the fact that lowering cutoff date for first enrollment can lead to gains in terms of costs for child care, parents' labor market participation, and earlier labor market entrance, it has been recognized that starting school too early can impede the learning process. In the light of this, governments tend to be very cautious about lowering the age for first enrollment and, instead, they prefer to fix a rule for mandatory education which gives parents the option to decide about early entry of their offspring. Although this type of normative setting is widespread in developed countries, its impacts on schooling attainment are unknown since empirical evaluation of this legislative frame is still missing. In this work, we evaluate the effects of giving parents some freedom to determine the entry of their children.

We exploit the specific nature of the Italian law which only gives some pupils the option to enroll in advance. We show that these pupils perform better in both reading and mathematics test scores with respect to those who do not have this option. This may happen because individual characteristics and intellectual maturity turn out to be better matched with the requirements of formal schooling. These advantages in test scores last throughout primary school.

Overall, our results have important implications for education policy. In particular, giving parents a role in the decision concerning early school entry may be beneficial in terms of educational performance. Our RD study identifies the mean impact of the program locally at the cutoff points; consequently, our policy implication is that optional early entry should involve pupils located next to the legal limits. Moreover, given that month of birth is random, local treatment effects can be generalized so that our empirical findings would apply to all those born in the same calendar year, implying the advisability of extending optional early entry. Further research is needed to assess this important aspect.

Finally, this policy issue also requires consideration of the dangers of opportunistic behavior by "pushy" parents. That said, the question for policymakers concerns the public financing of children's access to developmental and educational resources since school attendance also functions as a mechanism for nonparental care. In this, sense policy oriented toward leaving parents some margin of freedom in choosing the school entry age of their children should be considered in the light of the existing institutional environment and within a more general evaluation of social and educational policies.

APPENDIX
TABLE A1

Description of Variables

Dependent Variables

Standardized Test Sciores in Reading
and Mathematics: Continuous Variable
(Scale from 0 to 100).

REGRESSORS

Student           Age at school       Age in months
characteristics   entry               (65-80)

                  Gender              Dummy variable

                  Country of          Dummy variable
                  birth

                  Preschool           Dummy variable
                  attendance

School            School size         Continuous variable
characteristics
                  Class size          Continuous variable

                  Sample school       Dummy variable

                  School weekly       Multiple dummy
                  hours               variable

Parents'          Father's/           Dummy variable
background        Mother's
                  country of
                  birth

                  Father's/           Multiple dummy
                  Mother's            variable
                  educational
                  qualification

                  Father's/           Multiple dummy
                  Mother's            variable
                  employment
                  status

Territorial       Macrogeographical   Multiple dummy
characteristics   area                variable

                  Regions             Multiple dummy
                                      variable ordered
                                      according to the
                                      classification of the
                                      National Statistical
                                      Institute

Standardized Test Sciores in Reading
and Mathematics: Continuous Variable
(Scale from 0 to 100).

REGRESSORS                            Description

Student           Age at school       65-68 early enrollees
characteristics   entry               69-76 regular students
                                      77-80 students that have
                                      decided not to take the
                                      option of early entry

                  Gender              Male = 0
                                      Female = 1

                  Country of          Italy = 0
                  birth               Foreign country = 1

                  Preschool           Daycare (yes/no)
                  attendance          Kindergarten (yes/no)

School            School size         --
characteristics
                  Class size          --

                  Sample school       Sample school = 1 if the
                                      INVALSI test scores have
                                      been undertaken in the
                                      presence of an external
                                      inspector

                  School weekly       Up to 30 hours = 0
                  hours               From 31 to 39 hours = 1
                                      40 hours = 2

Parents'          Father's/           Italy = 0
background        Mother's            Foreign country = 1
                  country of
                  birth

                  Father's/           0 if none, 1 if primary
                  Mother's            school, 2 if lower
                  educational         secondary school, 3 if
                  qualification       vocational secondary
                                      school (3 years of
                                      study), 4 if upper
                                      secondary school
                                      (including Fine Arts
                                      Academy and Music
                                      Academy), 5 if
                                      university degrees
                                      or postgraduate

                  Father's/           0 if unemployed, 1 if
                  Mother's            homemaker, 2 if services
                  employment          personnel, 3 if trader,
                  status              farmer, craftsman,
                                      mechanic, 4 if military,
                                      5 if teacher or public
                                      employee (blue collar),
                                      6 if entrepreneur,
                                      landowner, 7 if manager,
                                      8 if public employee
                                      (white collar), 9 if
                                      professional employee
                                      or freelancer (doctor,
                                      lawyer, psychologist,
                                      researcher, etc)

Territorial       Macrogeographical   North = 0
characteristics   area                Center = 1
                                      South and Islands = 2

                  Regions             Abruzzo, Basilicata,
                                      Calabria, Campania.
                                      Emilia-Romagna,
                                      Friuli-Venezia Giulia,
                                      Lazio, Liguria,
                                      Lombardia, Marche,
                                      Molise, Piemonte,
                                      Puglia. Sardegna,
                                      Sicilia, Umbria, Valle
                                      d'Aosta, Veneto,
                                      Toscana, Trentino Alto
                                      Adige

TABLE A2

Descriptive Statistics of Students Attending the Second Grade in
School Year 2011/2012 by Age at School Entry (Reading Test
Score--N = 470,181)

                            Frequencies         Gender

Age in Months     Mean       N        %      Male    Female
at School        Score                       (%)      (%)
Entry

65               73.61     4,162    0.89    42.70    57.30
66               73.96     6,459    1.37    44.56    55.44
67               72.50    10,853    2.31    45.46    54.54
68               72.25    17,865    3.80    46.95    53.05
69#              70.35#   38,930#   8.28#   50.16#   49.84#
70#              70.80#   38,794#   8.25#   50.56#   49.44#
71#              71.42#   42,592#   9.06#   50.28#   49.72#
72#              71.80#   44,482#   9.46#   50.21#   49.79#
73#              72.36#   41,532#   8.83#   50.75#   49.25#
74#              72.85#   43,335#   9.22#   51.02#   48.98#
75#              73.51#   38,409#   8.17#   50.88#   49.12#
76#              74.12#   36,852#   7.84#   51.19#   48.81
77               74.23    31,046    6.60    51.62    48.38
78               74.50    29,547    6.28    52.67    47.33
79               74.29    24,072    5.12    52.67    47.33
80               73.86    21,251    4.52    54.36    45.64

                         Father's                 Mother's
                        Educational              Educational
                       Qualification            Qualification

Age in Months     Low     Medium    High     Low     Medium    High
at School         (%)      (%)      (%)      (%)      (%)      (%)
Entry

65               32.93    40.82    26.25    29.78    41.57    28.65
66               38.08    39.96    21.96    31.82    42.84    25.34
67               43.57    38.62    17.81    37.36    41.39    21.25
68               45.81    37.60    16.59    38.13    41.52    20.36
69#              49.76#   36.56#   13.69#   41.13#   42.33#   16.55#
70#              49.79#   36.60#   13.61#   41.53#   42.00#   16.46#
71#              50.34#   36.55#   13.11#   41.33#   42.43#   16.24#
72#              50.86#   35.99#   13.16#   42.00#   41.73#   16.28#
73#              51.52#   35.78#   12.70#   43.04#   41.41#   15.55#
74#              51.06#   36.03#   12.91#   42.12#   41.77#   16.11#
75#              51.14#   36.04#   12.82#   42.13#   42.04#   15.83#
76#              50.14    36.60    13.26    41.41    42.06    16.53
77               50.77    36.04    13.20    42.00    42.10    15.91
78               51.74    36.10    12.16    42.80    42.30    14.91
79               52.94    35.35    11.70    43.21    42.21    14.58
80               55.06    34.43    10.50    44.84    41.65    13.51

                          Father's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  5.99        0.68      19.46    42.42    31.45
66                  7.16        0.56      21.24    41.92    29.12
67                  8.00        0.86      25.74    40.85    24.55
68                  7.21        0.86      26.46    41.55    23.92
69#                5.73#        1.14#     29.28#   41.22#   22.63#
70#                5.81#        1.18#     29.51#   41.24#   22.26#
71#                5.70#        1.15#     29.97#   41.65#   21.53#
72#                5.73#        1.14#     31.25#   40.26#   21.63#
73#                5.73#        1.23#     30.72#   41.05#   21.27#
74#                5.94#        1.23#     30.62#   40.99#   21.22#
75#                5.57#        1.24#     30.60#   41.52#   21.07#
76#                 5.34        1.13      30.05    41.17    22.30
77                  5.08        1.15      30.42    41.82    21.52
78                  5.16        1.33      30.74    41.42    21.36
79                  4.69        1.27      31.20    42.03    20.80
80                  4.58        1.42      33.06    40.83    20.10

                             Mother's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  4.64        39.26      4.85    35.65    15.59
66                  5.39        41.57      6.16    32.50    14.37
67                  5.83        44.52      7.53    30.01    12.10
68                  5.46        41.86      9.11    31.16    12.41
69#                5.25#       36.78#     12.81#   33.29#   11.86#
70#                5.25#       36.98#     13.01#   33.32#   11.44#
71#                5.24#       37.10#     13.22#   33.21#   11.23#
72#                5.28#       37.41#     13.24#   32.75#   11.33#
73#                5.27#       37.35#     13.73#   32.59#   11.06#
74#                5.08#       37.44#     13.41#   32.41#   11.66#
75#                4.92#       37.56#     13.07#   33.33#   11.12#
76#                 5.01        36.36     13.14    34.11    11.37
77                  4.90        36.13     13.66    34.11    11.20
78                  4.69        35.02     14.38    34.40    11.51
79                  4.75        32.98     15.51    35.30    11.46
80                  5.07        32.61     16.42    35.31    10.58

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, and High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table Al: Unemployed if employment status =
0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A3

Descriptive Statistics of Students Attending the Second Grade in
School Year 2011/2012 by Age at School Entry (Mathematics Test
Score--N = 471,597)

                            Frequencies         Gender

Age in Months     Mean       N        %      Male    Female
at School        Score                       (%)      (%)
Entry

65               68.17     4,165    0.88    42.50    57.50
66               67.98     6,453    1.37    44.79    55.21
67               66.68    10,796    2.29    45.59    54.41
68               65.75    17,886    3.79    46.91    53.09
69               61.94#   39,071#   8.28#   50.04#   49.96#
70               62.61#   38,997#   8.27#   50.55#   49.45#
71               63.27#   42,696#   9.05#   50.47#   49.53#
72               63.82#   44,649#   9.47#   50.18#   49.82#
73               64.30#   41,709#   8.84#   50.92#   49.08#
74               64.98#   43,516#   9.23#   51.09#   48.91#
75               65.79#   38,431#   8.15#   50.99#   49.01#
76               66.41#   36,872#   7.82#   51.08#   48.92#
77               66.52    31,194    6.61    51.59    48.41
78               66.61    29,682    6.29    52.61    47.39
79               65.96    24,148    5.12    52.63    47.37
80               65.56    21,332    4.52    54.36    45.64

                        Father's                   Mother's
                       Educational                Educational
                      Qualification              Qualification

Age in Months     Low     Medium    High     Low     Medium    High
at School         (%)      (%)      (%)      (%)      (%)      (%)
Entry

65               33.21    40.56    26.24    30.13    41.09    28.78
66               38.18    39.92    21.90    32.00    42.80    25.21
67               43.76    38.53    17.71    37.52    41.37    21.11
68               45.83    37.67    16.50    38.20    41.57    20.23
69               49.90#   36.42#   13.68#   41.23#   42.20#   16.57#
70               49.81#   36.63#   13.55#   41.44#   42.09#   16.47#
71               50.34#   36.54#   13.12#   41.26#   42.53#   16.21#
72               50.91#   35.97#   13.12#   42.10#   41.68#   16.22#
73               51.65#   35.69#   12.67#   43.13#   41.32#   15.54#
74               21.55#   57.76#   20.69#   42.18#   41.72#   16.09#
75               51.19#   36.00#   12.82#   42.16#   41.99#   15.85#
76               50.23#   36.49#   13.28#   41.53#   41.96#   16.51#
77               50.81    36.03    13.16    42.04    42.09    15.87
78               51.77    36.06    12.17    42.89    42.26    14.85
79               52.98    35.35    11.67    43.21    42.16    14.63
80               55.09    34.37    10.54    44.86    41.59    13.55

                             Father's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  5.96        0.65      19.57    42.10    31.73
66                  2.44        0.19      72.80    14.51    10.05
67                  7.90        0.84      25.91    40.79    24.57
68                  7.15        0.82      26.60    41.46    23.96
69                 5.74#        1.15#     29.39#   41.18#   22.54#
70                 5.83#        1.20#     29.61#   41.18#   22.18#
71                 5.67#        1.16#     30.10#   41.61#   21.45#
72                 5.71#        1.14#     31.28#   40.26#   21.60#
73                 5.68#        1.24#     30.80#   41.09#   21.20#
74                 5.89#        1.25#     30.77#   40.87#   21.21#
75                 5.54#        1.22#     30.56#   41.56#   21.11#
76                 5.40#        1.13#     30.03#   41.17#   22.27#
77                  5.13        1.15      30.35    41.79    21.57
78                  5.18        1.36      30.85    41.37    21.24
79                  4.66        1.28      31.44    41.98    20.65
80                  4.56        1.47      33.13    40.78    20.05

                             Mother's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  4.61        39.27      4.76    35.64    15.73
66                  5.42        41.52      6.11    32.49    14.45
67                  5.93        44.55      7.41    30.15    11.95
68                  5.47        41.93      9.14    30.99    12.47
69                 5.26#       36.81#     12.89#   33.19#   11.85#
70                 5.32#       36.92#     13.07#   33.27#   11.42#
71                 5.20#       37.05#     13.33#   33.23#   11.19#
72                 5.25#       37.46#     13.29#   32.65#   11.36#
73                 5.27#       37.39#     13.78#   32.53#   11.02#
74                 5.09#       37.54#     13.45#   32.39#   11.55#
75                 4.93#       37.57#     13.08#   33.30#   11.12#
76                 5.02#       36.38#     13.11#   34.07#   11.41#
77                  4.89        36.10     13.64    34.18    11.19
78                  4.68        34.92     14.54    34.29    11.56
79                  4.74        32.94     15.59    35.28    11.45
80                  5.08        32.51     16.40    35.33    10.67

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table A1 : Unemployed if employment status
= 0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A4

Descriptive Statistics of Students Attending the Fifth Grade in
School Year 2013/2014 by Age at School Entry (Reading Test
Score--N = 460,798)

                             Frequencies         Gender

Age in Months     Mean       N        %      Male    Female
at School        Score                       (%)      (%)
Entry

65               67.26     4,244    0.92    43.54    56.46
66               66.69     6,324    1.37    44.35    55.65
67               65.35    11,474    2.49    45.08    54.92
68               65.21    18,682    4.05    46.34    53.66
69#              64.02#   36,492#   7.92#   49.83#   50.17#
70#              64.29#   35,070#   7.61#   49.32#   50.68#
71#              64.65#   40,063#   8.69#   49.73#   50.27#
72#              64.90#   41,200#   8.94#   49.62#   50.38#
73#              65.06#   39,166#   8.50#   50.47#   49.53#
74#              65.77#   42,219#   9.16#   50.78#   49.22#
75#              66.52#   39,299#   8.53#   50.38#   49.62#
76#              66.89#   41,336#   8.97#   50.00#   50.00#
77               66.67    32,265    7.00    51.68    48.32
78               66.86    30,144    6.54    51.54    48.46
79               66.44    22,451    4.87    52.00    48.00
80               65.84    20,369    4.42    53.13    46.87

                         Father's                  Mother's
                        Educational               Educational
                       Qualification             Qualification

Age in Months     Low     Medium    High     Low     Medium    High
at School         (%)      (%)      (%)      (%)      (%)      (%)
Entry

65               33.06    41.17    25.77    28.51    41.82    29.66
66               36.55    40.07    23.38    31.40    43.45    25.14
67               42.95    38.37    18.68    36.50    42.45    21.04
68               44.98    38.19    16.83    38.58    41.80    19.62
69#              49.52#   36.34#   14.15#   40.54#   42.78#   16.68#
70#              49.48#   36.89#   13.63#   41.10#   42.56#   16.34#
71#              49.47#   36.96#   13.57#   41.48#   42.32#   16.20#
72#              50.59#   36.19#   13.22#   42.54#   41.55#   15.91#
73#              50.87#   36.12#   13.00#   43.00#   41.81#   15.18#
74#              50.80#   36.29#   12.90#   42.67#   41.86#   15.47#
75#              50.51#   36.34#   13.16#   42.30#   42.14#   15.56#
76#              49.27#   37.03#   13.70#   41.10#   42.96#   15.94#
77               50.89    36.93    12.17    42.04    42.96    15.01
78               51.45    36.41    12.14    42.99    42.51    14.50
79               53.83    35.62    10.56    44.91    41.93    13.16
80               55.26    34.59    10.15    46.56    41.08    12.36

                             Father's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  6.51        0.35      18.97    43.52    30.65
66                  6.14        0.28      20.82    42.67    30.09
67                  7.69        0.31      23.74    41.59    26.66
68                  6.52        0.35      25.54    41.53    26.06
69#                5.80#        0.29#     29.68#   41.46#   22.76#
70#                5.74#        0.35#     29.68#   41.98#   22.25#
71#                5.84#        0.33#     29.41#   42.17#   22.25#
72#                5.83#        0.42#     30.35#   41.26#   22.14#
73#                6.19#        0.30#     30.61#   41.32#   21.57#
74#                5.83#        0.35#     29.93#   41.94#   21.95#
75#                5.77#        0.28#     29.83#   42.09#   22.03#
76#                5.43#        0.34#     29.49#   42.38#   22.36#
77                  5.23        0.32      30.50    42.56    21.39
78                  5.40        0.37      30.93    42.12    21.18
79                  5.05        0.27      32.51    42.11    20.06
80                  4.77        0.31      34.60    40.76    19.55

                            Mother's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  4.59        38.48      5.86    34.51    16.56
66                  4.88        40.28      6.33    34.76    13.75
67                  4.93        43.76      7.52    31.46    12.33
68                  4.98        42.36      8.85    31.17    12.64
69#                5.17#       37.15#     12.98#   33.56#   11.14#
70#                4.83#       36.95#     13.13#   34.31#   10.78#
71#                4.83#       37.73#     13.16#   33.38#   10.90#
72#                5.17#       37.94#     12.91#   33.27#   10.70#
73#                5.03#       37.96#     13.49#   32.97#   10.55#
74#                4.88#       38.01#     13.54#   33.01#   10.56#
75#                4.86#       37.68#     13.20#   33.37#   10.89#
76#                5.08#       37.10#     13.17#   33.90#   10.75#
77                  5.10        36.06     13.84    34.59    10.41
78                  4.98        35.60     14.49    34.52    10.41
79                  5.09        34.27     16.14    34.66     9.84
80                  5.15        33.85     17.12    34.38     9.50

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table Al: Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table Al: Unemployed if employment status =
0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #,
pupils with no early entry option.

TABLE A5

Descriptive Statistics of Students Attending the Fifth Grade in School
Year 2013/2014 by Age at School Entry (Mathematics
Test Score--N = 460,006)

                            Frequencies         Gender

Age in Months     Mean       N        %      Male    Female
at School        Score                       (%)      (%)
Entry

65               69.08     4,247    0.92    43.56    56.44
66               68.47     6,333    1.38    44.28    55.72
67               67.49    11,433    2.49    45.22    54.78
68               67.34    18,624    4.05    46.41    53.59
69               66.16#   36,409#   7.91#   49.89#   50.11#
70               66.56#   35,017#   7.61#   49.40#   50.60#
71               66.67#   40,062#   8.71#   49.79#   50.21#
72               66.89#   41,081#   8.93#   49.68#   50.32#
73               67.29#   39,085#   8.50#   50.54#   49.46#
74               67.76#   42,109#   9.15#   50.90#   49.10#
75               68.58#   39,151#   8.51#   50.32#   49.68#
76               68.90#   41,317#   8.98#   50.08#   49.92#
77               68.80    32,275    7.02    51.70    48.30
78               68.92    30,078    6.54    51.65    48.35
79               68.74    22,435    4.88    52.12    47.88
80               68.20    20.350    4.42    53.34    46.66

                       Father's                   Mother's
                      Educational                Educational
                     Qualification              Qualification

Age in Months     Low     Medium    High     Low     Medium    High
at School         (%)      (%)      (%)      (%)      (%)      (%)
Entry

65               33.06    40.85    26.09    28.42    41.75    29.82
66               36.58    40.08    23.33    31.38    43.49    25.13
67               42.93    38.40    18.67    36.32    42.67    21.01
68               44.76    38.26    16.98    38.32    41.98    19.70
69               49.44#   36.42#   14.14#   40.58#   42.82#   16.60#
70               49.49#   36.85#   13.66#   41.17#   42.50#   16.32#
71               49.43#   37.02#   13.55#   41.45#   42.32#   16.23#
72               50.62#   36.13#   13.26#   42.42#   41.64#   15.94#
73               50.89#   36.10#   13.01#   42.91#   41.90#   15.19#
74               50.75#   36.30#   12.95#   42.58#   41.96#   15.47#
75               50.44#   36.44#   13.12#   42.30#   42.23#   15.47#
76               49.24#   37.03#   13.73#   41.04#   43.06#   15.90#
77               50.94    36.95    12.11    42.06    43.04    14.90
78               51.43    36.43    12.15    42.99    42.49    14.52
79               53.88    35.49    10.63    44.96    41.85    13.19
80               55.33    34.54    10.13    46.55    41.09    12.37

                              Father's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  6.56        0.35      18.83    43.39    30.87
66                  6.11        0.27      20.77    42.76    30.09
67                  7.78        0.30      23.66    41.79    26.47
68                  6.40        0.34      25.50    41.45    26.31
69                 5.79#        0.30#     29.66#   41.47#   22.78#
70                 5.77#        0.36#     29.69#   41.96#   22.22#
71                 5.86#        0.34#     29.37#   42.21#   22.23#
72                 5.80#        0.42#     30.39#   41.28#   22.12#
73                 6.22#        0.30#     30.65#   41.27#   21.56#
74                 5.77#        0.35#     29.93#   41.99#   21.95#
75                 5.75#        0.29#     29.76#   42.19#   22.01#
76                 5.40#        0.34#     29.49#   42.43#   22.33#
77                  5.23        0.32      30.45    42.63    21.38
78                  5.38        0.37      30.94    42.18    21.14
79                  5.09        0.30      32.42    42.14    20.06
80                  4.80        0.32      34.50    40.80    19.58

                              Mother's Employment Status

Age in Months    Unemployed   Homemaker    Low     Medium    High
at School           (%)          (%)       (%)      (%)      (%)
Entry

65                  4.55        38.36      5.84    34.60    16.64
66                  4.82        40.26      6.37    34.95    13.61
67                  4.99        43.64      7.49    31.63    12.25
68                  4.91        42.27      8.85    31.27    12.71
69                 5.18#       37.07#     12.99#   33.63#   11.13#
70                 4.88#       36.87#     13.19#   34.25#   10.81#
71                 4.85#       37.77#     13.16#   33.32#   10.89#
72                 5.17#       37.91#     12.91#   33.27#   10.74#
73                 5.04#       37.93#     13.50#   32.96#   10.57#
74                 4.87#       37.90#     13.59#   33.03#   10.61#
75                 4.90#       37.62#     13.27#   33.34#   10.88#
76                 5.11#       36.99#     13.20#   33.94#   10.76#
77                  5.11        36.06     13.92    34.55    10.37
78                  4.93        35.57     14.52    34.59    10.38
79                  4.97        34.27     16.19    34.73     9.84
80                  5.20        33.73     17.13    34.41     9.54

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table Al: Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table A1 : Unemployed if employment status
= 0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A6

Descriptive Statistics of Students Attending the Second Grade in School
Year 2011/2012 by Month of Birth (Reading Test Score--N = 470,181)

                       Frequencies          Gender

Month        Mean       N        %      Male    Female
of Birth    Score                       (%)      (%)

January     73.12    39,116    8.32    50.98    49.02
February    73.74    34,925    7.43    50.43    49.57
March       74.41    36,006    7.66    51.21    48.79
April       74.16    35,208    7.49    50.56    49.44
May         74.12#   36,852#   7.84#   51.19#   48.81#
June        73.51#   38,409#   8.17#   50.88#   49.12#
July        72.85#   43,335#   9.22#   51.02#   48.98#
August      72.36#   41,532#   8.83#   50.75#   49.25#
September   71.80#   44,482#   9.46#   50.21#   49.79#
October     71.42#   42,592#   9.06#   50.28#   49.72#
November    70.80#   38,794#   8.25#   50.56#   49.44#
December    70.35#   38,930#   8.28#   50.16#   49.84#

                    Father's                  Mother's
                   Educational               Educational
                  Qualification             Qualification

Month        Low     Medium    High     Low     Medium    High
of Birth     (%)      (%)      (%)      (%)      (%)      (%)

January     50.90    35.86    13.24    41.82    41.59    16.59
February    50.09    36.35    13.56    41.43    41.96    16.61
March       49.36    36.77    13.87    40.88    42.39    16.72
April       48.77    36.57    14.66    40.62    42.04    17.34
May         50.14#   36.60#   13.26#   41.41#   42.06#   16.53#
June        51.14#   36.04#   12.82#   42.13#   42.04#   15.83#
July        51.06#   36.03#   12.91#   42.12#   41.77#   16.11#
August      51.52#   35.78#   12.70#   43.04#   41.41#   15.55#
September   50.86#   35.99#   13.16#   42.00#   41.73#   16.28#
October     50.34#   36.55#   13.11#   41.33#   42.43#   16.24#
November    49.79#   36.60#   13.61#   41.53#   42.00#   16.46#
December    49.76#   36.56#   13.69#   41.13#   42.33#   16.55#

                        Father's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.77        1.17      30.08    41.16    21.83
February       5.70        1.15      29.54    41.67    21.94
March          5.51        1.19      29.06    41.51    22.73
April          5.19        1.10      29.18    41.89    22.65
May           5.34#        1.13#     30.05#   41.17#   22.30#
June          5.57#        1.24#     30.60#   41.52#   21.07#
July          5.94#        1.23#     30.62#   40.99#   21.22#
August        5.73#        1.23#     30.72#   41.05#   21.27#
September     5.73#        1.14#     31.25#   40.26#   21.63#
October       5.70#        1.15#     29.97#   41.65#   21.53#
November      5.81#        1.18#     29.51#   41.24#   22.26#
December      5.73#        1.14#     29.28#   41.22#   22.63#

                       Mother's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.25        36.79     13.12    33.43    11.41
February       5.08        36.53     13.06    33.68    11.66
March          4.82        36.18     12.92    34.07    12.01
April          4.87        36.49     12.65    34.29    11.70
May           5.01#       36.36#     13.14#   34.11#   11.37#
June          4.92#       37.56#     13.07#   33.33#   11.12#
July          5.08#       37.44#     13.41#   32.41#   11.66#
August        5.27#       37.35#     13.73#   32.59#   11.06#
September     5.28#       37.41#     13.24#   32.75#   11.33#
October       5.24#       37.10#     13.22#   33.21#   11.23#
November      5.25#       36.98#     13.01#   33.32#   11.44#
December      5.25#       36.78#     12.81#   33.29#   11.86#

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table Al: Unemployed if employment status =
0. Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A7

Descriptive Statistics of Students Attending the Second Grade in School
Year 2011/2012 by Month of Birth (Mathematics Test Score--N = 471,597)

                       Frequencies         Gender

Month        Mean       N        %      Male    Female
of Birth    Score                       (%)      (%)

January     65.65    39,218    8.32    50.96    49.04
February    66.18    34,944    7.41    50.46    49.54
March       66.86    36,135    7.66    51.22    48.78
April       66.72    35,359    7.50    50.52    49.48
May         66.41#   36,872#   7.82#   51.08#   48.92#
June        65.79#   38,431#   8.15#   50.99#   49.01#
July        64.98#   43,516#   9.23#   51.09#   48.91#
August      64.30#   41,709#   8.84#   50.92#   49.08#
September   63.82#   44,649#   9.47#   50.18#   49.82#
October     63.27#   42,696#   9.05#   50.47#   49.53#
November    62.61#   38,997#   8.27#   50.55#   49.45#
December    61.94#   39,071#   8.28#   50.04#   49.96#

                   Father's                   Mother's
                  Educational                Educational
                 Qualification              Qualification

Month        Low     Medium    High     Low     Medium    High
of Birth     (%)      (%)      (%)      (%)      (%)      (%)

January     50.93    35.85    13.22    41.87    41.58    16.55
February    50.19    36.31    13.49    41.49    41.92    16.59
March       49.40    36.73    13.86    41.00    42.36    16.65
April       48.85    36.53    14.62    40.71    41.98    17.31
May         50.23#   36.49#   13.28#   41.53#   41.96#   16.51#
June        51.19#   36.00#   12.82#   42.16#   41.99#   15.85#
July        51.17#   35.95#   12.88#   42.18#   41.72#   16.09#
August      51.65#   35.69#   12.67#   43.13#   41.32#   15.54#
September   50.91#   35.97#   13.12#   42.10#   41.68#   16.22#
October     50.34#   36.54#   13.12#   41.26#   42.53#   16.21#
November    49.81#   36.63#   13.55#   41.44#   42.09#   16.47#
December    49.90#   36.42#   13.68#   41.23#   42.20#   16.57#

                      Father's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.73        1.18      30.19    41.09    21.81
February       5.64        1.15      29.76    41.62    21.84
March          5.52        1.22      29.14    41.50    22.64
April          5.22        1.09      29.14    41.83    22.72
May           5.40#        1.13#     30.03#   41.17#   22.27#
June          5.54#        1.22#     30.56#   41.56#   21.11#
July          5.89#        1.25#     30.77#   40.87#   21.21#
August        5.68#        1.24#     30.80#   41.09#   21.20#
September     5.71#        1.14#     31.28#   40.26#   21.60#
October       5.67#        1.16#     30.10#   41.61#   21.45#
November      5.83#        1.20#     29.61#   41.18#   22.18#
December      5.74#        1.15#     29.39#   41.18#   22.54#

                       Mother's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.26        36.76     13.13    33.37    11.48
February       5.10        36.49     13.09    33.71    11.60
March          4.81        36.09     13.06    33.97    12.07
April          4.86        36.47     12.62    34.35    11.70
May           5.02#       36.38#     13.11#   34.07#   11.41#
June          4.93#       37.57#     13.08#   33.30#   11.12#
July          5.09#       37.54#     13.45#   32.39#   11.55#
August        5.27#       37.39#     13.78#   32.53#   11.02#
September     5.25#       37.46#     13.29#   32.65#   11.36#
October       5.20#       37.05#     13.33#   33.23#   11.19#
November      5.32#       36.92#     13.07#   33.27#   11.42#
December      5.26#       36.81#     12.89#   33.19#   11.85#

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table Al: Unemployed if employment status =
0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A8

Descriptive Statistics of Students Attending the Fifth Grade in School
Year 2013/2014 by Month of Birth (Reading Test Score--N = 460,798)

                       Frequencies         Gender

Month        Mean                       Male    Female
of Birth    Score       N        %      (%)      (%)

January     65.54    39,051    8.47    49.89    50.11
February    66.07    33,925    7.36    49.66    50.34
March       66.83    36,468    7.91    50.30    49.70
April       66.74    36,509    7.92    50.73    49.27
May         66.89#   41,336#   8.97#   50.00#   50.00#
June        66.52#   39,299#   8.53#   50.38#   49.62#
July        65.77#   42,219#   9.16#   50.78#   49.22#
August      65.06#   39,166#   8.50#   50.47#   49.53#
September   64.90#   41,200#   8.94#   49.62#   50.38#
October     64.65#   40,063#   8.69#   49.73#   50.27#
November    64.29#   35,070#   7.61#   49.32#   50.68#
December    64.02#   36,492#   7.92#   49.83#   50.17#

                   Father's                  Mother's
                   Educational               Educational
                  Qualification             Qualification

Month        Low     Medium    High     Low     Medium    High
of Birth     (%)      (%)      (%)      (%)      (%)      (%)

January     50.42    36.29    13.30    42.81    41.42    15.77
February    50.24    36.53    13.23    42.14    42.10    15.75
March       48.95    37.02    14.02    41.06    42.67    16.27
April       48.94    37.40    13.67    40.55    42.83    16.61
May         49.27#   37.03#   13.70#   41.10#   42.96#   15.94#
June        50.51#   36.34#   13.16#   42.30#   42.14#   15.56#
July        50.80#   36.29#   12.90#   42.67#   41.86#   15.47#
August      50.87#   36.12#   13.00#   43.00#   41.81#   15.18#
September   50.59#   36.19#   13.22#   42.54#   41.55#   15.91#
October     49.47#   36.96#   13.57#   41.48#   42.32#   16.20#
November    49.48#   36.89#   13.63#   41.10#   42.56#   16.34#
December    49.52#   36.34#   14.15#   40.54#   42.78#   16.68#

                      Father's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.59        0.33      30.37    41.12    22.59
February       5.91        0.29      29.65    41.94    22.21
March          5.52        0.36      29.25    42.21    22.67
April          5.37        0.32      29.23    42.66    22.41
May           5.43#        0.34#     29.49#   42.38#   22.36#
June          5.77#        0.28#     29.83#   42.09#   22.03#
July          5.83#        0.35#     29.93#   41.94#   21.95#
August        6.19#        0.30#     30.61#   41.32#   21.57#
September     5.83#        0.42#     30.35#   41.26#   22.14#
October       5.84#        0.33#     29.41#   42.17#   22.25#
November      5.74#        0.35#     29.68#   41.98#   22.25#
December      5.80#        0.29#     29.68#   41.46#   22.76#

                      Mother's Employment Status

Month       Unemployed   Homemaker    Low     Medium    High
of Birth       (%)          (%)       (%)      (%)      (%)

January        5.07        37.84     13.24    32.88    10.97
February       5.03        37.39     13.31    33.61    10.66
March          4.96        36.38     13.13    34.56    10.97
April          5.04        36.33     12.96    34.58    11.09
May           5.08#       37.10#     13.17#   33.90#   10.75#
June          4.86#       37.68#     13.20#   33.37#   10.89#
July          4.88#       38.01#     13.54#   33.01#   10.56#
August        5.03#       37.96#     13.49#   32.97#   10.55#
September     5.17#       37.94#     12.91#   33.27#   10.70#
October       4.83#       37.73#     13.16#   33.38#   10.90#
November      4.83#       36.95#     13.13#   34.31#   10.78#
December      5.17#       37.15#     12.98#   33.56#   11.14#

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table Al: Unemployed if employment status =
0, Homemaker if employed status = 1, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.

TABLE A9

Descriptive Statistics of Students Attending the Fifth Grade in School
Year 2013/2014 by Month of Birth (Reading Test Score--N = 460, 798)

                       Frequencies         Gender

Month of     Mean       N        %      Male    Female
Birth       Score                       (%)      (%)

January     67.79    38,974    8.47    50.03    49.97
February    68.31    33,868    7.36    49.79    50.21
March       68.84    36,411    7.92    50.37    49.63
April       68.83    36,522    7.94    50.75    49.25
May         68.90#   41,317#   8.98#   50.08#   49.92#
June        68.58#   39,151#   8.51#   50.32#   49.68#
July        67.76#   42,109#   9.15#   50.90#   49.10#
August      67.29#   39,085#   8.50#   50.54#   49.46#
September   66.89#   41,081#   8.93#   49.68#   50.32#
October     66.67#   40,062#   8.71#   49.79#   50.21#
November    66.56#   35,017#   7.61#   49.40#   50.60#
December    66.16#   36,409#   7.91#   49.89#   50.11#

                    Father's                  Mother's
                   Educational               Educational
                  Qualification             Qualification

Month of     Low     Medium    High     Low     Medium    High
Birth        (%)      (%)      (%)      (%)      (%)      (%)

January     50.37    36.29    13.35    42.70    41.50    15.80
February    50.27    36.45    13.28    42.12    42.12    15.76
March       48.93    37.04    14.03    41.05    42.66    16.29
April       48.98    37.38    13.65    40.57    42.90    16.54
May         49.24#   37.03#   13.73#   41.04#   43.06#   15.90#
June        50.44#   36.44#   13.12#   42.30#   42.23#   15.47#
July        50.75#   36.30#   12.95#   42.58#   41.96#   15.47#
August      50.89#   36.10#   13.01#   42.91#   41.90#   15.19#
September   50.62#   36.13#   13.26#   42.42#   41.64#   15.94#
October     49.43#   37.02#   13.55#   41.45#   42.32#   16.23#
November    49.49#   36.85#   13.66#   41.17#   42.50#   16.32#
December    49.44#   36.42#   14.14#   40.58#   42.82#   16.60#

                       Father's Employment Status

Month of    Unemployed   Homemaker    Low     Medium    High
Birth          (%)          (%)       (%)      (%)      (%)

January        5.55        0.33      30.30    41.11    22.71
February       5.96        0.30      29.56    42.03    22.15
March          5.50        0.35      29.24    42.27    22.64
April          5.38        0.32      29.17    42.71    22.42
May           5.40#        0.34#     29.49#   42.43#   22.33#
June          5.75#        0.29#     29.76#   42.19#   22.01#
July          5.77#        0.35#     29.93#   41.99#   21.95#
August        6.22#        0.30#     30.65#   41.27#   21.56#
September     5.80#        0.42#     30.39#   41.28#   22.12#
October       5.86#        0.34#     29.37#   42.21#   22.23#
November      5.77#        0.36#     29.69#   41.96#   22.22#
December      5.79#        0.30#     29.66#   41.47#   22.78#

                        Mother's Employment Status

Month of    Unemployed   Homemaker    Low     Medium    High
Birth          (%)          (%)       (%)      (%)      (%)

January        5.06        37.72     13.26    32.94    11.02
February       4.98        37.34     13.34    33.71    10.63
March          4.91        36.35     13.16    34.65    10.92
April          5.05        36.31     13.02    34.55    11.07
May           5.11#       36.99#     13.20#   33.94#   10.76#
June          4.90#       37.62#     13.27#   33.34#   10.88#
July          4.87#       37.90#     13.59#   33.03#   10.61#
August        5.04#       37.93#     13.50#   32.96#   10.57#
September     5.17#       37.91#     12.91#   33.27#   10.74#
October       4.85#       37.77#     13.16#   33.32#   10.89#
November      4.88#       36.87#     13.19#   34.25#   10.81#
December      5.18#       37.07#     12.99#   33.63#   11.13#

Notes: Father/Mother's educational qualification grouped into three
groups according to the description reported in Table A1 : Low if
educational qualification <3, Medium if educational qualification =
4, High if educational qualification = 5. Father/Mother's
employment status grouped into five groups according to the
description reported in Table A1: Unemployed if employment status =
0, Homemaker if employed status = I, Low if employment status = 2
or 3, Medium if employment status = 4 or 5 or 6, High if employment
status >7. In bold, pupils with no early entry option.

Note: In #, pupils with no early entry option.


ABBREVIATIONS

BME: Bounded Misspecification Error

BSD: Bounded Second Derivative

CIs: Confidence Intervals

CRV: Clustering by the Running Variable

INVALSI: National Institute for the Evaluation of the

Education and Training System

IV: Instrumental Variable

RD: Regression Discontinuity

RDD: Regression Discontinuity Design

REFERENCES

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Armstrong, T. B., and M. Kolesar. "Simple and Honest Confidence Intervals in Nonparametric Regression." Cowles Foundation Discussion Paper No. 2044R, 2016. Available at: arXiv:1606.01200.

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Bertoni, M., G. Brunello, and L. Rocco. "When the Cat Is Near the Mice Won't Play: The Effect of External Examiners in Italian Schools." Journal of Public Economics, 104, 2013, 65-77.

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Buckles, K., and D. Hungerman. "Season of Birth and Later Outcomes: Old Questions, New Answers." Review of Economics and Statistics, 95(3), 2013, 711-24.

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Crawford, C., L. Dearden, and C. Meghir. "When You Are Born Matters: The Impact of Date of Birth on Educational Outcomes in England." DoQSS Working Paper No. 10-09, 2010.

Datar, A. "Does Delaying Kindergarten Entrance Give Children a Head Start?" Economics of Education Review, 25(1), 2006, 43-62.

Dobkin, C., and F. Ferreira. "Do School Entry Laws Affect Educational Attainment and Labor Market Outcomes?" Economics of Education Review, 29(1), 2010, 40-54.

Elder, T. E., and D. H. Lubotsky. "Kindergarten Entrance Age and Children's Achievement: Impacts of State Policies, Family Background, and Peers." Journal of Human Resources, 44, 2009, 641-83.

Fertig, M., and J. Kluve. "The Effect of Age at School Entry on Educational Attainment in Germany." RWI Discussion Papers No. 27, 2005.

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Fredriksson, P., and B. Ockert. "Is Early Learning Really More Productive? The Effect of School Starting Age on School and Labor Market Performance." IZA Discussion Papers No. 1659, 2005.

Hahn, J., P. Todd, and W. van der Klaauw. "Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design." Econometrica, 69(1), 2001, 201-9.

Italian Ministry of Education. "Anagrafe degli Studenti e Rilevazioni sulle Scuole." 2007. Accessed August 2017. http://www.miur.it.gov.it/web/guest/scuola.

Jones, M. M., and G. K. Mandeville. "The Effect of Age at School Entry on Reading Achievement Scores among South Carolina Students." Remedial and Special Education, 11(2), 1990, 56-62.

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Kolesar, M., and C. Rothe. "Inference in Regression Discontinuity Designs with a Discrete Running Variable". Mimeo, 2017. Available at arXiv:1606.04086v3.

Lee, D., and D. Card. "Regression Discontinuity Inference with Specification Error." Journal of Econometrics, 142, 2008, 615-35.

Mayer, S. E., and D. Knutson. "Does the Timing of School Affect How Much Children Learn?," in Earning and Learning: How School Matters, edited by S. E. Mayer and P. Peterson. Washington, DC: Brookings Institution Press, 1999,79-102.

McCrary, J. "Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test." Journal of Econometrics, 142(2), 2008, 698-714.

McEwan, P. J., and J. S. Shapiro. "The Benefits of Delayed Primary School Enrollment Discontinuity Estimates Using Exact Birth Dates." Journal of Human Resources, 43, 2008, 1-29.

Robertson, E. "The Effects of Quarter of Birth on Academic Outcomes at the Elementary School Level." Economics of Education Review, 30(2), 2011, 300-11.

Schwerdt, G., M. West, and M. Winters. "The Effects of Test-based Retention on Student Outcomes over Time: Regression Discontinuity Evidence from Florida." NBER Working Paper No. 21509.9, 2015.

Sharp, C. "What's Age Got to Do with It? A Study of Patterns of School Entry and the Impact of Season of Birth on School Attainment." Educational Research, 37(3), 1995, 251-65.

Strom, B. "Student Achievement and Birthday Effects." Mimeo, Norwegian University of Science and Technology, 2004.

Sweetland, J. D., and P. A. De Simone. "Age of Entry, Sex, and Academic Achievement in Elementary School Children." Psychology in the Schools, 24(4), 1987, 406-12.

PATRIZIA ORDINE, GIUSEPPE ROSE and DANIELA SPOSATO *

* We would like to thank the Editor Lars Lefgren and an anonymous referee for very important suggestions and comments. We are indebted to Hessel Oosterbeek for his insightful hints on a previous version of this paper. We also thank participants to the European Economic Association annual meeting (Toulouse, August 2014) and to the Italian Association of Labour Economists annual conference (Pisa, September 2014). The usual disclaimers apply. Ordine: Professor, Department of Economics, Statistics and Finance, University of Calabria, Rende 87036, Italy. Phone +39-0984-492458, Fax +39-0984-492421, E-mail patrizia.ordine@unical.it

Rose: Assistant Professor, Department of Economics, Statistics and Finance, University of Calabria, Rende 87036, Italy. Phone +39-0984-492446, Fax +39-0984-492421, E-mail giuseppe.rose@unical.it

Sposato: Ph.D. Student, Department of Economics, Statistics and Finance, University of Calabria, Rende 87036, Italy. Phone 0039-0984492458, Fax +39-0984-492421, E-mail daniela.sposato@unical.it

doi: 10.1111/ecin. 12568

(1.) Among others, see Sweetland and De Simone (1987), Jones and Mandeville (1990), Sharp (1995), Mayer and Knut son (1999), Str0m (2004), Datar (2006), Elder and Lubotsky (2009), McEwan and Shapiro (2008), Crawford, Dearden, and Meghir (2010), and Robertson (2011). The state of the art on child development is presented in Francesconi and Heckman (2016).

(2.) Fertig and Kluve (2005). Dobkin and Ferreira (2010), Fredriksson and Ockert (2005), Kawaguchi (2011), and Bedard and Dhuey (2012) find evidence in support of higher educational attainment and wages for students who enter school at an older age.

(3.) Barua and Lang (2016) question the use of instrumental variable (IV) techniques arguing that the required monotonicity assumption is likely to fail when dealing with pupils' age and attainment. These authors propose an instrument that satisfies this assumption and gives rise to consistent estimates of the policy-relevant treatment effect. Their results show that the effect of entry age on educational attainment appears to be very small, but still significant. Black, Devereux, and Sal-vanes (2011) remark that the issue of age-at-test needs to be addressed to provide unbiased estimates of the effect of entry age on test scores. Using data on IQ test scores undertaken by a sample of adults in Norway, these authors implement IV procedures showing that age at test is a determinant of individuals' performance while school entry age seems to play a minor role. Finally, Buckles and Hungerman (2013) show that in the United States, children born at different times in the year are conceived by women with different socioeconomic characteristics, posing concerns on results obtained by comparing pupils born in different quarters when data on parental background are not available.

(4.) As discussed in detail in Section II.A, we rely on these specific grades and cohorts since they involve pupils who had first enrollment under a normative frame that allows us to identify the issue at hand. These two cohorts will be used separately in order to evaluate the impact of optional early enrollment on scores at both the beginning and the end of primary education.

(5.) Data are available from INVALSI upon request. The reading test is divided into three main sections: (1) reading comprehension of a narrative text, (2) reading comprehension of expository text, and (3) grammatical knowledge and skills. The mathematics test is divided into four areas: (1) numbers, (2) space and figures, (3) data and forecasts, and (4) relations and functions. At the second grade of primary school, the math test is limited to the first three sections.

(6.) Starting from 2012/2013, the sixth grade has been excluded from the INVALSI investigation.

(7.) For pupils in the second grade, the law regulating first enrollment is the Ministerial Circular n. 4/2010. For pupils in the fifth grade, first enrollment is regulated by the Ministerial Circular n. 74/2009.

(8.) Reading and mathematics tests take place at different dates so for each single grade we have two different datasets containing a slightly different number of observations.

(9.) INVALSI test scores are intended to evaluate pupils' as well as teachers' performance. Hence, it could be reasonable to suspect that some teachers could help their students when doing tests in order to achieve a better evaluation for the entire classroom (see Bertoni, Brunello, and Rocco 2013 and Angrist, Battistin, and Vuri 2014 for relevant studies on the topic using INVALSI data). In our specific case, however, the issue is likely to play a very minor role since our main strategy relies on comparison between groups of pupils that differ from each other only by 1 month of age, so that there is no reason to think that one group is affected by cheating phenomena differently from the other one. In this view, we prefer to use the entire dataset at our disposal instead of only schools with the external examiner since in the latter case we would lose about 99.5% of our observations and this would be a very major loss in our setup where the number of observations below and above the cutoff is important to undertake valid statistical tests. Finally, it should be remarked that there are no official documents providing information on how examiners are assigned to schools for specific grades and cohorts, so that if we were to rely only on this specific subset of schools, we would probably add some bias. In the light of the above, when appropriate, we will use this information only as a control in our specifications.

(10.) The issue of same-grade and same-age comparisons is important in the grade retention literature where same-grade comparisons are problematic because individuals of different age are compared. See Schwerdt, West, and Winters (2015) for a detailed discussion.

(11.) Lee and Card (2008) discuss the use of a discrete rating variable in RD designs. A very important recent contribution on the topic is Kolesar and Rothe (2017).

(12.) Since in our case, an additional month of age at school entry generates about 0.3 points difference in scores, we set K equal to 0.04.

(13.) A normalized standard error of a CI [a, b] with nominal level 95% is defined as (b-a)/(2 x 1.96), so that the CI is given by adding and subtracting the normalized standard error times the usual 1.96 critical value from its midpoint.

(14.) We are indebted to a referee for this insightful suggestion.

(15.) Since we cannot control for mobility across regions, the estimate of the effect of optional early enrollment can be biased downward.

(16.) We could not use standardized scores directly, since the two grades have been rescaled in a different manner by the IN VALS I.

Caption: FIGURE 1 Second Grade--Reading Standardized Test Scores against Age at School Entry

Caption: FIGURE 2 Second Grade--Mathematics Standardized Test Scores against Age at School Entry

Caption: FIGURE 3 Fifth Grade--Reading Standardized Test Scores against Age at School Entry

Caption: FIGURE 4 Fifth Grade--Mathematics Standardized Test Scores on the Vertical Axis against Age at School Entry

Caption: FIGURE 5 Second Grade--Fitted Values of Reading Standardized Test Scores against Age at School Entry

Caption: FIGURE 6 Second Grade--Fitted Values of Mathematics Standardized Test Scores against Age at School Entry

Caption: FIGURE 7 Fifth Grade--Fitted Values of Reading Standardized Test Scores against Age at School Entry

Caption: FIGURE 8 Fifth Grade--Fitted Values of Mathematics Standardized Test Scores against Age at School Entry

Caption: FIGURE 9 Pupils Born in 2001 Enrolled in either Fifth or Sixth Grade in School Year 2011/2012 in Schools Located in Veneto

Caption: FIGURE 10 Pupils Born in 2001 Enrolled in either Fifth or Sixth Grade in School Year 2011/2012 in Schools Located in Veneto
TABLE 1

RDD Nonparametric Estimates of Discontinuities, Second Grade--Reading

                       (1)                (2)              (3)

                            Reading Standardized Test Score

                                     No Covariates

Dependent          [[beta].sup.      [[beta].sup.      [[beta].sup.
Variable           JD.sub.RDD]        AM.sub.RDD]      JD.sub.RDD]/
                                                       [[beta].sup.
                                                       AM.sub.RDD]

Estimate            2 358 ***       -0.462 **(BSD)       1.896"WW
                    (BSD&BME)

BSD normalized        0.191              0.221            0.292
SE (K = 0.04)

BSD CI            (1.983,2.733)    (-0.897, -0.0273)

BME                   0.599              0.669            0.897
normalized SE

BME CI            (1.246, 3.595)    (-1.220, 0.358)

CRV SE                0.171              0.199            0.262

Implied                 4                  4
bandwidth

Effective obs.        45,084            54,322

Notes: RDD local nonparametric estimates. The dependent variable is
reading standardized test score. No covariates included.
[[beta].sup.JD.sub.RDD] compares pupils with age at school entry of
either 68 or 69 months. [[beta].sup.AM.sub.RDD] compares pupils
with age at school entry of either 76 or 77 months. BSD and BME CI
refer to the bounded second derivative and bounded misspecification
error "honest" confidence intervals as discussed in Kolesar and
Rothe (2017). BSD and BME normalized standard errors also reported.
A normalized SE of a CI [a, b] with nominal level 95% is defined as
(b-a)/(2 x 1.96), so that the CI is given by adding and subtracting
the normalized standard error times the usual 1.96 critical value
from its midpoint. K is the bound imposed to the second derivative
of the conditional expectation function determined as suggested by
Kolesar and Rothe (2017). Bandwidth selected according to the
fixed-length-confidence-interval optimal bandwidth criterion
discussed in Armstrong and Kolesar (2016). CRV SE indicates
clustering running variable standard errors as in Lee and Card
(2008). Effective obs. indicates the total number of observations
just above and below the cutoff. In column 3, standard errors are
evaluated as the square root of the sum of the squared standard
errors of the two parameters and [[beta].sup.JD.sub.RDD] and
[[beta].sup.AM.sub.RDD]

*** Significant at 1%; ** significant at 5%; * significant at 10%.

TABLE 2

RDD Nonparametric Estimates of Discontinuities,
Second Grade--Mathematics

                       (1)               (2)              (3)

                         Mathematics Standardized Test Score

                                    No Covariates

Dependent          [[beta].sup.      [[beta].sup.     [[beta].sup.
Variable           JD.sub.RDD]       AM.sub.RDD]      JD.sub.RDD]+
                                                      [[beta].sup.
                                                      AM.sub.RDD]

Estimate            4.475 ***      -0.641 ** (BSD)     3.834 ***
                    (BSD&BME)                          (BSD&BME)

BSD normalized        0.260             0.302            0.397
SE (K = 0.04)

BSD CI            (3.964, 4.987)   (-1.647, -0.431)

BME                   0.563             0.461            0.727
normalized SE

BME CI            (3.411, 5.619)   (-1.299, 0.550)

CRV SE                0.238             0.268            0.358

Implied                 4                 8
bandwidth

Effective obs.        38,959            69,823

Notes: RDD local nonparametric estimates. The dependent variable
is mathematics standardized test score. See notes of Table 1.

*** Significant at 1% level; ** significant at 5% level.

TABLE 3

RDD Nonparametric Estimates of Discontinuities, Fifth Grade--Reading

                       (1)               (2)              (3)

                             Reading Standardized Test Score

                                    No Covariates

Dependent          [[beta].sup.      [[beta].sup.     [[beta].sup.
Variable           JD.sub.RDD]       AM.sub.RDD]      JD.sub.RDD]+
                                                      [[beta].sup.
                                                      AM.sub.RDD]

Estimate            1.389 ***          0.414 **        0.975 ***
                    (BSD&BME)           (BSD)         (BSD)*(BME)

BSD normalized        0.188             0.173            0.254
SE (K = 0.04)

BSD CI            (1.019, 1.758)   (-0.793, -0.038)

BME                   0.435             0.390            0.583
normalized SE

BME CI            (0.308,2.016)     (-1.380,0.152)

CRN SE                0.170             0.154            0.229

Implied                 4                 3
bandwidth

Effective obs.        44,450            61,845

Notes: RDD local nonparametric estimates. The dependent variable
is reading standardized test score. See notes of Table 1.

*** Significant at 1% level; ** significant at 5% level;
* significant at 10% level.

TABLE 4

RDD Nonparametric Estimates of Discontinuities,
Fifth Grade--Mathematics

                       (1)               (2)               (3)

                           Reading Standardized Test Score

                                   No Covariates

Dependent          [[beta].sup.      [[beta].sup.      [[beta].sup.
Variable           JD.sub.RDD]       AM.sub.RDD]       JD.sub.RDD]+
                                                       [[beta].sup.
                                                       AM.sub.RDD]

Estimate            1.492 ***         -0.449 ***          1.043
                                                      ***(BSD)*(BME)

BSD normalized        0.192             0.197             0.273
SE (K = 0.04)

BSD CI            (0.851,1.607)    (-0.897, -0.122)

BME normalized        0.443             0.285             0.526
SE

BME CI            (0.346, 2.085)   (-1.020, 0.098)

CRY SE                0.159             0.149             0.216

Implied                 4                 3
bandwidth

Effective obs.        47,132            61.850

Notes: RDD local nonparametric estimates. The dependent variable is
mathematics standardized test score. See notes of Table 1.

*** Significant at level; * significant at 10% level.

TABLE 5

RDD Nonparametric Estimates of Discontinuities, Second and Fifth Grade:
Fitted Values of Standardized Test Scores against Age at School Entry

                               (1)            (2)            (3)

                                  Fitted Values of Test Scores

                                          No Covariates

Dependent Variable         [[beta].sup.   [[beta].sup.   [[beta].sup.
                           JD.sub.RDD]    AM.sub.RDD]    JD.sub.RDD]+
                                                         [[beta].sup.
                                                         AM.sub.RDD]

Reading second grade
  Estimate                   0.602 **      -0.419 **        0.183

  BSD normalized SE           0.184          0.142          0.232
  (K = 0.04)

  Effective obs.              26,667         40,093

Mathematics second grade
  Estimate                   0.402 **      -0.449 **        -0.047

  BSD normalized SE           0.179          0.166          0.242
  (K = 0.04)

  Effective obs.              27,464         55,586

Reading fifth grade
  Estimate                   0.310 **      -0.343 **        -0.033

  BSD normalized SE           0.111          0.102          0.148
  (K = 0.04)

  Effective obs.              44,450         61,845

Mathematics fifth grade
  Estimate                   0.220 **       -0.190 *        0.030

  BSD normalized SE           0.101          0.100          0.141
  (K = 0.04)

  Effective obs.              47,132         61,850

Notes: RDD local nonparametric estimates. The dependent variable is
fitted values of either reading or mathematics standardized test
score. [[beta].sup.JD.sub.RDD] compares pupils with age at school
entry of either 68 or 69 months. [[beta].sup.AM.sub.RDD] compares
pupils with age at school entry of either 77 or 76 months. Fitted
values derived from a linear model where standardized scores are
explained by: gender, country of birth, type of preschool care,
class size, school size, school weekly hours, father's country of
birth, mother's country of birth, 8 dummies for mother's employment
status, 8 dummies for father's employment status, 4 dummies for
mother's educational level, 4 dummies for father's educational
level, 20 dummies for geographical regions and a dummy variable for
pupils whose examination has been undertaken in the presence of an
external supervisor. In column 3, standard errors are evaluated as
the square root of the sum of the squared standard errors of the
two parameters and [[beta].sup.JD.sub.RDD] and
[[beta].sup.AM.sub.RDD]

** Significant at 5%; * significant at 10%.

TABLE 6

RDD Nonparametric Estimates of Discontinuities, Second Grade--Reading
and Mathematics: Covariates Included

                       (1)              (2)               (3)

                           Reading Standardized Test Score

                                 Covariates Included

                  [[beta].sup.      [[beta].sup.      [[beta].sup.
                   JD.sub.RDD]      AM.sub.RDD]       JD.sub.RDD]+
                                                      [[beta].sup.
                                                      AM.sub.RDD]

Estimate              2.092        -0.579 **(BSD)    1.513 ***(BSD)
                  ***(BSD&BME)                          **(BME)

BSD normalized        0.252            0.242             0.349
SE (K = 0.04)

BSD CI            (1.599,2.586)   (-1.036, -0.086)

BME                   0.694            0.429             0.793
normalized SE

BME CI            (0.574,3.190)   (-1.430, 0.252)

CRW SE                0.231            0.219             0.318

Implied                 4                4
bandwidth

Effective obs.       37,667            40,693

                         Mathematics Standardized Test Score

                               Covariates Included

Estimate          [[beta].sup.      [[beta].sup.      [[beta].sup.
                   JD.sub.RDD]      AM.sub.RDD]       JD.sub.RDD]+
                                                      [[beta].sup.
                                                      AM.sub.RDD]

BSD normalized        0.291            0.175         0.339 ***(BSD)
SE (K = 0.04)     ***(BSD&BME)        **(BSD)           **(BME)

BSD CI            (3.454,4.598)   (-0.764, -0.075)

BME                   0.750            0.500             0.901
normalized SE

BME CI            (2.619,5.599)   (-1.416, 0.544)

CRN SE                0.269            0.155             0.310

Implied                 4                6
bandwidth

Effective obs.       29,464            55,586

Notes: RDD local nonparametric estimates. The dependent variable is
standardized test score in either mathematics or reading. In each
specification control variables (pupil's gender, country of birth,
type of preschool care, class size, school size, school weekly
hours, father's country of birth, mother's country of birth, 8
dummies for mother's employment status, 8 dummies for father's
employment status, 4 dummies for mother's educational level, 4
dummies for father's employment status, 20 dummies for geographical
regions, a dummy variable for pupils whose examination has been
undertaken in the presence of an external supervisor, school fixed
effect, and class fixed effect) included. See notes of Table 1.

*** Significant at 1% level; ** significant at 5% level.

TABLE 7

RDD Nonparametric Estimates of Discontinuities, Fifth Grade--Reading
and Mathematics: Covariates Included

                       (1)               (2)               (3)

                            Reading Standardized Test Score

                                   Covariates Included

                   [[beta].sup.      [[beta].sup.      [[beta].sup.
                   JD.sub.RDD]       AM.sub.RDD]       JD.sub.RDD]/
                                                       [[beta].sup.
                                                       AM.sub.RDD]

Estimate              1.389             0.414             0.975
                  *** (BSD&BME)        **(BSD)        ***(BSD)*(BME)

BSD normalized        0.188             0.173             0.254
SE (K = 0.04)

BSD CI            (1.019, 1.758)   (-0.793, -0.038)

BME                   0.435             0.390             0.583
normalized SE

BME CI            (0.308,2.016)     (-1.380,0.152)

CRV SE                0.170             0.154             0.229

Implied                 4                 3
bandwidth

Effective obs.        44,450            61,845

                         Mathematics Standardized Test Score

                                Covariates Included

                   [[beta].sup.      [[beta].sup.      [[beta].sup.
                   JD.sub.RDD]       AM.sub.RDD]       JD.sub.RDD]/
                                                       [[beta].sup.
                                                       AM.sub.RDD]

Estimate            1.492 ***           -0.449            1.043
                    (BSD&BME)          **(BSD)        ***(BSD)*(BME)

BSD normalized        0.192             0.197             0.273
SE (K = 0.04)

BSD CI            (0.851. 1.607)   (-0.897. -0.122)

BME                   0.443             0.285             0.526
normalized SE

BME CI            (0.346, 2.085)    (-1.020,0.098)

CRV SE                0.159             0.149             0.216

Implied                 4                 3
bandwidth

Effective obs.        47,132            61,850

Notes: RDD local nonparametric estimates. The dependent variable is
standardized test score in either mathematics or reading. In each
specification control variables (pupil's gender, country of birth,
type of preschool care, class size, school size, school weekly
hours, father's country of birth, mother's country of birth, 8
dummies for mother's employment status, 8 dummies for father's
employment status, 4 dummies for mother's educational level, 4
dummies for father's employment status, 20 dummies for geographical
regions, a dummy variable for pupils whose examination has been
undertaken in the presence of an external supervisor, school fixed
effect, and class fixed effect) included. See notes of Table 1.

Table 8

RDD Nonparametric Estimates of Discontinuity: Same-Age
Comparisons. Fifth and Sixth Grade in School
Year 2011/2012 in Veneto, Reading and Mathematics

                        (1)               (2)

                  Fifth and Sixth Grade 2011/2012

                       Panel A: RDD Estimates

                        Normalized Test Score

Dependent             Reading         Mathematics
Variable

Estimate               0.015             0.009
                  *** (BSD)**(BME)   **(BSD)*(BME)

BSD normalized         0.002             0.003
SE (K = 0.04)

BME                    0.006             0.005
normalized SE

CRV SE                 0.001             0.001

Implied                  4                 4
bandwidth

Effective obs.         6,307             6,855

                         Panel B: McCrary Test

                    Frequency of Count of Each Bin

Dependent             Reading         Mathematics
Variable

Estimate               5.149             7.876

BSD Normalized         14.198            18.65
SE (K = 0.04)

Effective obs.         37,543           37,842
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Author:Ordine, Patrizia; Rose, Giuseppe; Sposato, Daniela
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