Estimation of Educational Borrowing Constraints Using Returns to ...

educational borrowing constraints 169
and
′ ′ ′ ′ ′
X iG 2 { XTibT1 a 1(XWibW XL0ib L0) a 2 a 1 XTib T 0, (29)
˜
e { a log (R ) n av n , (30)
2i 3 i 1i 1 0i 0i
so that G1and G2are defined appropriately. Individuals choose schooling
option 1 if V1i V0i 1 0, and option 0 otherwise. Note that e1i
enters the
expression inside log (7) in equation (26) and is correlated with the
residual e2i both through the correlation between v0i and v1i
and because
log (R i ) enters both residuals.
The vector (X i, X Ci) is observable to the analyst but (v 1i, v 0i,
log (R i), n 1i, n 0i) is not. We assume that (X i, X Ci)
is independent of
(v 1i, v 0i, log(R i), n 1i, n 0i). The selection equation (26) has three indices
′ ′ ′
(X iG 1, X iG 2, X Cib C) and two error terms (e 1i, e 2i)
. The three degrees of
freedom represented by these index functions are sufficient to trace out
the joint distribution of (e 1i, e 2i)
.
Equations (28) and (30) show that these indices cannot also identify
the distribution of log (R i)
from the distribution of v1i. However, since
wages depend on v1i
but not on R i, the wage equation given by (18)
provides sufficient variation to trace out the distribution of log (R i ) up
to a location normalization.
Theorem 1 contained in the appendix (available at http://www
.econ.northwestern.edu/faculty/taber) proves nonparametric identification
of the joint distribution of the error terms from the schooling
choice and wage equations, (e 1i, e 2i, u 1it)
, using two exclusion restrictions,
one for opportunity costs and one for direct costs. 31 Once this joint
distribution is identified, we can identify the conditional expectation
E(u 1itFe 1i) p E(v1iFv 1i log (R i)).
The appendix further demonstrates that this conditional expectation
can be used to estimate the distribution of Ri
when Ri
is independent
of v 1i . Thus the entire distribution of interest rates faced by students is
identified up to a normalization.
Intuition for this result follows from the fact that the form of the
selection bias
′ ′ ′
E(v1iFa 1 log {exp [X iG1 v1i log (R i)] X Cib C} X iG2 e2i
1 0,
′ ′ ′
X iG 1, X Cib C, X iG 2) (31)
depends on the realized value of college costs. This expression is complicated
and depends on the full distribution of (v 1i, R i, e 2i)
. Under the
same logic outlined in Section III, the term v1i log (R i)
is more im-
31 ′
With the notation used above, the first type of exclusion restriction affects X i G 2 , but
′ ′ ′ ′ ′
not X G or X b ; the second enters X b but not X G or X G .
i 1 Ci C Ci C i 2 i 1