Estimating the Technology of Cognitive and Noncognitive Skill Formation

where M ≥ 3 and where the indicator Zj is observed while the latent factor θ and the distur-
bance εj are not. The variables Zj , θ, and εj are assumed to be vectors of the same dimension.
In our application, the vector of observed indicators and corresponding disturbances is

Zj =    {Z1,C,t,j }t^T=1 , {Z1,N,t,j }t^T=1 , {Z2,C,t,j }t^T=1 , {Z2,N,t,j }t^T=1 , Z3,C,1,j , Z3,N,1,j

εj =     {ε1,C,t,j}t^T=1 , {ε1,N,t,j}t^T=1 , {ε2,C,t,j}t^T=1 , {ε2,N,t,j }t^T=1 , ε3,C,1,j , ε3,C,N,1,j
while the vector of unobserved latent factors is:

θ=  {θC,t}_t^T₌₁ , {θN,t}_t^T₌₁ , {IC,t}_t^T₌₁ , {IN,t}_t^T₌₁ , θC,P , θN,P   .

The functions α_j∙ (∙, ∙) for j ∈ {1,..., M} in Equations (3.7) are unknown. It is necessary to
normalize one of them (e.g., a₁ (∙, ∙)) in some way to achieve identification, as established in
the following theorem.

Theorem 2 The distribution of θ in Equations (3.7) is identified under the following con-
ditions:

1. The joint density of θ, Z₁ , Z₂, Z₃ is bounded and so are all their marginal and condi-
tional densities.¹⁷

2. Z1 , Z2 , Z3 are mutually independent conditional on θ.

3. PZ₁∣Z₂ (Z1 | Z₂) and pθ∣z₁ (θ | Z1) form a bounded complete family of distributions in-
dexed by Z₂ and Z₁, respectively.

4. Whenever θ = θ, pZ₃∣θ (Z₃ | θ) and pZ₃∣θ ZZ₃ | θ^ differ over a set of strictly positive
probability.

5. There exists a known functional Ψ, mapping a density to a vector, that has the property
that Ψ [pz₁∣θ (∙ | θ)] = θ.

Proof. See Web Appendix, Part 3.2.¹⁸

The proof of Theorem 2 proceeds by casting the analysis of identification as a linear
algebra problem analogous to matrix diagonalization. In contrast to the standard matrix

¹⁷This is a density with respect to the product measure of the Lebesgue measure on R^L × R^L × R^L and
some dominating measure μ. Hence θ, Z₁, Z₂ must be continuously distributed while Z₃ may be continuous
or discrete.

¹⁸ A vector of correctly measured variables C can trivially be added to the model by including C in the
list of conditioning variables for all densities in the statement of the theorem. Theorem 2 then implies that
Pθ∣C(0|C) is identified. Since p_c(C) is identified it follows that pθ,C(θ, C) = pθ∣_c^|C)p_c(C) is also identified.

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