Estimation of Educational Borrowing Constraints Using Returns to ...

educational borrowing constraints 175
insensitive to the number of children in the family (number of siblings).
Once again, confidence intervals are tight, indicating that the null is
rejected because of small coefficients, not lack of power.
D. Evidence When R i Is Not Observed
Failing to find evidence of heterogeneity in borrowing rates in observables,
we extend the structural model to capture borrowing rate heterogeneity
in unobservables. We assume that individuals come in one
of two types: those borrowing at the market rate 1/d and those who are
“constrained” and borrow at rate R c 1 1/d. We estimate the borrowing
rate R c and the fraction of the population that borrows at the higher
rate, denoted as P c . The distribution of borrowing rates is restricted to
be independent of the observables and other error terms.
Unobserved ability enters the model through a single standard normal
factor v i , which is known by the agent at the time schooling choices are
made and is independent of observables. The error terms in the wage
equations are defined as
u p f v q , (32)
Sit WS i Sit
where fWS
is the wage equation factor loading term, and qSit
is inde-
pendently and identically distributed over time and is orthogonal to
v i for all t. Ability may also be correlated with unobserved tastes for
schooling. The taste residual for school level S is now defined as
˜
n p f v n , (33)
Si TS i Si
where f TS is the tastes equation factor loading for school level S, and
ñ Si has a GEV distribution, which yields the nested logit model from the
previous section.
A major goal of the approach in this subsection is to identify the
extent of borrowing constraints from economic influences regulating
the behavioral interaction between interest rates and schooling costs
and not from functional form assumptions on the distribution of the
error terms. This goal led to two nonstandard estimation strategies. First,
because we have no presumption about the form of the selection bias,
we retain flexibility in its modeling. Ideally, we would place no restrictions
on the joint distribution of the taste and wage errors (v 2i, v 4i,
v , u , u , u , u ). However, full nonparametric estimation of this
6i 0it 2it 4it 6it
distribution is not computationally feasible. Even after we restrict the
distribution of the error terms to specific functional forms with a onefactor
representation, there are still seven factor loading terms on v i
that determine the form of the selection bias. We purposely overparameterize
the selection bias function in order to force identification from