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 }tT=1 , {Z1,N,t,j }tT=1 , {Z2,C,t,j }tT=1 , {Z2,N,t,j }tT=1 , Z3,C,1,j , Z3,N,1,j
εj = {ε1,C,t,j}tT=1 , {ε1,N,t,j}tT=1 , {ε2,C,t,j}tT=1 , {ε2,N,t,j }tT=1 , ε3,C,1,j , ε3,C,N,1,j
while the vector of unobserved latent factors is:
θ= {θC,t}tT=1 , {θN,t}tT=1 , {IC,t}tT=1 , {IN,t}tT=1 , θC,P , θN,P .
The functions αj∙ (∙, ∙) 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 RL × RL × RL 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 pc(C) is identified it follows that pθ,C(θ, C) = pθ∣c^|C)pc(C) is also identified.
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