{1, . . . , T }, k ∈ {C, N }, which may depend on the X .
3.2 Identification of the Factor Loadings and of the Joint Distri-
butions of the Latent Variables
We first establish identification of the factor loadings under the assumption that the εa,k,t,j
are uncorrelated across t and that the analyst has at least two measures of each type of child
skills and investments in each period t, where T ≥ 2. Without loss of generality, we focus on
α1,C,t,j and note that similar expressions can be derived for the loadings of the other latent
factors.
Since Z1,C,t,1 and Z1,C,t+1,1 are observed, we can compute Cov (Z1,C,t,1 , Z1,C,t+1,1) from the
data. Because of the normalization α1,C,t,1 = 1 for all t, we obtain:
Cov (Z1,C,t,1 , Z1,C,t+1,1) = Cov (θC,t, θC,t+1) .
(3.4)
In addition, we can compute the covariance of the second measurement on cognitive skills
at period t with the first measurement on cognitive skills at period t + 1:
(3.5)
Cov (Z1,C,t,2, Z1,C,t+1,1) = α1,C,t,2Cov (θC,t, θC,t+1) .
If Cov (θC,t, θC,t+1) 6= 0, one can identify the loading α1,C,t,2 from the following ratio of
covariances:
Cov (Z1,C,t,2, Z1,C,t+1,1) _
Cov (Zι,c,t,ι, Zι,c,t+I,ι) 1,c,t,2
If there are more than two measures of cognitive skill in each period t, we can identify α1,C,t,j
for j ∈ {2, 3, . . . , M1,C,t}, t ∈ {1, . . . , T} up to the normalization α1,C,t,1 = 1. The assumption
that the εa,k,t,j are uncorrelated across t is then no longer necessary. Replacing Z1,C,t+1,1 by
Za0,k0,t0,3 for some (a0, k0, t0) which may or may not be equal to (1, C, t), we may proceed in
the same fashion.15 Note that the same third measurement Za0,k0,t0,3 can be reused for all a, t
and k implying that in the presence of serial correlation, the total number of measurements
needed for identification of the factor loadings is 2L + 1 if there are L factors.
15The idea is to write
α1,c,t,2αa0,k0,t0,3C0v (θc,t, θk0,t0) _ α1,c,t,2 __
Cov (Z1,C,t,2, Za0,k0,t0,3)
Cov (Z1,C,t,1, Za0,k0,t0,3)
α1,C,t,1αa' ,k0,t0,3Cov (θC,t ,θk0,t0 ) α1,C,t,1 , ,,
This only requires uncorrelatedness across different j but not across t.
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