Estimating the Technology of Cognitive and Noncognitive Skill Formation

indirect effect through πt = q_k^-_,t¹ (θt, θP , yt, Ik,t). To accomplish this, we exploit our knowledge
of q_k^-_,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)^ly_t.q^-1 (θ_t,θ_p ,y_t,I_k,_t)=π_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 q_k^-_,t¹ (θ_t, θ_P, y_t, I_k,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 y_t conditional on θ_t , θ_P , I_k,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 f_k,s and q_k,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 π_t , 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 (θt,θP,yt,πt) ,θP,πt, νk,t) .

Eliminating I_k,t solves the endogeneity problem because the two disturbances π_t and ν_k,t are
now independent of all explanatory variables, by assumption if the π_t 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 fπ_t (πt), the
density of π_t . We can then write:

E [θk,t+ι∖θt^,θP^,yt] = У ^fk,s (θt^,qk,t (θt^,θp^,yt^,πt) ^,θp^,πt^, 0) ^f∏t (^πt) d^πt ≡ ^fk,s (θt^,θp^,yt^,β) .

The right-hand is now known up to a vector of parameters β which will be (at least) locally
identified if it happens that ∂f_k,_s (θ_t,θP,y_t,β) /∂β evaluated at the true value of β is a
vector function of θ_t, θ_P , y_t 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 .

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