tablish identification of the technology of skill formation. We relax the strong independence
assumptions for error terms in the measurement equations that are maintained in Cunha and
Heckman (2008) and Carneiro, Hansen, and Heckman (2003). The assumption of linearity of
the technology in inputs that is used by Cunha and Heckman (2008) and Todd and Wolpin
(2003, 2005) is not required because we allow inputs to interact in producing outputs. We
generalize the factor-analytic index function models used by Carneiro, Hansen, and Heckman
(2003) to allow for more general functional forms for measurement equations. We solve the
problem of defining a scale for the output of childhood investments by anchoring test scores
using adult outcomes of the child, which have a well-defined cardinal scale. We determine
the latent variables that generate test scores by estimating how these latent variables pre-
dict adult outcomes.6 Our approach sets the scale of test scores and latent variables in an
interpretable metric. Using this metric, analysts can meaningfully interpret changes in out-
put and conduct interpretable value-added analyses.7 We also solve the problem of missing
inputs in estimating technologies in a way that is much more general than the widely used
framework of Olley and Pakes (1996) that assumes perfect proxies for latent factors. We
allow for imperfect proxies and establish that measurement error is substantial in the data
analyzed in this paper.
The plan of this paper is as follows. Section 2 briefly summarizes the previous literature
to motivate our contribution to it. Section 3 presents our identification analysis. Section 4
discusses the data used to estimate the model, our estimation strategy, and the model esti-
mates. Section 5 concludes.
2 A Model of Cognitive and Noncognitive Skill For-
mation
We analyze a model with multiple periods of childhood, t ∈ {1, 2, . . . , T }, T ≥ 2, followed
by A periods of adult working life, t ∈ {T + 1, T + 2, . . . , T + A}. The T childhood periods
are divided into S stages of development, s ∈ {1, . . . , S}, with S ≤ T. Adult outcomes are
produced by cognitive skills, θC,T +1 , and noncognitive skills, θN,T+1 at the beginning of the
adult years.8 Denote parental investments at age t in child skill k by Ik,t, k ∈ {C, N}.
6Cawley, Heckman, and Vytlacil (1999) anchor test scores in earnings outcomes.
7Cunha and Heckman (2008) develop a class of anchoring functions invariant to affine transformations.
This paper develops a more general class of monotonic transformations and presents a new analysis of joint
identification of the anchoring equations and the technology of skill formation.
8This model generalizes the model of Becker and Tomes (1986), who assume only one period of childhood
(T = 1) and consider one output associated with “human capital” that can be interpreted as a composite of
cognitive (C) and noncognitive (N) skills. We do not model post-childhood investment.