Estimating Distributions of Counterfactuals with an Application ... - UCL

382 CARNEIRO, HANSEN, AND HECKMAN
random variable, and the random variables possess nonvanishing (a.e.) characteristic
functions, then the densities of (θ 1 ,v 1 ,v 2 ), g(θ 1 ), g 1 (v 1 ), g 2 (v 2 ), respectively,
are identified.
PROOF. See Kotlarski (1967). See also Rao (1992).
Applied to our context, consider the first two equations of T and suppose that
the components depend only on θ 1 . We use our notation for the factor loadings to
write
T s
1 = λs 11 θ 1 + ε s 1
where λ s 11 = 1
T s
2 = λs 21 θ 1 + ε s 2
where λ s 21 ≠ 0
Here we use a notation associating the subscript of εi s with its position in the T s
vector. Applying Theorem 4, we can identify λ s 21
(subject to the normalization
λ s 11 = 1).24 Thus we can rewrite these equations as
T s
1 = θ 1 + ε s 1
T2
s
λ s = θ 1 + ε ∗,s
2
21
where ε ∗,s
2
= ε2 s/λs 21
. Applying Kotlarski’s theorem, we can nonparametrically
identify the densities g(θ 1 ), g 1 (ε1 s), and g 2(ε ∗,s
2 ). Since we know λs 21
we can identify
g(ε2 s). Let B 1 denote the number of measurements (elements of T s ) that depend
only on θ 1 . Proceeding through the first B 1 measurements, we can identify g(εi s),
i = 1,...,B 1 .
Proceeding to equations B 1 + 1 and B 1 + 2 (corresponding to the first two measurements
in the next set of equations that depend on θ 1 and θ 2 ), we may use the
normalization adopted in Theorem 4 to write
T s B 1 +1 = λs B 1 +1,1 θ 1 + θ 2 + ε s B 1 +1
T s B 1 +2 = λs B 1 +2,1 θ 1 + λ s B 1 +2,2 θ 2 + ε s B 1 +2
Rearranging, we may write these equations as
T s B 1 +1 − λs B 1 +1,1 θ 1 = θ 2 + ε s B 1 +1
T s B+2 − λs B 1 +2,1 θ 1
λ s B 1 +2,2
= θ 2 + ε ∗,s
B 1 +2
24 To be able to identify λ s 21 we need a third measurement on this factor, which we can get from
equation B 1 + 1. Since there is no equation B K+1 , we require that B K − B K−1 ≥ 3 in order to be able
to identify the loadings on θ K .