Estimating the Technology of Cognitive and Noncognitive Skill Formation

heterogeneity.

To see how this can be done, suppose that we observe at least three adult outcomes, so
that J ≥ 3. We can then write outcomes as functions of T + 1 skills as well as unobserved
(by the economist) time-invariant heterogeneity component, π , on which parents make their
investment decisions:

Z4,j = α4,C,j θC,T +1 + α4,N,jθN,T+1 + α4,π,j π + ε4,j, for j ∈ {1, 2, . . . , J}.

We can use the analysis of section 3.2, suitably extended to allow for measurements Z4,j ,
to secure identification of the factor loadings α_4,C,j , α_4,N,j , and α_4,π,j . We can apply the
argument of section 3.4 to secure identification of the joint distribution of (θt, It, θP , π).²⁴
Write ηk,t = (π, νk,t). Extending the preceding analysis, we can identify a more general
version of the technology:

θk,t+1 = fk,s (θt, Ik,t, θP , π, νk,t) .

π is permitted to be correlated with the inputs (θ_t, I_t, θ_P) and ν_k,t is assumed to be indepen-
dent from the vector (θt, It, θP , π) as well as νl,t for l 6= k. The next subsection develops a
more general approach that allows π to vary over time.

3.6.2 More General Forms of Endogeneity

This subsection relaxes the invariant heterogeneity assumption by using exclusion restrictions
based on economic theory to identify the technology under more general conditions. πt
evolves over time and agents make investment decisions based on it. Define y_t as family
resources in period t (e.g., income, assets, constraints). As in Sections 3.2 and 3.3, we
assume that suitable multiple measurements of θP , {θt , IC,t, IN,t , yt}_t^T₌₁ are available to
identify their (joint) distribution. In our application, we assume that y_t is measured without
error²⁵ We further assume that the error term η_k,t can be decomposed into two components:
(πt , νk,t) so that we may write the technology as

θ_k,t+1 = f_k,s (θ_t, I_k,t, θ_P , π_t, ν_k,t) .                                (3.10)

πt is assumed to be a scalar shock independent over people but not over time. It is a
common shock that affects all technologies, but its effect may differ across technologies.
The component ν_k,t is independent of θ_t, I_k,t, θ_P , y_t and independent of ν_k,t0 for t⁰ 6= t. Its
realization takes place at the end of period t, after investment choices have already been

²⁴We discuss the identification of the factor loadings in this case in Web Appendix 4.

²⁵Thus the “multiple measurements”on yt are all equal to each other in each period t.

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