Estimating the Technology of Cognitive and Noncognitive Skill Formation



indirect effect through πt = qk-,t1 (θt, θP , yt, Ik,t). To accomplish this, we exploit our knowledge
of
qk-,t1 (θt, θP , yt , Ik,t) to write:

fk,s (θt, Ik,t,θP ,πt,νk,t) = fkfs (θt,Ik,t,θP ,yt,νk,t)lyt.q-1 (θtp ,yt,Ik,t)t

where, on the right-hand side, we set yt so that the corresponding implied value of πt matches
its value on the left-hand side. This does not necessarily require
qk-,t1 (θt, θP, yt, Ik,t) to be
invertible with respect to
yt , since we only need one suitable value of yt for each given
(
θt , θP , Ik,t , πt) and do not necessarily require a one-to-one mapping. By construction, the
support of the distribution of
yt conditional on θt , θP , Ik,t , is sufficiently large to guarantee
the existence of at least one solution because, for a fixed
θt , Ik,t , θP , variations in πt are
entirely due to
yt . We present a more formal discussion of our identification strategy in
Section 3.3 of the Web appendix.

In our empirical analysis, we make further parametric assumptions regarding fk,s and qk,t ,
which open the way to a more convenient estimation strategy to account for endogeneity.
The idea is to assume that the function
qk,t (θt, θP, yt, πt) is parametrically specified and
additively separable in
πt , so that its identification follows under standard instrumental
variables conditions. Next, we replace
Ik,t by its value given by the policy function in the
technology

θk,t+1 = fk,s (θt, qk,t (θtP,ytt) Pt, νk,t) .

Eliminating Ik,t solves the endogeneity problem because the two disturbances πt and νk,t are
now independent of all explanatory variables, by assumption if the
πt are serially indepen-
dent. Identification is secured by assuming that
fk,s is parametric and additively separable
in
νk,t (whose conditional mean is zero) and by assuming a parametric form for fπt (πt), the
density of
πt . We can then write:

E [θk,t+ιθtP,yt] = У fk,s (θt,qk,t (θt,θp,ytt) ,θpt, 0) ft (πt) dπt fk,s (θt,θp,yt) .

The right-hand is now known up to a vector of parameters β which will be (at least) locally
identified if it happens that
∂fk,s (θtP,yt) /∂β evaluated at the true value of β is a
vector function of
θt, θP , yt that is linearly independent. Section 4.2.5 below describes the
specific functional forms used in our empirical analysis, and relaxes the assumption of serial
independence of the
πt .

21



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