Once the parameters α1,C,t,j are identified, we can rewrite (3.1), assuming α1,C,t,j 6= 0, as:
Z = μ + θc t + ε1ctj, j ∈ {1, 2,..., Mι,c,t}. (3.6)
α1,C,t,j α1,C,t,j α1,C,t,j
In this form, it is clear that the known quantities ZCt’j play the role of repeated error-
contaminated measurements of θC,t. Collecting results for all t = 1, . . . , T , we can identify the
joint distribution of {θC,t}tT=1 . Proceeding in a similar fashion for all types of measurements,
a ∈ {1, 2, 3}, on abilities k ∈ {C, N}, using the analysis in Schennach (2004a,b), we can
identify the joint distribution of all the latent variables. Define the matrix of latent variables
by θ , where
θ = {θC,t}tT=1 , {θN,t}tT=1 , {IC,t}tT=1 , {IN,t}tT=1 , θC,P , θN,P .
Thus, we can identify the joint distribution of θ, p(θ).
Although the availability of numerous indicators for each latent factor is helpful in im-
proving the efficiency of the estimation procedure, the identification of the model can be
secured (after the factor loadings are determined) if only two measurements of each latent
factor are available. Since in our empirical analysis we have at least two different measure-
ments for each latent factor, we can define, without loss of generality, the following two
vectors
Wi
(ʃ Z1,C,t,i
α1,C,t,i
i∈{1,2}.
T
t=1
Z1,N,t,i
α1,N,t,i
T
t=1
Z2,C,t,i
α2,C,t,i
T
t=1
Z2,N,t,i
α2,N,t,i
T
t=1
0
Z3,C,1,i Z3,N,1,i ʌ
, I
α3,C,1,i α3,N,1,i
These vectors consist of the first and the second measurements for each factor, respectively.
The corresponding measurement errors are
|
TTTT 0 ε1,C,t,i ε1,N,t,i ε2,C,t,i ε2,N,t,i ε3,C,1,i ε3,N,1,i | |
|
ωi |
Ii f,∣ f,∣ f,∣ I , , I α1,C,t,i t=1 α1,N,t,i t=1 α2,C,t,i t=1 α2,N,t,i t=1 α3,C,1,i α3,N,1,i i∈{1,2}. |
Identification of the distribution of θ is obtained from the following theorem. Let L
denote the total number of latent factors, which in our case is 4T + 2.
Theorem 1 Let W1, W2, θ, ω1, ω2 be random vectors taking values in RL and related through
W1 = θ + ω1
W2 = θ + ω2 .
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