Estimating the Technology of Cognitive and Noncognitive Skill Formation

gC,j (θC,T+1)  =  ^μ4,j ^{+ α}4,C,j^θC,T +1

gN,j (^θN,T+1) = μ4,j ^{+ α}4,N,j^θN,T +1.

We can also anchor using nonlinear functions. One example would be an outcome pro-
duced by a latent variable Z4j, for j ∈ {J₁ + 1,..., J}:

Z4j = gj (^θC,T+1^,θN,T +1) ^{- ε}4,j.

Note that we do not observe Zjj, but we observe the variable Z₄,_j∙ which is defined as:

^{1, if g}j (^θC,T +1^{, θ}N,T+1) ^{- ε}4,j ≥ 0

0, otherwise.

In this notation

P^r ( Z4,j = ¹I ^θC,T+1^{, θ}N,T +1) = P^r [^ε4,j ≤ ^gj (^θC,T+1^{, θ}N,T + 1)∣ ^θC,T +1^{, θ}N,T+1]
= Fε4,j [gj (θC,T +1 ^,θN,T +1)∣ θC,T+1^,θN,T +1]
= gj (θC,T+1, θN,T+1) .

Adult outcomes such as high school graduation, criminal activity, drug use, and teenage
pregnancy may be represented in this fashion.

To establish identification of gj (θ_{C,T +1} , θ_{N,T +1}) for j ∈ {J₁ + 1, . . . , J}, we include the
dummy Z4,j in the vector θ. Assuming that the dummy Z4,j is measured without error, the
corresponding element of the two repeated measurement vectors W1 and W2 are identical
and equal to Z_4,j . Theorem 1 implies that the joint density of Z_4,j , θ_C,t and θ_N,t is identified.
Thus, it is possible to identify Pr [Z4,j = 1 ∣ θC,T+1, θN,T+1].

We can extract two separate “anchors” gC,j (θC,T +1) and gN,j (θN,T +1) from the function
gj (θC_{,T +1} , θ_N,T+1), by integrating out the other variable, e.g.,

gC,j (θC,T +1) ≡     gj (θC,T +1, θN,T+1)pθN,T+1 (θN,T+1) dθN,T +1,            (3.9)

gN,j (θN,T+1) ≡     gj (θC,T +1, θN,T+1)pθC,T+1 (θC,T +1) dθC,T+1,

where the marginal densities, pθ_j,T (θN,T +1), j ∈ {C, N} are identified by applying the pre-
ceding analysis. Both g_C,j (θ_{C,T +1}) and g_N,j (θ_{N,T +1}) are assumed to be strictly monotonic in

17

<<   15   16   17   18   19   20   21   >>

More intriguing information