Estimating the Technology of Cognitive and Noncognitive Skill Formation

these features of the model does not prevent identification of the distribution of θ.

Nevertheless, various normalizations ensuring that the functions aj (θ, εj) are fully iden-
tified are available. For example, if each element of εj is normalized to be uniform (or any
other known distribution), the aj (θ, εj) are fully identified. Other normalizations discussed
in Matzkin (2003, 2007) are also possible. Alternatively, one may assume that the aj (θ, εj)
are separable in εj with zero conditional mean of εj given θ.²¹ We invoke these assumptions
when we identify the policy function for investments in Section 3.6.2 below.

The conditions justifying Theorems 1 and 2 are not nested within each other. Their dif-
ferent assumptions represent different trade-offs best suited for different applications. While
Theorem 1 would suffice for the empirical analysis of this paper, the general result established
in Theorem 2 will likely be quite useful as larger sample sizes become available.

Carneiro, Hansen, and Heckman (2003) present an analysis for nonseparable measurement
equations based on a separable latent index structure, but invoke strong independence and
“identification-at-infinity” assumptions. Our approach for identifying the distribution of θ
from general nonseparable measurement equations does not require these strong assumptions.
Note that it also allows the θ to determine all measurements and for the θ to be freely
correlated.

3.4 Nonparametric Identification of the Technology Function

Suppose that the shocks η_k,t are independent over time. Below, we analyze a more general
case that allows for serial dependence. Once the density of θ is known, one can identify
nonseparable technology function (2.1) for t ∈ {1, . . . , T}; k ∈ {C, N}; and s ∈ {1, ...., S}.

Even if (θ_t, I_t, θ_P) were perfectly observed, one could not separately identify the distribution
of η_k,t and the function f_k,s because, without further normalizations, a change in the density
of ηk,t can be undone by a change in the function fk,s.²²

One solution to this problem is to assume that (2.1) is additively separable in η_k,t. An-
other way to avoid this ambiguity is to normalize η_k,t to have a uniform density on [0, 1].
Any of the normalizations suggested by Matzkin (2003, 2007) could be used. Assuming ηk,t
is uniform [0, 1] , we establish that fk,s is nonparametrically identified, by noting that, from
the knowledge of pθ we can calculate, for any θ ∈ R,

Pr [θk,t+ι ≤ θ∣θt, Ik,t, θp] ≡ G (⅜, Ik,t, θp) .

²¹Observe that Theorem 2 covers the identifiability of the outcome (Qj) functions (2.2) even if we supple-
ment the model with errors εj , j ∈ {1, . . . , J} that satisfy the conditions of the theorem.

²²See, e.g, Matzkin (2003, 2007).

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