Estimating the Technology of Cognitive and Noncognitive Skill Formation

We identify technology (2.1) using the relationship

fk,s (θt, Ik,t, θp,ηk,t) = G-¹ (ηk,t | θt, Ik,t, θp),

where G^-1 (η_k,_t | θ_t,I_k,_t,θ_p) denotes the inverse of G (θ | θ_t, I_k,_t,θ_p) with respect to its first
argument (assuming it exists), i.e., the value θ such that η_k,_t = G (θ | θ_t, I_k,_t,θ_p). By con-
struction, this operation produces a function f_k,s that generates outcomes θ_k,t+1 with the
appropriate distribution, because a continuously distributed random variable is mapped into
a uniformly distributed variable under the mapping defined by its own cdf.

The more traditional separable technology with zero mean disturbance, θ_k,t+1

= fk,s (θt, Ik,t, θp) + ηk,t, is covered by our analysis if we define

fk,s (θt,Ik,t,θp) ≡ E [θk,t+ι | θt,Ik,t,θp] ,

where the expectation is taken under the density p_θk _t+1∣_θt,I_{k t},_θP, which can be calculated from
pθ . The density of ηk,t conditional on all variables is identified from

^pnk,t∣θt,ik,t,θp (^ηk,t | θt^, Ik,t^{, θ}P) = ^pθk,t+ι∣θt,ik,t,θp (^ηk,t ⁺ E [θk,t+ι | θt^, Ik,t^{, θ}P^] | θt^, Ik,t^{, θ}P) ^,

since pθ_{k t+1}∣_θt,I_{k t},_θP is known once p_θ is known. We now show how to anchor the scales of
θ_C,t+1 and θ_N,t+1 using measures of adult outcomes.

3.5 Anchoring Skills in an Interpretable Metric

It is common in the empirical literature on child schooling and investment to measure out-
comes by test scores. However, test scores are arbitrarily scaled. To gain a better under-
standing of the relative importance of cognitive and noncognitive skills and their interactions
and the relative importance of investments at different stages of the life cycle, it is desirable
to anchor skills in a common scale. In what follows, we continue to keep the conditioning on
the regressors implicit.

We model the effect of period T + 1 cognitive and noncognitive skills on adult outcomes
Z4,j , for j ∈ {1, . . . , J}.²³ Suppose that there are J1 observed outcomes that are linear
functions of cognitive and noncognitive skills at the end of childhood, i.e., in period T :

Z4,j = μ4,j ^{+ α}4,C,j^θC,T +1 ^{+ α}4,N,j^θN,T +1 ^{+ ε}4,j^{, for j} ∈ {^1, . . . ^, J1}.

When adult outcomes are linear and separable functions of skills, we define the anchoring
²³The Z4,j correspond to the Qj of Section 2.

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