to differ at different stages of the life cycle and for both to be different from the elasticities
of substitution for cognitive and noncognitive skills in producing adult outcomes. We test
and reject the assumption that φs,C = φs,N for s ∈ {1, . . . , S}.
3 Identifying the Technology using Dynamic Factor
Models
Identifying and estimating technology (2.1) is challenging. Both inputs and outputs can
only be proxied. Measurement error in general nonlinear specifications of technology (2.1)
raises serious econometric challenges. Inputs may be endogenous and the unobservables in
the input equations may be correlated with unobservables in the technology equations.
This paper addresses these challenges. Specifically, we execute the following tasks: (1) De-
termine how stocks of cognitive and noncognitive skills at date t affect the stocks of skills at
date t + 1, identifying both self productivity (the effects of θN,t on θN,t+1, and θC,t on θC,t+1)
and cross productivity (the effects of θC,t on θN,t+1 and the effects of θN,t on θC,t+1) at each
stage of the life cycle. (2) Develop a non-linear dynamic factor model where (θt, It, θP) is
proxied by vectors of measurements which include test scores and input measures as well as
outcome measures. In our analysis, test scores and personality evaluations are indicators of
latent skills. Parental inputs are indicators of latent investment. We account for measure-
ment error in these proxies. (3) Estimate the elasticities of substitution for the technologies
governing the production of cognitive and noncognitive skills. (4) Anchor the scale of test
scores using adult outcome measures instead of relying on test scores as measures of output.
This allows us to avoid relying on arbitrary test scores as measurements of output. Any
monotonic function of a test score is a valid test score. (5) Account for the endogeneity
of parental investments when parents make child investment decisions in response to the
characteristics of the child that may change over time as the child develops and as new
information about the child is revealed.
Our analysis of identification proceeds in the following way. We start with a model where
measurements are linear and separable in the latent variables, as in Cunha and Heckman
(2008). We establish identification of the joint distribution of the latent variables without
imposing conventional independence assumptions about measurement errors. With the joint
distribution of latent variables in hand, we nonparametrically identify technology (2.1) given
alternative assumptions about ηk,t . We then extend this analysis to identify nonparametric
measurement and production models. We anchor the latent variables in adult outcomes
to make their scales interpretable. Finally, we account for endogeneity of inputs in the