Estimating the Technology of Cognitive and Noncognitive Skill Formation

{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 Z_1,C,t,1 and Z_1,C,t+1,1 are observed, we can compute Cov (Z_1,C,t,1 , Z_1,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) .

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:

Cov (Z1,C,t,2, Z1,C,t+1,1) = α1,C,t,2Cov (θC,t, θC,t+1) .

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 α_1,C,t,j
for j ∈ {2, 3, . . . , M_1,C,t}, t ∈ {1, . . . , T} up to the normalization α_1,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 Z_a0_,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 __

^α1,C,t,1^αa' ,k⁰,t⁰,3C^ov (^θC,t ^,θk⁰,t⁰ )     ^α1,C,t,1         , ^,,

This only requires uncorrelatedness across different j but not across t.

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