Estimation of Educational Borrowing Constraints Using Returns to ...

educational borrowing constraints 167
and
′ ′ ′
V1i p a1 log {exp [ log (R i) XWibW XL1ibL1 v 1i] X Cib C}
′
a 2 a 3log (R i) XTibT1 n 1i, (25)
where a1, a2, and a3
are scalars (as defined in [23]) that do not vary
across individuals.
Though (24) and (25) are complicated, these expressions are close
to a familiar linear index model. A nonlinearity arises inside the logarithm
in the first term of (25) and captures the interaction between the
′
borrowing rate R i and schooling costs X CibC. No such interaction exists
between interest rates and forgone earnings, which are represented by
′ ′
X ˜
WibW XL0ibL0 v0i. This feature of the model delivers identification
of the parameters of interest.
We estimate two versions of the model. The first restricts variation in
R i to be determined through particular sets of observables, such as race
or family income. The second version treats R i as a variable known to
the individual but not observed by the econometrician. Identification
of each case is discussed separately.
˜
1. Case I: R i Determined by Observed Characteristics
The first approach assumes the absence of persistent, unobserved individual
influences on wages ( vSi
p 0) and assumes that R i varies only
with observables. The empirical content of this version of the model is
closely related to the regression approach presented in Section VF. To
see why, suppose that local labor market and wage variables are not
′ ′
determinants of schooling value ( X ˜
WibW XLSibSL
p 0 for both S p 0
′
and S p 1). Let R i be parameterized by the index XRibR, and assume
′
that the schooling costs index, X CibC, varies only with the presence of
a local college. Equation (25) shows that log (R i ) enters the value func-
′
tion through the terms log {exp [ log (R i)] X Cib C} and a 3 log (R i)
.
Since the observable variables that determine R i are also included in
the set of observables governing tastes, X Ti , and since they are all dummy
variables (such as race), a 3 log (R i)
is not separately identified from
′
XTibT1. Therefore, identification of log (R i)
comes completely from the
′
term log {exp [ log (R i)] X Cib C}
. In addition, since X Ci contains only
a constant and an indicator variable, testing for interest rate heterogeneity
(i.e., bR
p 0) is identical to testing for interactions between the
presence of a local college and the variables determining the borrowing
rate (such as race) and almost identical to tests for the ad hoc interactions
estimated in the schooling regressions of table 7. The main
difference is that the schooling outcome is modeled here as a discrete
variable.