Estimating the Technology of Cognitive and Noncognitive Skill Formation

made and implemented. The shock πt is realized before parents make investment choices, so
we expect Ik,t to respond to it.

We analyze a model of investment of the form

Ik,t=qk,t(θt,θP,yt,πt), k∈ {C,N},t∈ {1,...,T}.                 (3.11)

Equation (3.11) is the investment policy function that maps state variables for the parents,
(θt, θP , yt, πt), to the control variables Ik,t for k ∈ {C, N}.²⁶

Our analysis relies on the assumption that the disturbances πt and νk,t in Equation (3.10)
are both scalar, although all other variables may be vector-valued. If the disturbances π_t are
i.i.d., identification is straightforward. To see this, impose an innocuous normalization (e.g.,
assume a specific marginal distribution for πt). Then, the relationship Ik,t = qk,t (θt, θP , yt, πt)
can be identified along the lines of the argument of Section 3.2 or 3.3, provided, for instance,
that π_t is independent from (θ_t, θ_P , y_t).

If πt is serially correlated, it is not plausible to assume independence between πt and θt ,
because past values of πt will have an impact on both current πt and on current θt (via the
effect of past π_t on past I_k,t). To address this problem, lagged values of income y_t can be used
as instruments for θt (θP and yt could serve as their own instruments). This approach works
if πt is independent of θP as well as past and present values of yt . After normalization of
the distribution of the disturbance π_t , the general nonseparable function q_t can be identified
using quantile instrumental variable techniques (Chernozhukov et al., 2007), under standard
assumptions in that literature, including monotonicity and completeness.²⁷

Once the functions q_k,t have been identified, one can obtain q_k^-_,t¹ (θ_t, θ_P , y_t , I_k,t), the inverse
of q_k,t (θ_t, θ_P , y_t, π_t) with respect to its last argument, provided q_k,t (θ_t, θ_P , y_t, π_t) is strictly
monotone in πt at all values of the arguments. We can then rewrite the technology function
(3.11) as:

^θk,t+ι = ^fk,s (θt,Ik,t^,θP^,q-1 (θt^,θp^,yt^, Ik,t) ^,νk,t} ≡ ^ff (θt^, Ik,t^,θP^,yt^,νk,t) .

Again using standard nonseparable identification techniques and normalizations, one can
show that the reduced form f^rf is identified. Instruments are unnecessary here, because
the disturbance νk,t is assumed independent of all other variables. However, to identify
the technology fk,s , we need to disentangle the direct effect of θt, Ik,t, θP on θt+1 from their

²⁶The assumption of a common shock across technologies produces singularity across the investment
equations (3.11). This is not a serious problem because, as noted below in Section 4.2.5, we cannot distinguish
cognitive investment from noncognitive investment in our data. We assume a single common investment so
qk,t(∙) = qt(∙) for k ∈{C,N}.

²⁷Complete regularity conditions along with a proof are presented in Web Appendix 3.3.

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