Estimating the Technology of Cognitive and Noncognitive Skill Formation

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 f_k,_s,j (θj,_t, I_k,_t, θP, η_k,_t
— is also identified. To do so, redefine the technology function to be

^fk,s,j θθj,c^,t, θj,N,t, h,t^{, θ}C,P^{, θ}N,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 g_k,j (∙). Invertibility follows from the assumed
monotonicity. It is straightforward to show that

^fk,s,j θθcc,t, ^θj,N,t^, Ik,t^{, θ}C,P^{, θ}N,P^, ŋk,t^

= ^fk,s,j (gC,j (^θC,t) ^, gN,j (^θN,t) ^, Ik,t^{, θ}C,P^{, θ}N,P^, ηk,t)

= ^gk,j (^fk,s (^gC,¹j (^gC,j (θC,t)) ^{, g}N1j (^gN,j (θN,t)) ^, Ik,t^{, θ}C,P^{, θ}N,P^{, η}k,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, f_k,_s,j is the equation of motion for the anchored skills θ'_k,j,_t+₁ that is
consistent with the equation of motion f_k,s for the original skills θ_k,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 (θ_t,I_k,_t,θ_p) 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

<<   16   17   18   19   20   21   22   >>

More intriguing information