ηk,t+1. Define the tth row of θ as θtr where r stands for row. Thus
ln θtr = (lnθC,t,lnθN,t,lnIt,lnθC,P,lnθN,P,lnπ) .
Identification of this model follows as a consequence of Theorems 1 and 2 and results in
Matzkin (2003, 2007). We estimate the model under different assumptions about the distri-
bution of the factors. Under the first specification, ln θtr is normally distributed with mean
zero and variance-covariance matrix Σt . Under the second specification, ln θtr is distributed
as a mixture of T normals. Let φ (x; μt,τ, Σt,τ) denote the density of a normal random vari-
able with mean μt,τ and variance-covariance matrix Σt,τ. The mixture of normals writes the
density of ln θtr as
T
P (ln θt ) = X ωτφ (ln θr ; μt,τ, ¾,τ )
τ=1
subject to: PT=1 ωτ = 1 and PT=1 ωτμt,τ = 0.
Our anchored results allow us to compare the productivity of investments and stocks of
different skills at different stages of the life cycle on the anchored outcome. In this paper,
we mainly use completed years of education by age 19, a continuous variable, as an anchor.
4.2 Empirical Estimates
This section presents results from an extensive empirical analysis that estimates the multi-
stage technology of skill formation accounting for measurement error, non-normality of the
factors, endogeneity of inputs and family investment decisions. The plan of this section is as
follows. We first present baseline two stage models that anchor outcomes in terms of their
effects on schooling attainment, that correct for measurement errors, and that assume that
the factors are normally distributed. These models do not account for endogeneity of inputs
through unobserved heterogeneity components or family investment decisions. The baseline
model is far more general than what is presented in previous research on the formation of
child skills that uses unanchored test scores as outcome measures and does not account for
measurement error.32
We present evidence on the first order empirical importance of measurement error. When
we do not correct for it, the estimated technology suggests that there is no effect of early
investment on outcomes. Controlling for endogeneity of family inputs by accounting for
unobserved heterogeneity (π), and accounting explicitly for family investment decisions has
substantial effects on estimated parameters.
32An example is the analysis of Fryer and Levitt (2004).
24
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