their arguments.
The “anchored” skills, denoted by θj,k,t, are defined as
r , - - ___ - __
θj,k,t = gk,j (θk,t) , k ∈ {C, N}, t ∈ {1, . . . ,T}.
The anchored skills inherit the subscript j because different anchors generally scale the same
latent variables differently.
We combine the identification of the anchoring functions with the identification of the
technology function fk,s (θt, Ik,t, θP , ηk,t) established in the previous section to prove that the
technology function expressed in terms of the anchored skills — denoted by fk,s,j (θj,t, Ik,t, θP, ηk,t
— is also identified. To do so, redefine the technology function to be
fk,s,j θθj,c,t, θj,N,t, h,t, θC,P, θN,P ,ηk,t^
≡ gk,j fkk,s gCGJ (θj',C,t} ,gΝ1j (θj∙,N,t} ,Ik,t,θC,P ,θN,P ,ηk,t^ ) ,k ∈ {C, N}
where g-j (∙) denotes the inverse of the function gk,j (∙). Invertibility follows from the assumed
monotonicity. It is straightforward to show that
fk,s,j θθcc,t, θj,N,t, Ik,t, θC,P, θN,P, ŋk,t^
= fk,s,j (gC,j (θC,t) , gN,j (θN,t) , Ik,t, θC,P, θN,P, ηk,t)
= gk,j (fk,s (gC,1j (gC,j (θC,t)) , gN1j (gN,j (θN,t)) , Ik,t, θC,P, θN,P, ηk,ty}
= gk,j (fk,s (θC,t , θN,t , Ik,t , θC,P , θN,P , ηk,t))
∖
= gk,j (θk,t+1) = θk,j,t+1 ,
as desired. Hence, fk,s,j is the equation of motion for the anchored skills θ'k,j,t+1 that is
consistent with the equation of motion fk,s for the original skills θk,t .
3.6 Accounting for Endogeneity of Parental Investment
3.6.1 Allowing for Unobserved Time-Invariant Heterogeneity
Thus far, we have maintained the assumption that the error term ηk,t in the technology (2.1)
is independent of all the other inputs (θt,Ik,t,θp) as well as n`,t,k = `. This implies that
variables not observed by the econometrician are not used by parents to make their decisions
regarding investments Ik,t . This is a very strong assumption. The availability of data on
adult outcomes can be exploited to relax this assumption and allow for endogeneity of the
inputs. This subsection develops an approach for a nonlinear model based on time-invariant
18