In their justification for using entrance cutoff dates and compulsory education laws as a natural experiment, the authors of Angrist and Krueger (1991) rightly give much attention to the effectiveness of compulsory attendance laws. However, the authors do not give proper attention to the decisions made by parents. If redshirting is commonplace and nonrandom, as it is in the ECLS-K data set, then the identifying assumption of monotonicity does not hold, and their identification scheme does not work. This problem is distinct from those discussed in Bound and Jaeger (2000).

Aliprantis, Dionissi

English

Research Papers in Economics

A NOTE ON WHY QUARTER OF BIRTH IS NOT A VALID
INSTRUMENT FOR EDUCATIONAL ATTAINMENT
DIONISSI ALIPRANTIS
1. Angrist and Krueger (1991)
The goal in Angrist and Krueger (1991) is to use variation in schooling attain-
ment unrelated to ability to identify the eﬀect of attending an additional year of
school. The paper uses date of birth and compulsory education laws as a natural
experiment to assign one group of students an extra year of education. While the
paper rightly focuses much attention on demonstrating the eﬀectiveness of com-
pulsory education laws, it does not give suﬃcient consideration to the problems
generated for the IV framework if there is signiﬁcant redshirting
1 around entrance
cutoﬀ dates. This problem is distinct from those discussed in Bound and Jaeger
(2000), which assumes “the typical compulsory school attendance law requires stu-
dents to start ﬁrst grade in the autumn of the calendar year in which they turn 6,”
precluding the possibility of redshirting.
1.1. Monotonicity. The implicit assumption needed to identify the eﬀect of at-
tending an additional year of school is that children enroll in school when ﬁrst
eligible. Without this implicit assumption, the explicit assumption of Monotonic-
ity (Angrist and Imbens (1995)) does not hold. As reported in Aliprantis (2007),
in 1998 about 30% of parents in the US chose to delay their children’s enrollment
in primary school if their child’s birthday was within two months of their school’s
entrance cutoﬀ date. This rate was almost one in four for the youngest quarter of
eligibility to enroll. Furthermore, this selection was not random, as will be discussed
in Section 2.
To illustrate the problem with this implicit assumption suppose, for the sake
of simplicity, that all children born in quarter 1 (Q1) are born on February 15th,
quarter 2 (Q2) on May 15th, quarter 3 (Q3) on August 15th, and quarter 4 (Q4) on
November 15th. Assume that school starts on August 15th and the cutoﬀ date for
all children to turn six is December 31st of the academic school year. Then Table
1 shows the pattern of ages at entry of ﬁrst grade we will observe.
Now assume that all children must attend school until they reach 16 years of
age and that we are looking only at those children who would like to drop out once
they turn 15 years old. Then Table 2 shows the years of completed schooling at
which students are eligible to drop out conditional on redshirting and quarter of
birth. The spirit of the identiﬁcation scheme used in Angrist and Krueger (1991) is
that since laws apply to a student’s age, and not their years of completed schooling,
Date: October 5, 2007.
Email: jalipran@econ.upenn.edu
I acknowledge ﬁnancial support from the Institute of Education Statistics.
1Redshirting is the term given to parents voluntarily delaying their children’s enrollment
1A NOTE ON QUARTER OF BIRTH AND EDUCATIONAL ATTAINMENT 2
variation in students’ birth dates means they face diﬀerent laws regarding the years
of schooling they must complete. In our example, students born in Q4 must stay
in school until they complete 10.25 years of schooling, while individuals born in Q1
only have to complete 9.5 years of schooling before being eligible to drop out. Thus
comparing these groups will tell us the treatment eﬀect (ie, the eﬀect of attending
an extra year of school against one’s will).
There is a problem with this scheme if there is signiﬁcant redshirting that is
not random with respect to expected educational attainment. Let us begin with
the framework in Angrist and Imbens (1995), comparing only those children born
in the ﬁrst and last quarters of the year. Deﬁne treatment as being eligible to
enroll when young (ie, being born in Q4). Let S1 be the years completed for
those exposed to treatment, and S0 the years of completed schooling for those not
exposed to treatment. Then if all children enroll when eligible and comply with
the compulsory education law, we have from Table 2 that S1 = 10.25 and S0 = 9.5.
Thus
E[Yi|Q4] − E[Yi|Q1] = E[Yi|S1] − E[Yi|S0] = E[Yi|S0 + 0.75] − E[Yi|S0].
However, consider what happens if not all parents enroll their child when ﬁrst
eligible. Suppose that there are two types, H and L. Assume that both types enroll
when eligible when born in Q1, but that only those of type H enroll when eligible
if they are born in Q4. Let Bj be the cohort of children born j months before
their school’s cutoﬀ date. Table 3 shows that this assumption is reasonable in the
ECLS-K data set. For those of type H, S1 = S0 + 0.75. However, for those of type
L, S1 = S0 − 0.25. Thus for those of type H, S1 − S0 = 0.75 > 0, but for those of
type L, S1 − S0 = −0.25 < 0. This violates the assumption of Monotonicity.
To illustrate why this creates a problem, consider what would happen if those
who would drop out when 15 years of age are all of type L. In this case we would
expect that those entering school when younger tend to complete more schooling -
precisely what is reported in Dobkin and Ferreira (2007). Then comparing outcomes
of groups Q4 and Q1 would yield something very diﬀerent from the parameter of
interest:
E[Yi|Q4] − E[Yi|Q1] = E[Yi|S1] − E[Yi|S0] = E[Yi|S0 − 0.25] − E[Yi|S0].
This leaves us unable to interpret the IV estimates, similarly to other cases discussed
in Rosenzweig and Wolpin (2000).
2. Redshirting in the ECLS-K Data Set
I am not aware of any historical data on redshirting. Thus I cannot draw any
conclusions from the above discussion about the validity of the estimates presented
in papers such as Angrist and Krueger (1991), Acemoglu and Angrist (2000), or
Lochner and Moretti (2004). Nevertheless, a ﬁrst glance at this issue can be seen in
Table 1, which shows the average age at enrollment for boys in the 1960 Census by
quarter of birth (as reported in Angrist and Krueger (1991)), including the diﬀer-
ence between the census average and the age predicted by our simple assumptions.
Note that there is a large discrepancy in Q4. This diﬀerence could be generated
by schools with cutoﬀ dates before January 1st, but it could also be evidence of
signiﬁcant redshirting in the fourth quarter.A NOTE ON QUARTER OF BIRTH AND EDUCATIONAL ATTAINMENT 3
In the ECLS-K data set we observe four groups of children whose birth dates
are within one month of their school’s cutoﬀ date, as shown in Figure 1:
Cohort C1 : Children eligible in the fall of 1997 but who chose to wait until the
fall of 1998 to enroll.
Cohort C2 : Children ineligible in the fall of 1997 who waited until the fall 1998 to
enroll.
Cohort C3 : Children eligible in the fall of 1998 who chose to enroll.
Cohort C4 : Children ineligible in the fall of 1998 but who chose to enroll anyway.
As shown in Table 3, very few parents would choose to delay their child when ﬁrst
eligible if they are in Cohort C2. In contrast, cohort C1 is made up entirely of
children whose parents chose to delay their enrollment when eligible in 1997, while
C3 is made up entirely of children whose parents chose to enroll them when eligible
in 1998. This allows us to exploit the diﬀerence in the composition of groups C1
and C3 to obtain information about those children whose parents would choose to
delay their enrollment. Tables 4–9 show how these cohorts diﬀer. The children who
delayed enrollment were disproportionately wealthy, white, male, English-speaking,
and had better-educated parents. These characteristics are correlated with educa-
tional attainment, casting doubt on the validity of the instrument generated by
entrance cutoﬀ dates and compulsory education laws.
References
Acemoglu, D. and J. Angrist (2000). How large are human-capital externalities?
Evidence from compulsory schooling laws. NBER Macroeconomics Annual 15,
9–59.
Aliprantis, D. (2007). When should children start school? Working Paper.
Angrist, J. D. and G. W. Imbens (1995). Two-stage least squares estimation of
average causal eﬀects in models with variable treatment intensity. Journal of the
American Statistical Association 90(430), 431–442.
Angrist, J. D. and A. B. Krueger (1991). Does compulsory attendance aﬀect school-
ing and earnings? The Quarterly Journal of Economics 106(4), 979–1014.
Bound, J. and D. A. Jaeger (2000). Do compulsory school attendance laws alone
explain the association between quarter of birth and earnings? In S. W. Polachek
(Ed.), Worker Well Being, Volume 19, pp. 83–108. Research in Labor Economics.
Dobkin, C. and F. Ferreira (2007, July). Do school entry laws aﬀect educational
attainment and labor market outcomes? Working Paper.
Lochner, L. and E. Moretti (2004). The eﬀect of education on crime: Evidence from
prison inmates, arrests, and self-reports. The American Economic Review 94(1),
155–189.
Rosenzweig, M. R. and K. I. Wolpin (2000). Natural “Natural Experiments” in
Economics. Journal of Economic Literature 38, 827–874.Tables 4
Tables
Table 1: Entrance Age (in Years) by Quarter of Birth
Entrance Age (in Years)
Entry Q1 Q2 Q3 Q4
When Eligible 6.5 6.25 6 5.75
Redshirted 7.5 7.25 7 6.75
1960 Census Avg 6.45 6.28 6.08 6.07
’60 – Eligible -0.05 0.03 -0.08 -0.32
Table 2: Years of Schooling Completed before Eligible to Drop Out by Quarter of
Birth
Years of Schooling Completed
Entry Q1 Q2 Q3 Q4
When Eligible 9.5 9.75 10 10.25
Redshirted 8.5 8.75 9 9.25
Table 3: Redshirting in the ECLS-K
Percent Redshirting
Cohort B1 (C1 + C3) B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 (C2)
% Redshirting 28.8 29.8 13.9 14.0 11.3 7.7 8.0 6.8 6.1 5.2 5.4 3.7Tables 5
Table 4: Descriptive Statistics by Cohort
Diﬀerent Variables Summarized by Cohort
Variable Statistic ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
Mean 52,946 61,098 52,456 48,531 61,912
Income Std. Dev. 36,375 40,389 34,959 34,466 40,146
n 12,326 241 702 619 57
Mean 74.35 86.89 75.11 69.74 76.84
# of Books Std. Dev. 59.87 60.64 57.57 57.45 65.18
n 14,910 300 849 744 63
Mean 15.37 18.26 16.09 13.99 17.70
# of Children’s Std. Dev. 17.86 20.33 18.81 17.42 17.68
Records, Tapes, or CD’s n 15,008 303 851 750 63
Mean 65.56 68.34 70.63 60.07 62.00
Age at entry to K Std. Dev. 4.26 5.53 2.91 1.67 5.25
(in months) n 15,042 302 851 751 63
Mean 23.98 25.26 23.93 23.64 24.12
Mother’s Age Std. Dev. 5.44 5.74 5.58 5.37 5.49
(At First Birth) n 14,054 288 793 696 57
Mean 43.82 51.43 41.50 39.74 59.38
Cost Nonparental Childcare Std. Dev. 51.72 49.18 46.17 44.66 77.77
(Pre-K, in $) n 11,465 223 653 559 54
Mean 22.28 24.08 22.80 20.69 17.71
Age at ﬁrst Std. Dev. 19.59 20.95 20.72 18.24 17.71
Nonparental Care (in Months) n 13,047 266 754 637 58
Mean 25.23 22.05 26.66 25.33 27.08
Hours Spent in Std. Dev. 21.73 19.79 23.31 22.15 20.32
Nonparental Care (Pre-K) n 14,858 298 839 744 63
Table 5: WIC Beneﬁts
Did the Child Receive any WIC Beneﬁts as a Child (in %)
ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
Yes 42.88 31.35 40.66 47.87 39.68
No 56.00 67.66 58.05 51.06 60.32
n 15,064 303 851 752 63
Table 6: Child’s Language to Mother
Language to Mother in %
Speaks Non-English Language ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
Never 79.53 93.07 80.85 79.26 77.78
Sometimes, Often, or Very Often 18.53 5.28 16.93 18.36 17.46
n 15,064 303 851 752 63Tables 6
Table 7: Race
The Composition of Cohorts by Race (in %)
Race ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
White, Non-Hispanic 56.31 75.29 60.95 53.94 63.01
Black or African-American, Non-Hispanic 14.20 9.30 13.18 13.87 12.33
Hispanic, Race Speciﬁed 8.52 3.49 7.51 6.58 1.37
Hispanic, Race Not Speciﬁed 8.91 5.52 7.71 11.40 4.11
Asian 6.35 2.03 5.68 7.17 15.07
Native Hawaiian or Other Paciﬁc Islander 1.14 – 1.12 2.47 –
Native American or Native Alaskan 1.80 1.74 1.01 2.00 1.37
More than one race, Non-Hispanic 2.55 2.62 2.64 2.59 2.74
n 17,527 344 984 851 73
Table 8: Gender
The Composition of Cohorts by Gender (in %)
Gender ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
Male 51.15 63.66 53.25 49.82 32.88
Female 48.78 36.34 46.75 50.18 67.12
n 17,554 344 986 851 73
Table 9: Parent’s Highest Education Level
Parent’s Highest Education Level by Cohort in %
ECLS-K Cohort 1 Cohort 2 Cohort 3 Cohort 4
8th Grade or Below 3.09 0.95 2.76 3.05 1.43
9th-12th Grade 6.55 3.15 6.62 6.62 4.29
High School Diploma/Equivalent 23.80 17.67 25.69 27.61 21.43
Voc/Tech Program 6.77 5.36 8.27 7.12 4.29
Some College 26.50 27.13 25.69 27.86 28.57
Bachelor’s Degree 18.53 26.81 16.43 17.05 21.43
Grad/Professional School - No Degree 2.56 3.15 2.65 1.65 2.86
Master’s Degree (MA, MS) 7.50 7.89 6.73 6.49 12.86
Doctorate or Professional Degree 4.71 7.89 5.18 2.54 2.86
n 15,961 317 907 786 70Figures 7
Figures
Figure 1: ECLS-K Cohorts

Do compulsory school attendance laws alone explain the association between quarter of birth and earnings?

Do school entry laws aﬀect educational attainment and labor market outcomes? Working Paper.

Does compulsory attendance aﬀect schooling and earnings?

How large are human-capital externalities? Evidence from compulsory schooling laws.

Imbens

Natural “Natural Experiments” in Economics.

The eﬀect of education on crime: Evidence from prison inmates, arrests, and self-reports.

When should children start school? Working Paper.

A Note on Why Quarter of Birth is Not a Valid Instrument for Educational Attainment

A Note on Why Quarter of Birth is Not a Valid Instrument for Educational Attainment

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