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

368 CARNEIRO, HANSEN, AND HECKMAN
Aggregating over choices s ′ = 1,..., ¯S; s ′ ≠ s, we may define the marginal treatment
effect over all origin states as
(7)
MTE s
(a, {V s,s ′} ¯S
s ′ =1,s ′ ≠s
)
=
¯S∑
s ′ =1
s ′ ≠s
(
/ (
))
MTE(a, V s,s ′) f (V s , V s ′ | V s = V s ′ = V s,s ′ ≥ V j , j ≠ s, s ′ ) ψ a, {V s,s ′}
s S ′ =1,s ′ ≠s
the weighted average of the pairwise marginal treatment effects from all source
states to s (at a given level of utility V s,s ′) with the weights being the density of
persons at each relevant margin for specified values of utility where
ψ ( a, {V s,s ′} ¯S )
¯S∑
s ′ = 1,s ′ ≠s = f (V s , V s ′ | V s = V s ′ = V s,s ′ ≥ V j , j ≠ s, s ′ )
s ′ = 1
s ′ ≠s
is a normalizing constant (the population density of people at all margins given a
and {V s,s ′}
s S ′ =1,s ′ ≠s
), assumed to be positive.
We next present a framework for estimating the distributions of the treatment
effects and the parameters derived from them, which allows us to estimate the
parameters defined in this section as well as other parameters. To simplify the
notation, we suppress the ω argument in the rest of the article.
4. FACTOR STRUCTURE MODELS
The strategy adopted in this article identifies the distribution of counterfactuals
by postulating a low-dimensional set of factors θ so that, conditional on them
and the covariates X and Z, the Y s,a and V s ′ are jointly independent for all s, s ′ ,
and a. The distributions of the components of θ are nonparametrically identified
under the conditions specified below. With these distributions in hand, it is possible
to construct the distribution of counterfactuals. Under the conditions specified in
Section 5, it is possible with low-dimensional factors to nonparametrically identify
the counterfactual distributions and to estimate all of the treatment effects in the
literature suitably extended to multidimensional versions.
Throughout this article we analyze a separable-in-the-errors system. Thus, preferences
can be described by
(8)
V s = µ s (Z) − e s
s = 1,..., ¯S.
It is conventional to assume that µ s (Z) = Z ′ β s with s = 1,..., ¯S. Linear approximations
to value functions are advocated by Heckman (1981) and Eckstein and
Wolpin (1989) and are developed systematically in Geweke et al. (2001). Our approach
does not require linearity but critically relies on separability between the
deterministic portion of the model and the errors e s . Following Heckman (1981),