20 Years after the Angrist-Krueger Analysis on Return to Education:
Multiyear Re-estimates and Updated Tests on Instrument Quality
Keywords: Return to Education; Compulsory Education Laws; Instrumental Variables
Author(s): Haogen Yao and Yuen Ting Liu
Email: hy2264@columbia.edu
Abstract
Angrist and Krueger’s 1991 study (AK-1991) introduces the use of birth quarter (QOB) as an
instrumental variable (IV) for education attainment in measuring the returns to schooling. The
controversy over the strength and validity of the IV inspired this paper to replicate their model
with the multi-year 1968-1971 Current Population Survey and 2005-2008 American Community
Survey data in addition to AK’s 1970 and 1980 Census sample. We find that the return to
education is higher in the recent cohorts. But, even with higher gender equity today, education is
still a weaker determinant for women’s earning than for men’s earning. Our additional tests find
weak association between QOB and schooling, inconclusive result for the QOB-race correlation,
and negative evidence for the exclusion restriction assumption. More comprehensive Jackknife
Two Stage Least Square results also demonstrate the low performance of the IV after controlling
for the age factors. We cast doubts on the strength and validity of the QOB IV, and suggest
directions for future research.
I. Introduction
Although education is often one of the biggest items in the national budget for most
developed countries, social scientists have yet to conclude the most appropriate way to measure
returns to schooling. The early studies mostly based their framework on Jacob Mincer’s model of
earnings (1974), in which earnings are directly regressed against education level and other
covariates using Ordinary Least Square (OLS) estimation (e.g. Psacharopoulos 1985, Ashenfelter
and Krueger 1994). These studies suffer from omitted variable bias because unobservables such
as ability, motivation, and socioeconomic status affect both education and earnings. As a result,
the endogenous variable, education, prevents an unbiased estimate of the causal impact of itself.
To overcome the problem of omitted variable bias, Angrist and Krueger (1991) (AK1991) suggests using the interactions of year of birth and quarter of birth (QOB) as instrumental
variables (IV) for educational attainment. Most states have policies regarding the minimum age
to start school, which usually has a cutoff at the beginning of the year. Students born in different
seasons would, therefore, start school at different ages. Their rationale is that since the
compulsory schooling law mandates the students to stay until their 16th or 17th birthday,
individuals born in certain seasons can drop out earlier if they want to; while the younger ones
are compelled to stay longer in schools. The schools’ start age policies and the compulsory
schooling laws then create a natural experiment for researchers to look at the impact of
compulsory schooling laws and returns to education for these students.
There are two major findings in AK-1991. First, compulsory schooling laws increase
educational attainment for those covered by the law. Second, the Two-Stage Least Squares
(2SLS) estimates of monetary return to an additional year of schooling is about 7.5 %, which is
slightly larger than the OLS estimates. The differences between the 2SLS and OLS estimates
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increase after controlling for cross-state seasonal variations in education. AK-1991’s findings
also demonstrate lower returns to schoolings for blacks when compared to whites.
In addition, AK-1991 includes an extensive section to test the strength and validity of the
IV. To prove the association between QOB and years of completed schooling, AK-1991 first
graphs the years of completed education against QOB, which shows that people born later in the
year tend to have higher education. They then conduct descriptive as well as regression analyses
of the de-trended schooling and other education outcomes on QOB. The above results and the
enrollment rate by birthday and legal dropout age all support using QOB as an instrument for
schooling. Secondly, AK-1991 claims that QOB satisfies the exclusion restriction assumption,
which rules out any other path linking the IV to earnings except through compulsory schooling
laws. To illustrate that QOB only affects earnings through the difference in schooling level
induced by compulsory schooling laws, AK-1991 presents several arguments. First, they show
that seasonal pattern in education is not evident in college graduation rates nor graduate school
completion rates. Second, AK-1991’s finding indicates greater decline in the enrollment of
sixteen-year olds in states that allow sixteen-year olds to drop out than states that requires them
to be in school. Third, while socioeconomic status or psychological reasons fail to explain the
decline of QOB effect on more recent cohort’s education, the compulsory law explanation seems
to be workable since more recent cohorts are likely to be less constrained by the compulsory
schooling requirement. Fourth, the coefficients of QOB dummies are insignificant in the OLS
regression of earning on education, implying that their influence may be fully captured by the
education variable. And fifth, there is no relationship between QOB and the earning of college
graduates, who should not be bound by the compulsory schooling laws.
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Despite AK-1991’s effort in affirming their QOB IV, many scholars are still concerned
about the strength and exogeneity of the IV. Incorporating the suggestions in the literature for
various falsification checks and robust estimators, our paper will contribute to the return to
schooling and IV literature in the followings ways: (1) replicating of AK-1991’s model for two
multi-year data sets in addition to the Census data; (2) testing for the validity of the IV with more
recent data; and (3) applying the Jackknife Two Stage Least Square (JK2SLS) model with the
inclusion of all covariates to the original data.
The replication focuses on the 40-49 age cohort as recommended by AK-1991.
Comparing our multiyear re-estimates, we demonstrate that the returns to schooling is higher for
recent cohorts (2005-2008) than for that in the 1970 and 1980 Census assuming that the IVs are
valid. As gender equality has increased for both schooling and employment opportunity, we look
at both male and female samples in the more recent American Community Survey (ACS) data,
but find that returns to education is statistically much less significant for women than for men.
For the pre-1970 Current Population Survey (CPS) data, the 2SLS results are statistically
insignificant. It may be due to the small sample size, the loose implementation of the compulsory
school laws, and/or other historical events.
In response to the concerns raised in the literature, we conduct four tests to examine the
IV validity. The first test looks at the statistics of the first stage results to examine the strength of
the QOB-education association. The inconsistent and high p value indicate the weak IV problem
and hence the possibility of finite-sample bias. The second test regresses the first and last QOB
dummy respectively on the race dummy. Results on the original samples used by AK-1991
demonstrate with statistical significance that black is more likely to give birth during the first
quarter and less likely to give birth in the forth quarter. In the third test, we compare the
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correlation of QOB and earning for individuals that should not be affected by the compulsory
schooling law to those that should be affected by it. Our results cast doubt on AK-1991’s strong
belief in the QOB-education association. The last test is to apply the JK2SLS to the 1970 and
1980 Census data with all covariates controlled. The JK2SLS estimates are highly statistically
insignificant once the age factors, which are jointly insignificant in our sample, are controlled for.
Overall, the paper shows that QOB is not a strong and convincing IV for education in measuring
return to schooling. Even though the IV appears to work relatively well for the 1980 Census data,
the JK2SLS illustrates the otherwise.
The paper unfolds as followed. Section II reviews the QOB and returns to schooling
literature related to our topics. Section III summarizes the data source and the data samples.
Section IV outlines the empirical strategies while section V discusses the results. Section VI
concludes the paper and suggests directions for future research.
II. Literature Review
AK-1991 is the first to use QOB as an IV for educational attainment in looking at the
causal relationship between schooling level and earnings. This publication has led to multiple
debates and new developments in the returns to education and IV literature. The IV validity
debates focus on the endogeneity issue of QOB, violation of the IV’s exclusion restriction
assumption, and the strength of the IV. The literature has further developed certain falsification
checks and estimators to detect and correct for some of these problems. Even though our paper
will not focus on alternative IV development, AK-1991 has also inspired many researchers to
develop other IVs for schooling such as distance to education institutions (Kane and Rouse 1993),
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interaction between location and cohort dummies (Lumieux and Card 1998, Currie and Moretti
2003), and smoking history (Evans and Montgomery 1994).
Some researchers cast doubt on the exogeneity of QOB and suggested that other factors
like socioeconomic status (SES) of the mother may influence the season of birth. If QOB is
indeed endogenous, it violates one of the crucial assumptions for IV estimation that the IV
should be as good as randomly assigned. Early studies of seasonality in the U.S. find inconsistent
results. Seasonality increases with lower social status in Georgia through 1967-1977 with a
popular trend of mothers in middle classes giving birth during July to December (Warren and
Tyler 1979). Erhardt et al. (1971) find no distinct pattern in New York between QOB and
socioeconomic status (SES) for 1954-1963. With birth records from Baltimore, Pasamanick et al.
(1960) conclude that mothers of minority and low SES experience the highest summer birth rates
while the highest SES category shows the smallest seasonality.
More of the recent studies agree that mothers with higher SES tend to give birth during
the warmer seasons. Using data of all live births registered in Czech Republic for 1989-1991,
Bobak and Gjonca (2001) find the seasonality is especially strong among older, married, and
better-educated mothers, who tend to give birth between March and May. Their result is
consistent with data from Britain (James 1971). Similar to the above studies, Bunkles and
Hungerman (2008) examine live birth certificates and the Census data from the U.S. and confirm
that a disproportionally high percentage of women with high SES give birth in the warmer
seasons. As a result, teenage, single, and less educated women are more likely to give birth to
children in winter in the U.S.
The potential association between QOB and SES gives rise to another concern among
scholars – violation of the exclusion restriction assumption. Students from low SES background
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tend to have less education resource and lower level of schooling, which contribute to worse
academic outcomes and lower earning. If lower SES students tend to be born in winter, QOB
will be associated with earnings directly, instead of only through the effect of compulsory
schooling laws. Other than the channel of SES, researchers have found other paths linking QOB
to either educational attainment or earnings. For example, QOB seems to relate to health
outcomes, which may affect education outcomes or earnings. Some simple correlation studies
point out that people being born early in the year have a high probability of getting
Schizophrenia (Torrey et al. 1997, Jablensky 1995, Suvisaari, Haukka, and Lonnqvist 2001).
Others show that season of birth is associated to manic-depression (Hare, Price, and Slater 1974,
Videbech 1974). Some evidences show a relationship between season of birth and performance
in school, such as attendance (Carroll 1992), cognitive development (McGrath et al. 2006),
performance in reading, writing, and math (Mortimore et al. 1988), and university examination
outcomes (Fieder et al. 2006). Even though an overwhelming amount of literature tries to link
QOB to other factors that may affect schoolings and earnings, most of them are from at least a
decade ago and uses methods such as simple correlation or mean comparison, which do not
allow a casual interpretation.
Even though AK-1991’s falsification test (1991) with the 1980 Census data illustrates
that QOB does not affect people that do not leave school as soon as the law allows for it, some
researchers find the otherwise. Using the 1900 Census data, Bound and Jaeger (1996) find a
distinct pattern between QOB and weekly earning for white men born in 1840-1855, who are not
affected by compulsory school laws. Their study shows that people with summer birth tend to
have a higher wages than those born in winter. Hoogerheide and Van Dijk (2006) actually
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demonstrate that the QOB IV is stronger for people with less than 9 years or more than 13 years
of schooling than people with 9-13 years of education.
Furthermore, QOB may be a weak IV since it only affects a small proportion of students,
who leave school immediately as soon as the law permits it, and has a limited variation with
maximum one year of education. Bound, Jaeger and Baker (1995) found that the QOB IV is so
weak that even when they re-estimate the model with randomly generated QOB data, their
results are similar to AK-1991 with the standard error being about 50% larger. In response to the
above studies, Angrist and Kruger (1995) introduce the Split Sample Instrumental Variables
(SSIV) approach to obtain more unbiased estimates. SSIV split the sample into half randomly
and use only half of the sample to estimate parameters in the first stage. It then uses these
estimates to construct fitted values and second stage estimates using data from the other half of
the sample. Though asymptotically less efficient than traditional IV, SSIV is biased toward zero
and unlike IV estimates, which tend to biased in the same way as OLS in finite samples if the IV
are weak. Angrist and Kruger (1995) redo their 1991 study using SSIV. They get estimates that
are close to the conventional IV and OLS estimates with “reasonable” standard error. Their result
is also supported by Staiger and Stock (1997).
The situation of weak IV is more complex if the IV is correlated with the stochastic
disturbances of the structural equation. Bound, Jaeger and Baker (1996) cautions the use of IV
that explains little of the variation in the endogenous predictor as it can induce large
inconsistencies in the IV estimates even if the IV is weakly associated with the error in the
estimation equation. They suggest routinely reporting partial R2 and the F-statistic of the IV in
the first-state estimation to indicate the quality of the IV. On the other hand, Cruz and Moreira
(2005) find that AK-1991’s model with many IVs is reliable despite the low first-stage F-statistic.
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Stock, Wright and Yogo (2002) suggest using a fully robust or a partially robust method if the
first-stage F statistic is smaller than or equals to 10 given that the error terms are homoscedastic
and serially uncorrelated. If F statistic is larger than 10, they suggests checking the results with
various estimators, including Jackknife 2SLS (JK2SLS) explained and used in this study later,
especially when there are many IVs. Hahn and Hausman (2003) find that JK2SLS performs
consistently in the weak IV situation. JK2SLS has finite sample moments up to the degree of
over-identification and solves the “moments problem” caused by too many IVs in a weak IV
situation. It will therefore eliminate the second-order finite sample bias of 2SLS. Poi (2006) also
confirms the high performance of Jackknife estimators through Monte Carlo simulations. Angrist,
Imbens, and Krueger (1999) apply the JK2SLS to the original data and get slightly higher
coefficient that remains statistically significant. Nevertheless, they only use JK2SLS for the
parsimonious model with year-of-birth dummies controlled.
This paper will contribute to the literature by replicating AK-1991’s model, applying
different tests, and utilizing robust estimator with more recent data to address the debates in the
literature. With the inclusion of women, the ACS from 2005-2008 would not only allow us to
evaluate the strength of QOB as an IV and whether return to education has changed over time,
but also permit the comparison of returns to education across genders. Considering the weak IV
arguments, the paper examines the first-stage results for every sample. In response to the
endogeneity discussion, the paper looks at the relationship between QOB and race, which is
often linked to SES. To examine the exclusion restriction assumption, we will examine the effect
of QOB on earnings for cohorts with less than 6 and more than 16 years of education, who are
not supposed to be affected by the compulsory schooling law, as well as, inspired by
Hoogerheide and Van Dijk (2006), the effect on those with education between 9 to 13 years.
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Lastly, since the returns to schooling literature has yet to estimate with JK2SLS including all
covariates, the paper attempts to see how the JK2SLS work in a full model.
III. Data and Summary Statistics
The data are from the 1970 & 1980 U.S. Decennial Census, 1968-1971 Current
Population Survey (CPS), and 2005-2008 American Community Survey (ACS), all available
from the Integrated Public Use Microdata Series website. All of the data sets have a similar
format. For the outcome of interest, log of weekly earning, we computed using information on
wage and the number of working week from the CPS and ACS data. The data sets also contain
educational attainment, QOB, and year of birth, as well as covariates including race, martial
status, age, and the indication of central city residency. We have recoded the data sets so that
they all have the same assortment and scale of variables.
The 1970 and 1980 Census 5 percent sample is the cleaned dataset from the Angrist Data
Archive1. The U.S. Census Bureau surveyed the U.S. population and collected demographic and
household information decennially to assist policy planning and research analysis. The two
cohorts of interest are 40-49 year-old men born between 1920 and 1929 from the 1970 census,
and 40-49 year-old men born in the period of 1930 – 1939 from the 1980 census. For the above
cohorts, these data sets contain more than half a million observations.
CPS is the primary source of information on the labor force characteristics of the U.S.
population jointly conducted by the Bureau of the Census for the Bureau of Labor Statistics.
Since QOB data is only available for 1968-1971, the paper cannot evaluate the returns to
schooling during the 1990s. Another limitation to the 1968-1971 CPS data is the small sample
1
http://econ-www.mit.edu/faculty/angrist/data1/data/angkru1991
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size when compared to the Census and the ACS data. The numbers of observation is about 7300
per year for the cohort of interest – 40-49 year-old male with positive earning. Despite the small
sample size, we try to give CPS data a more accurate scaling for the educational attainment
variable when compared with the Census data. While the years of education are integers in the
AK-1991 sample, the CPS data are recoded to have an interval of 0.5 year to show the
educational level of a person who has dropped out before attaining a full year of education. As
for the ACS data, we created the education attainment variable also with a 0.5 year interval from
the string variable, ‘highest level of education’. Another major difference of the CPS data is the
maximum number of education being 18 instead of 20 as in the other two data sets. In our
analysis, compulsory education laws are not supposed to affect any individuals with more than
13 years of education; therefore, our tests treated this group of people equally whether they have
18 or 20 years of education. As a result, the difference in the maximum education attainment
would not affect our results.
ACS randomly selects three million addresses each year to participate in the survey to
collect information about social, economic, demographic, and housing characteristics of the U.S.
population annually. This data set contains information of the household from all 50 states and
the District of Columbia for year 2005-2008. When compared to the 1970 and 1980 Census data
used by Angrist and Krueger, the ACS data set is more recent and is much larger since it
contains a longer time period. Although the data cover ages from less than one to 96 year old, we
only look at the 40-49 year-old observations, which reduces the sample size to about 1.4 millions.
The education of the household members ranges from no schooling completed, K-12, high
school, to bachelor, masters, and doctoral degree. A significant difference of our ACS data is the
inclusion of female in addition to male.
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Table 1 summarizes the variables of interest from the recoded data sets. As one would
anticipate, both wage and schooling have been increasing over years. However, man’s years of
education have not changed much between 1980 (12.77) and 2008 (13.64). Despite the one year
difference in mean, the difference is not statistically significant. Such small change justifies the
attempt of applying AK-1991’s method to recent data sets2. If the recent educational attainment
far exceeds the length of compulsory schooling, then the complier group will be too small to be
worthy of research. In fact, all of the variables with the exception of the central city status
(SMSA)3 dummy are not different across the data sets with statistical significance. It is also
noticeable that men have much higher mean wage than women based on the ACS data, but
women have higher standard deviation of wage than men.
The descriptive statistics for other variables look normal and do not raise any red flag.
About 8% of the men are African American in the census data and ACS data. The percentage of
African American is 9% for the CPS data and 11% for female in the ACS data. Around 24% and
29% of the men lived in a central city in the Census and CPS data respectively. This percentage
has significantly dropped to less than 12% during 2005-2008. This is hardly surprising since
technological advances have contributed to better transportation, telecommunication, and
Internet usage. In addition, combining with the development in non-central cities, more people
are working out of central cities. The proportion that married has also fallen from the 1970s to
2000s. 88% of men are married with spouse present in the Census and CPS data, but only about
71% of male and 65% of female are married with spouse present in the 2005-2008 ACS data.
2
We notice the possibility of higher response bias in the old times. Participants got the census form in the mail, answered it, and
mailed it back. People with lower education level would less likely to complete this process. From 1970 to 1980, the man’s year
of education had raised 1.27 in ten years. Response bias should be a reason for such big jump. For the 1980~2008 period,
however, the impact of this bias seems to be negligible since there is educational disparity, only 0.87 for 30 year, is already very
small.
3
It is noticeable that we have the least confidence on the replication of SMSA dummy. In AK1991, it has been sometimes stated
as a dummy about residence and sometimes a dummy about work location, but these are two different concepts. In our replication,
we obtain SMSA from the variable METRO, which indicates whether the residency was within a metropolitan area's central city.
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Such percentage drop is consistent with the higher marital age and increased divorce rate in
recent years. For all samples, the age variable averages at 45 with a standard deviation of 2.9,
which indicate the normal distribution of our samples with respects to age. Lastly, the QOB
variable shows that slightly more people were born in the first half of the year in earlier samples,
but slightly more people are born in the second half of the year in the more recent ACS sample.
Figure 1 documents the relationship between QOB and education attainment in the ACS
data. The graph shows a decreasing trend of education for people born in the period 1955 and
1961. An increasing trend of school level also occurs in the period 1962-1968 and 1969 – 1973.
QOB also seems to correlate with level of education. With the exception of the individuals born
in 1959 and between 1944 and 1977, schooling level tends to be lower for those born in the first
quarter. Despite the consistent pattern between QOB and schooling, the difference of education
is no more than a month, which is much smaller than what Angrist and Kruger (1991) found in
the 1980 Census data.
IV. Empirical Strategy
For simplification, this paper mainly focuses on the 40-49 age group as suggested by AK1991. We first compare the 2SLS results using three different data sets. The availability of
women information in the ACS data also allows us to compare the results across genders, who
have more similar educational level and employment opportunity in recent years. Four tests are
used to examine the IV validity. The first test examines the strength of the IV-education
correlation with the first stage statistics. In the second test, we investigate whether the IV is as
good as randomly assigned by regressing respectively the QOB dummies on the major control
variables. The third test utilizes different samples to test the IV-earning correlation for people of
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different education attainments. Lastly, the study applies JK2SLS to the full model in AK-1991,
while it was only applied to the simplest model in existent literatures with only birth-year
dummies. JK2SLS is only done for the AK-1991 data, but the other three tests are all replicated
for multiyear.
A. Multi-year Re-estimation
A simple but compelling way to prove the consistency of the 2SLS estimates is to
replicate it for different years assuming that compulsory schooling laws are equally binding in
alternative years. Assuming the IVs are valid, multiyear re-estimates could also help unveil the
variation trend of the returns to education. Here we replicate the primary model in AK-1991 with
1970 and 1980 Census samples for men, the 1968-1971 CPS data for men, and the 2005-2008
ACS data for both genders owing that women have more comparable educational attainment and
career choice in recent years. Recapping the model:
(1)
(2)
Ei = X i π + ∑c Yicδ c + ∑c ∑ j Yic Qijθ jc + ε i
ln Wi = X i β + ∑c Yicξ c + ρEˆ i + µ i
The first stage regression estimates the education level of the ith individual, Ei , as a
function of X i , Yic , and YicQij . X i is the vector of covariates including race, central city status,
marry status, age, and region dummies. Yic is the birth year dummy in year c while YicQij is the IV
for education - thirty interaction terms between quarter-of-birth and year-of-birth. The second
stage estimation examines the effect of education attainment from (1) on weekly earning, Wi ,
including all the covariates from (1). We are interested in ρ , the return to education. We will
have the same four specifications as the AK-1991’s (i.e. column (2), (4), (6), (8) in its Table IV
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to VI). Since the CPS and ACS data are not as representative as the Census data, we have
personally weighted their re-estimates.
The sample size of the CPS samples are much smaller than that of the Census or ACS
samples. The average CPS sample size for replication is about 7300 per year, but for Census and
ACS are about 288,000 and 177,000 respectively. Therefore, we also try to pool the four CPS
samples in our analyses to increase the statistical power and capture the year fixed effects with
year dummies. Though the ACS data set is large, the analysis will pooling the ACS sample too
for comparison. We do not expect the results to be very different using separate regressions or
pooling methods for the ACS data.
B. Four Tests on Instrument Quality
Based on previous literature and making use of the multiyear data sets, we conduct four
additional analyses. Though they are not methodologically advanced, they examine the IV
quality in different ways. The first test is to look at the first-stage results for multiple years. For
each year/group, we run equation (1) and check the joint significance of the thirty IVs with F-test.
Bound et al. (1995) recommend using the partial R2 and the F statistic of the identifying IVs in
the first-stage estimation as indicators of the estimates’ quality, here we make it more
straightforward by reporting the p value of the F statistics, following Hoogerheide and van Dijk’s
method (2006) for their first-stage check. Using Bound et al.’s term, the larger the p value, the
more qualitatively important the finite-sample bias. Since p value can be affected by the sample
size and therefore misleading, we also need to check the coefficients of the thirty IVs for
statistical significance. From a policy perspective, we expect compulsory schooling laws to be
less binding in recent years, which leads to the attenuation of seasonal pattern. More exactly, the
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coefficients for the interactions between first quarter birth and earlier birth year should be
negative and have relatively large absolute values, while the coefficients for all IVs should be
approaching zero and becoming less significant for more recent cohorts.
The second test is to regress respectively the QOB dummies on the only pre-treatment
control, race, for different years. A statistically significant association between birth quarter and
this variable might suggest that the graphic, single variable regression, and basic comparison
evidence presented in AK-1991 are in fact revealing the association between education and nonQOB factor. This would weaken AK-1991’s argument for using QOB as an IV for education.
More specifically, we are interested in how race is associated with the chance of birth in the first
and the fourth quarter. AK-1991 found that people born in the first quarter has the least
education while people born in the fourth quarter has the most. On the other hand, race is often
associated with SES and culture, whose correlation with QOB is still up to debate in the
literature. A statistically significant coefficient of race would point out the endogeneity of QOB,
which shows that this instrument is not as good as just randomly assigned.
The third test is to use our samples to test the statistical significance of IV-earning
association for people of different education attainments. This test is inspired by the falsification
test, which is regarded as the “perhaps most convincing” IV exclusion restriction evidence in
AK-1991. They look at whether birth quarter has influences on the earnings of college graduates,
who should not be affected if birth quarter impacts earnings only through education. We expand
the test to not only people who should not be influenced but also people who should be
influenced. We test three groups of people: those with less than 6 years of education, more than
16 years of education, and with 9 to 13 years of schoolings. The 9 to 13 span is suggested by
Hoogerheide and van Dijk (2006). The procedure of this test is to regress log weekly earning on
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the three QOB dummies, with birth years controlled, then conduct F test for the QOB dummies
estimates and report the p value for the F statistics.
Lastly, we apply JK2SLS to the full model in AK-1991, while in existent literatures it
was only applied to the simplest model without covariates. Blomquist and Dahlberg (1999) has a
concise summary of the JK2SLS’s four-step algorithm— (1) Use all but the ith observation to
estimate parameters for the first-stage equation; (2) Combine those parameters with the IVs for
the ith observation to construct a fitted value for the ith observation; (3) Repeat the first two
steps for all i; and (4) Regress the outcome variable (here the log weekly wage) on the fitted
values and the exogenous regressors. Mathematically, the bias of the 2SLS estimator arises from
the correlation between the fitted value from the first-stage regression and the error term in
second-stage regression for each observation. A straightforward way to eliminate the bias is to
guarantee a zero correlation, namely to avoid using the ith observation in constructing the
optimal IV for itself, which is what JK2SLS achieves. In other words, JK2SLS can be regarded
as an extreme version of SSIV, but it is even better since it does not compromise the sample size.
We will use the unbiased JIVE1 estimator by Angrist et al. (1999), which is regarded as a
relatively good JK2SLS estimator based on Monte Carlo evidence (Poi, 2006). Since JK2SLS
does not allow weights, we will only run it for the self-weighting Census data.
V. Results
The multi-year re-estimates bring up three major findings. First, 2SLS results are
statistically insignificant for pre-1970s, which could be caused by the small CPS sample size,
loose implementation of the compulsory laws, and/or other historical events. Second, assuming
the IVs are valid, the returns to education for male workers are higher for recent cohorts than for
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the cohorts in 1970 and 1980 Census. Third, despite our belief that men and women should be
comparable in more recent cohort, it seems the education-earning association remains weaker for
women than for men.
The tests on IV quality using different samples (except for JK2SLS) come to four
findings. First, finite-sample bias is highly possible in our regressions since our IVs are very
weak. We actually find no constructive support of the IV strength in all cohorts with respect to
the first-stage coefficient size, direction, and significant level, which is why we have reservation
about the validity of the IVs for the above multi-year re-estimates findings. Second, tests in AK1991 probably exaggerate the QOB-education association. The revealed association in AK1991’s preliminary tests could be actually the associations between race and education. Third,
QOB may influence earnings differently in recent years, and hence make the IVs inappropriate.
And finally, JK2SLS results are highly statistically insignificant once age factors are controlled
for, but it is also worth noticing that age factors are jointly insignificant in our samples.
Overall, our results point out that returns to education cannot be convincingly measured
with the IVs Angrist and Krueger suggested. The estimates based on the 1980 Census sample are
relatively trustworthy, but they are also called into questions by the corresponding JK2SLS
estimates.
A. Results for Multi-year Re-estimates
Results for multi-year re-estimates are displayed in Table 2. For 40-49 year-old men in
the Census sample, our replications achieve almost the same results as AK-1991’s, i.e. Column
(2), (4), (6) and (8) of their Table IV and Table V.4 Using CPS sample, we do not find
4
Here we use the word “almost” because our coefficients on SMSA are exactly the same but of opposite sign to that from AK-
17
statistically significant effect except for 1971. As for the ACS sample, we get larger estimates for
both men and women than for the men in the 1970 and 1980 Census; however, those estimates
are statistically less significant, especially for women.
Rows 5-9 of Table 2 show the results using the CPS sample. We only can get significant
estimates for the year 1971 (column 3 and 4). The small sample size (about 7300 per year) may
be the main contributor of the small statistical power. This explanation is supported by the same
regressions using pooling CPS. Though the pooled sample size, 29,205, is still quite small when
compared with the Census or ACS data, the 2SLS provides very stable estimates for all four
specifications. According to the full model estimates, the monetary return to an additional year of
schooling is about 6.8 % (coefficient equals to 0.0681). Some might question the value of those
small-size CPS samples because while the estimates for 1970 CPS tell little, the estimates for
1970 Census offer very strong result, suggesting 10.1% rate of return and it is statistically
significant at 1% level in the full model. We then pool the CPS 1970 and 1971 samples for 2SLS
and find a coefficient 0.084 with the z value equals to 2.67. Therefore, though CPS samples
induce small sample size problem, they are far from meaningless.
The level of law enforcement and history could also be the reasons for those insignificant
CPS estimates. The mean education level of 40-49 years old men in the 1970 Census sample is
only 11.5, with 25.86% observations have only 9 years or less education, who probably did not
obey the compulsory schooling law. In comparison, these two figures are 12.77% and 13.72% in
the 1980 Census. In addition, many 40-49 years old men in the 1968-1971 samples were
experiencing either the great depression or the World War Two when they were teenagers. The
1991. In addition, when using the 1980 sample, one standard error for age and one for education are also different from theirs.
Since the rest of the coefficients and standard errors are exactly the same as that of the AK-1991’s and we used the clean data and
Stata code that the authors used, these differences should not be an issue of the data set or analysis command. We suspect that the
authors might have typos in their tables. But we also note that these typos will not have an impact on the main conclusions.
18
level of law enforcement could be impaired by these events. And these historical events could
add large uncertainties to one’s education decision, making the laws less relevant to one’s final
education attainment, and the education attainment less relevant to one’s future earning.
As for the ACS samples, the 2SLS estimates are overall larger than the estimates from the
Census samples, indicating larger returns to education. Assuming the IVs are valid for these
more recent data, the results are consistent with the trend of increasing wage premium by
education (Goldin and Katz 2008). Temporarily disregarding the statistically significance, our
results show similar returns to education for men and women. Focusing on the full model results,
the four years’ average coefficient for men is 0.1023, and for women 0.1094. Compared with
men, the returns to education are lower for 40-49 year-old women in the 2006 and 2008 samples,
but higher in the 2005 and 2007 samples.
The coefficients for women are mostly smaller than those for men, especially for the
2007-2008 data, which could be due to sample size problem or the fact that education is
actually not as influential for women’s earning as for men’s. When pooling the four years’
samples, we find a statistically significant return to education for men. The full model
coefficient is 0.0878 with standard error 0.0279, and this estimate has been consistent under the
other three specifications. For women, however, the coefficient turns out to be small and very
insignificant, 0.0686 with standard error 0.0519 in the full model. And no specification
achieves a coefficient that is statistically at 5% level. In short, the pooling regressions confirm a
higher return to education for men, but cast doubt on the role of education in determining
women’s wage.
B. Results for Tests on Instrument Quality
19
Our additional tests question the IV validity for the estimates of all years. We report the
JK2SLS results in Table 2 for comparison with other 2SLS results. Multi-year results for the
other three tests are reported in Table 3, with coefficients boldfaced when p value is smaller than
0.2.
With the multi-year full-model first-stage results, we conclude that finite-sample bias is
very likely to be presented in every sample. The 1970 and 1980 Census samples have almost 0 p
values for the F statistics on IVs. These two F statistics, 2.92 and 2.91, are not big enough for
claiming a negligible bias, especially when they are compared with other F statistics, say, almost
4000 for RACE, SMSA and MARRIED. Moreover, 2.92 is the biggest F statistics we could
obtain from our samples. P values for other samples imply that the joint effect of IVs on
education is significant at 10% level for the pooling CPS (marginally), the 2005, 2006, 2008 and
pooling men sample, and the 2006 women sample.
Note that we do not report the first-stage coefficients. The reason is that the attained
figures are very confusing. They are of different sizes and directions, with no identifiable trend
across the samples. The most interesting findings include the positive and insignificant effects
when applying 1970 and 1980 Census sample and the negative, mostly significant, and larger
effects when applying the 2008 ACS sample, which contradicts to what one would expect since
more recent cohorts should be less binding by the compulsory schooling laws.
The second test finds interesting relationship between QOB and race. Using the 1970 and
1980 Census data, we found that the probability of being born in the first quarter is about 1%
higher for Blacks than for Whites, and it is statistically significant at the 1% level. But the same
coefficients are much less statistically significant for all other data sets. For the association
between race and being born in the fourth quarter, they are relatively small and less significant.
20
Since race was controlled in the full model, such finding does not indicate the failure of 2SLS,
but it does show that QOB is not as good as randomly assigned and this association may
comprises the validity of QOB as an appropriate IV to education. Furthermore, it leads to the
worry that the graphic, single variable regression, and basic comparison evidence presented by
Angrist and Krueger were in fact revealing the correlation between education and a series of
factors rather than just the IV-education correlation.
The IV-earning association test indicates the IVs work best for year 1980. AK1991 only
tested whether QOB affects college graduates for that year, and then claim the fulfillment of
exclusion restriction assumption because the F tests on birth quarter dummies are insignificant
for a group that should not be influenced. But, what if we also get no significant joint effect for
those should be influenced? That happens to the 1970 Census sample. Among the samples we
have, only the 1980 sample supports the argument that QOB affects earnings only through
forcing some people stay in school longer than they want. Its p value for 9-13 education group is
0.0459, while for the other two groups 0.2893 and 0.9192. From several other samples (e.g. 1969
CPS, 2008 ACS and ACS pooling for men), we find evidences that the IVs have weaker effect
on people with education between 9 to 13 than on people with education not in that range
(Hoogerheide and van Dijk 2006). Such findings question the validity of our multi-year reestimates. In addition, we look at the coefficient sizes and directions for the three QOB dummies.
Again, just like what happens in our first-stage regression test, those coefficients are too erratic
to offer any information.
Finally, we find that the standard error blows up once the AGEQ and AGEQSQ are
controlled when applying JK2SLS to the Census data (row 3 and 4 of Table 2). Controlling for
only the year-of-birth dummies, we get the same estimates as in Angrist et al. (1999), which
21
implies that our code and data are correct. But in the full model, the p values of estimates for the
two samples are respectively 0.465 and 0.684, indicating statistically insignificant estimates of
returns to education. As long as the age factors are left out in the regression (column 1 and 3), we
get similar JK2SLS results as the ones from 2SLS; otherwise we get very insignificant results
(column 2 and 4). On the other hand, the two age variables are neither individually nor jointly
significant in the JK2SLS. The full-model F statistics for AGEQ and AGEQSQ is just 1.68 for
1970 sample, and 1.16 for the 1980 sample. Therefore, age variables might not belong to the true
model. Of course, if QOB really matter for education and if education is a main determinant of
earnings, the 2SLS estimates should be robust even with irrelevant variables included in the
model. Age argument is not a strong defense.
VI. Conclusion
This paper finds higher returns to education for male workers in the 2005-2008 ACS data
than in the 1968-1971 CPS and 1970 & 1980 Census data, but detects weaker education-earning
association for female workers, and, more importantly, calls into question the strength and
validity of QOB as an IV for education in measuring return to education. Even though AK-1991
seems to make a strong case for QOB, our results demonstrate that the performance of QOB is
inconsistent. Our additional tests and unbiased estimator JK2SLS point out the IV may not be
robust. Due to the scope of this paper, we cannot include all of our ideas on this topic. For future
research directions, the 2SLS model may include covariates for field of occupation, which can
also be created for all of the data sets used in this paper, to increase the accuracy of the model.
Returns to education in different field are very different especially in recent years. For example,
the field of law has much higher returns than that in the field of journalism. The results would be
22
more precise if we include the field of occupation. Besides, redoing these analyses with more
data, such as the 2010 Census, can also increase the statistical power and provide more evidence
to evaluate our conclusions. Lastly, regression discontinuity may be another way to compare the
results using the QOB IV. It would be interesting to employ one month band before and after the
cutoff to look at the impact of schooling on earnings through QOB.
VII. Reference
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Currie, Janet, and Enrico Moretti, “Mother’s Education and the Intergenerational Transmission
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Erhardt, Carl, Frieda Nelson, and Jean Pakter, “Seasonal Patterns of Conception in New York
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Evans, William, and Edward Montgomery, “Education and Health: Where There’s Smoke
There’s an Instrument,” NBER Working Paper No. 4949, 1994.
Fieder, Martin et al., “Season of Birth Contributes to Variation in University examination
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Goldin, Claudia, and Lawrence Katz, “The Race between Education and Technology,” Belknap
Press, 2008.
24
Hahn, Jinyong, and Jerry Hausman, “Weak Instruments: Diagnosis and Cures in Empirical
Econometrics,” American Economic Review, 93 (2003), 118-125.
Hare, Edward, John Price, and Eliot Slater, “Mental Disorder and Season of Birth: A National
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Hoogerheide, Lennart, and Herman K. van Dijk, “Reconsideration of the Angrist-Krueger
Analysis on Returns to Education,” Econometric Institute report EI, 2006.
Hoogerheide, Lennart, Frank Kleibergen and Herman K. van Dijk, “Natural Conjugate Priors for
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Credit a Credit and Do Degrees Matter?,” NBER Working Paper NO. 4268, 1993.
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26
VIII. Tables and Figures
Table 1: Descriptive Statistics for Major Variables
Census 1970 &
1980 for Men
CPS 1968-1971
for Men
ACS 2005-2008
for Men
ACS 2005-2008
for Women
Log Weekly Wage
(LWKLYWGE)
5.580708
(.7621963)
5.053324
(.6826205)
6.840141
(.8277077)
6.343795
(.8713411)
Years of Education
(EDUC)
12.22273
(3.375168)
11.44902
(3.388506)
13.59271
(2.96578)
13.78735
(2.669117)
=1 if Black
(RACE)
.0818438
(.2741269)
.0901216
(.2863607)
.0811741
(.2731025)
.1054323
(.3071099)
=1 if Residence in a Central City
(SMSA)
.2359131
(.4245685)
.2851909
(.451513)
.1129815
(.3165704)
.1155282
(.3196586)
=1 if Married with Spouse Present
(MARRIED)
.875528
(.3301195)
.8831707
(.3212223)
.7063398
(.4554384)
.6537465
(.4757755)
Age with Quarter of Birth Considered
(AGEQ)
45.05477
(2.902725)
45.06899
(2.884917)
45.18298
(2.863572)
45.23961
(2.862976)
AGEQ Square
(AGEQSQ)
2038.358
(261.9626)
2039.536
(260.3241)
2049.702
(258.652)
2054.819
(258.7157)
Quarter of Birth for IV Construction
(QOB)
2.498559
(1.112412)
2.489368
(1.112368)
2.519941
(1.114882)
2.521764
(1.117048)
576708
29205
723605
694341
Variable
Number of Observations
Note: The displayed statistics are mean and standard deviation (in parenthesis). The Census samples are downloaded
from the Angrist Data Archive. Census 1970 sample is drawn from the State, County, and Neighborhoods 1 percent samples of
the 1970 Census (15 percent form). Census 1980 sample is drawn from the 5 percent sample of the 1980 Census. CPS stands for
Current Population Survey. ACS stands for American Community Survey. Both CPS and ACS samples are drawn from the
Integrated Public Use Microdata Series (IPUMS). We omit the statistics for IVs, year dummies and region dummies, and we use
40-49 age group samples for our analyses.
27
Year
Table 2: Multiyear Re-Estimates of the Return to Education
(2)
(3)
(4)
(1)
Obs
Census 1970 & 1980 for Men
1970
1980
0.0769**
(0.015)
0.0891**
(0.0161)
0.131**
(0.0334)
0.076**
(0.029)
0.0669**
(0.0151)
0.0806**
(0.0164)
0.1007**
(0.0334)
0.06**
(0.029)
247199
0.0755**
(0.0212)
0.0959**
(0.0222)
0.0803
(0.1365)
0.115
(0.2644)
0.065**
(0.0242)
0.0904**
(0.0258)
0.031
(0.0424)
0.086
(0.211)
247199
0.0488
(0.0387)
0.019
(0.0336)
0.0289
(0.0408)
0.0423
(0.0373)
0.0691**
(0.0275)
0.0576
(0.0472)
0.0204
(0.0351)
0.0307
(0.0406)
0.0528
(0.0385)
0.0702**
(0.277)
0.0501
(0.0373)
-0.0025
(0.0336)
0.0384
(0.0365)
0.0652**
(0.0308)
0.0673**
(0.255)
0.0566
(0.0463)
-0.0018
(0.0349)
0.0411
(0.0363)
0.0761**
(0.0325)
0.0681**
(0.026)
7554
0.1038**
(0.0346)
0.154**
(0.0344)
0.111**
(0.0418)
0.114**
(0.0341)
0.0911**
(0.0278)
0.0984**
(0.0348)
0.1511**
(0.0347)
0.1292**
(0.0425)
0.1214**
(0.0369)
0.0993**
(0.0281)
0.0919**
(0.0343)
0.139**
(0.0339)
0.051**
(0.0449)
0.1023**
(0.034)
0.0830**
(0.0271)
0.0844**
(0.0348)
0.1377**
(0.034)
0.0778*
(0.046)
0.1093**
(0.0365)
0.0878**
(0.0279)
181662
0.1699**
(0.0487)
0.1277**
(0.0435)
0.0869*
(0.0491)
0.0837
(0.0517)
0.0814*
(0.0483)
0.1597**
(0.05)
0.103**
(0.0488)
0.0865*
(0.0493)
0.0854
(0.0536)
0.0844*
(0.0507)
0.1716**
(0.0493)
0.128**
(0.0438)
0.0807
(0.0502)
0.0902*
(0.0517)
0.0706
(0.0489)
0.1623**
(0.0505)
0.0992**
(0.0499)
0.0824
(0.0504)
0.0938*
(0.0532)
0.0686
(0.0519)
175423
√
√
√
√
√
√
√
√
√
329509
Census 1970 & 1980 for Men (Jackknife)
1970
1980
329509
CPS 1968-1971 for Men
1968
1969
1970
1971
Pooling (Three survey year dummies added)
7480
7133
7038
29205
ACS 2005-2008 for Men
2005
2006
2007
2008
Pooling (Three survey year dummies added)
183370
180681
177892
723605
ACS 2005-2008 for Women
2005
2006
2007
2008
Pooling (Three survey year dummies added)
175916
172858
170144
694341
Control Variable
RACE, SMSA, and MARRIED
Year-of-birth dummies
Region-of-residence dummies
AGEQ and AGEQSQ
√
Note: * if p<0.1, ** if p<0.05. Standard errors are in parentheses. For sample and variable introductions see Table 1.
Instruments are a full set of quarter-of-birth times year-of-birth interactions. The dependent variable is LWKLYWGE. The
independent variable of interest is EDUC. Each equation also includes an intercept. Regressions for the CPS/ACS samples are
personal weighted.
28
Table 3: New Tests on IV Quality
Year
First-Stage (p)
1970 Census_Men
0
1980 Census_Men
0
1968 CPS_Men
0.8066
1969 CPS_Men
0.211
1970 CPS_Men
0.7663
1971 CPS_Men
0.3813
Pooling CPS_Men
0.1040
2005 ACS_Men
0.0363
2006 ACS_Men
0.0331
2007 ACS_Men
0.458
2008 ACS_Men
0.0208
Pooling ACS_Men
0.0005
2005 ACS_Women
0.2012
2006 ACS_Women
0.0574
2007 ACS_Women
0.2663
2008 ACS_Women
0.4149
Pooling ACS_Women
0.6091
Birth Quarter-RACE
Earning-Birth Quarter (p)
1st Quarter
4th Quarter
9<EDUC<13
EDUC>16
EDUC<6
0.0122
(0.0032)
0.0071
(0.0028)
-0.014
(0.0186)
-0.0151
(0.0193)
-0.0302
(0.0172)
0.0019
(0.0183)
-0.0145
(0.0092)
0.0119
(0.0051)
0.0001
(0.0044)
0.0014
(0.0051)
0.0004
(0.0044)
0.0034
(0.0023)
0.0075
(0.0044)
-0.0011
(0.004)
0.0041
(0.004)
0.0102
(0.0041)
0.0052
(0.0021)
-0.0045
(0.0031)
-0.0042
(0.0027)
0.0169
(0.0184)
0.0069
(0.0199)
0.0186
(0.0186)
0.0144
(0/0188)
0.0142
(0.0095)
-0.0058
(0.0049)
-0.0036
(0.0044)
-0.0016
(0.0045)
0.0011
(0.0045)
-0.0025
(0.0023)
0.0019
(0.0045)
0.0034
(0.0041)
0.0008
(0.0041)
-0.0033
(0.004)
0.0007
(0.0021)
0.1907
0.6708
0.5571
0.0459
0.2893
0.9192
0.6801
0.9711
0.6605
0.9521
0.2204
0.2539
0.6994
0.6672
0.5923
0.4094
0.7315
0.4472
0.6328
0.5816
0.1450
0.9711
0.3349
0.3192
0.6999
0.2775
0.1368
0.1455
0.0551
0.3622
0.8871
0.6108
0.0013
0.9569
0.1087
0.1674
0.1587
0.2562
0.567
0.1177
0.1580
0.5984
0.1981
0.6551
0.2074
0.8707
0.1802
0.6393
0.1054
0.1850
0.1483
Note: Standard errors are in parentheses. For sample and variable introductions see Table 1. For first stage, the
dependent variable is EDUC and the independent variables are the IVs plus all covariates (full model). The p values for the F test
of the 30 IVs (39 for CPS pooling and ACS pooling) are reported. For Birth Quarter- RACE, the dependent variables are the first
and fourth birth-quarter dummies, and the independent variable is RACE. Coefficients and robust standard errors (in parentheses)
are displayed. For Earning- Birth Quarter, the regressions are respectively done for subsamples of three different EDUC level.
The dependent variable is LWKLYWGE and independent variables include three birth-quarter dummies and the nine year-ofbirth dummies (also the year dummies for CPS pooling and ACS pooling). The p values for the F test of quarter dummies are
reported. Regressions for the CPS/ACS samples are personal weighted. Boldface is used for results with p value smaller than 0.2.
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