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 I
k,t to respond to it.

We analyze a model of investment of the form

Ik,t=qk,ttP,ytt), k {C,N},t {1,...,T}.                 (3.11)

Equation (3.11) is the investment policy function that maps state variables for the parents,
t, 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 π
t). Then, the relationship Ik,t = qk,t t, 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 (θt, θP , yt).

If πt is serially correlated, it is not plausible to assume independence between πt and θt ,
because past values of π
t will have an impact on both current πt and on current θt (via the
effect of past π
t on past Ik,t). To address this problem, lagged values of income yt can be used
as instruments for θ
t P and yt could serve as their own instruments). This approach works
if π
t 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 qt 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 qk,t have been identified, one can obtain qk-,t1t, θP , yt , Ik,t), the inverse
of q
k,t t, θP , yt, πt) with respect to its last argument, provided qk,t t, θP , yt, πt) is strictly
monotone in π
t at all values of the arguments. We can then rewrite the technology function
(3.11) as:

θk,t+ι = fk,s (θt,Ik,tP,q-1 t,θp,yt, Ik,t) k,t} fft, Ik,tP,ytk,t) .

Again using standard nonseparable identification techniques and normalizations, one can
show that the reduced form frf is identified. Instruments are unnecessary here, because
the disturbance ν
k,t is assumed independent of all other variables. However, to identify
the technology f
k,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.

20



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