Estimating the Technology of Cognitive and Noncognitive Skill Formation

outcomes available in the CNLSY data. They solve several problems in estimating skill
formation technologies. First, there are many proxies for parental investments in children’s
cognitive and noncognitive development. Using a dynamic factor model, we let the data
pick the best combinations of family input measures that predict levels and growth in test
scores. Measured inputs that are not very informative on family investment decisions will
have negligible estimated factor loadings. Second, our models help us solve the problem of
missing data. Assuming that the data are missing at random, we integrate out the missing
items from the sample likelihood.

In practice, we cannot empirically distinguish investments in cognitive skills from invest-
ments in noncognitive skills. Accordingly, we assume investment in period t is the same for
both skills although it may have different effects on those skills. Thus we assume IC,t = IN,t
and define it as I_t .

4.1 Empirical Specification

We use the separable measurement system (3.1). We estimate versions of the technology
(2.3)-(2.4) augmented to include shocks:

1

φs,k             φs,k             φs,k             φs,k             φs,k ^φs,k η_{k t+1}

θk,t+1 = γs,k,1θC,t + γs,k,2θN,t + γs,k,3It + γs,k,4θC,P + γs,k,5θN,P e ^,   ,      (4.1)

where γs,k,l ≥ 0 and ^P_l⁵₌₁ γs,k,l = 1, k ∈ {C, N}, t ∈ {1, 2}, s ∈ {1, 2}. We assume that
the innovations are normally distributed: η_k,_t ~ N (θ, δ2,_s). We further assume that the ηk,_t
are serially independent over all t and are independent of n`,t for k = '. We assume that
measurements Za,k,t,j proxy the natural logarithms of the factors. In the text, we report only
anchored results.³⁰ For example, for a = 1,

Z1,k,t,j = ^μ1,k,t,j ^{+ α}1,k,t,j ^ln θk,t ^{+ ε}1,k,t,j

j∈{1,...,Ma,k,t},t∈{1,...,T},k∈{C,N}.

We use the factors (and not their logarithms) as arguments of the technology.³¹ This keeps
the latent factors non-negative, as is required for the definition of technology (4.1). Collect
the ε terms for period t into a vector ε_t. We assume that ε_t ~ N (θ, Λ_t), where Λ_t is a
diagonal matrix. We impose the condition that εt is independent from εt0 for t 6= t⁰ and all

³⁰Web Appendix 11.1 compares anchored and unanchored results.

³¹We use five regressors (X) for every measurement equation: a constant, the age of the child at the
assessment date, the child’s gender, a dummy variable if the mother was less than 20 years-old at the time
of the first birth, and a cohort dummy (one if the child was born after 1987 and zero otherwise).

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