This paper tests the hypothesis that compulsory school attendance
laws, which typically require school attendance until a specified birthday,
induce a relationship between years of schooling and age at school entry.
Variation in school start age created by children's date of birth provides
a natural experiment for estimation of the effect of age at school entry.
Because no large data set contains information on both age at school entry
and educational attainment, we use an Instrumental Variables (IV) estimator
with data derived from the 1960 and 1980 Censuses to test the age-at-entry/compulsory schooling model. In most IV applications, the two
covariance matrices that form the estimator are constructed from the same
sample. We use a method of moments framework to discuss IV estimators that
combine moments from different data sets. In our application, quarter of
birth dummies are the instrumental variables used to link the 1960 Census,
from which age at school entry can be derived for one cohort of students,
to the 1980 Census, which contains educational attainment for the same
cohort of students. The results suggest that roughly l0 percent of
students were constrained to stay in school by compulsory schooling laws.

This paper presents evidence showing that individuals’ season of birth is
related to their educational attainment because of the combined effects of
school start age policy and compulsory school attendance laws. In most school
districts, individuals born in the beginning of the year start school at a
slightly older age, and therefore are eligible to drop out of school after
completing fewer years of schooling than individuals born near the end of the
year. Our estimates suggest that as many as 25 percent of potential dropouts
remain in school because of compulsory schooling laws. We estimate the impact
of compulsory schooling on earnings by using quarter of birth as an
instrumental variable for education in an earnings equation. This provides a
valid identification strategy because date of birth is unlikely to be
correlated with omitted earnings determinants. The instrumental variables
estimate of the rate of return to education is remarkably close to the
ordinary least squares estimate, suggesting that there is little ability bias
in conventional estimates of the return to education. The results also imply
that individuals who are compelled to attend school longer than they desire by
compulsory schooling laws reap a substantial return for their extra schooling.

Instrumental Variables (IV) estimates tend to be biased in the same direction as
Ordinary Least Squares (OLS) in finite samples if the instruments are weak. To address
this problem we propose a new IV estimator which we call Split Sample Instrumental
Variables (SSIV). SSIV works as follows: we randomly split the sample in half, and use
one half of the sample to estimate parameters of the first-stage equation. We then use these
estimated first-stage parameters to construct fitted values and second-stage parameter
estimates using data from the other half sample. SSIV is biased toward zero, rather than
toward the plim of the OLS estimate. However, an unbiased estimate of the attenuation
bias of SSIV can be calculated. We use this estimate of the attenuation bias to derive an
estimator that is asymptotically unbiased as the number of instruments tends to infinity,
holding the number of observations per instrument fixed. We label this new estimator
Unbiased Split Sample Instrumental Variables (USSIV). We apply SSIV and USSIV to the
data used by Angrist and Krueger (1991) to estimate the payoff to education.