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 α4,C,j , α4,N,j , and α4,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 (θt, It, θP) and νk,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 yt 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}tT=1 are available to
identify their (joint) distribution. In our application, we assume that yt is measured without
error25 We further assume that the error term ηk,t can be decomposed into two components:
(πt , νk,t) so that we may write the technology as
θk,t+1 = fk,s (θt, Ik,t, θP , πt, νk,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 νk,t is independent of θt, Ik,t, θP , yt and independent of νk,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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