used for the analysis.3 The overall models and estimation procedure are described in the
following sections, and draw heavily on the theoretical expositions of Heckman, Urzua and
Vytlacil (2006a; 2006b).
Parametric model with heterogeneous treatment effects
The parametric model with essential heterogeneity adopts the familiar latent variable
framework shown:
|
D* = α Z - ε = μ(Z ) - ε D = 1 if D* ≥ 0 D=0 |
Choice model/decision rule | |
|
(if the worker opts for legal status ) |
(10) | |
|
lnY1 =β1'Xi+u1i |
Wage outcome for treated group | |
|
lnY0 =β0'Xi+u0i |
Wage outcome for untreated group |
(11) (11) |
where the Z and X are vectors of observable characteristics and ε,u1i,u0i are error terms that
encapsulate the unobservable characteristics of individuals. The decision to accept treatment
(legal status) is defined by a choice model that allows for two separate log wage
outcomes(lnY1,lnY0).4 The choice model may be interpreted as a net utility for individuals with
the characteristics Z and ε. Similarly, the (log wage) outcomes are functions of the ith worker’s
characteristics denoted by Xi and uji (j = 0,1), respectively. The error of the choice model (ε) is
assumed to be independent of Z given X. The parametric model assumes joint normality of the
errors(ε,u1i,u0i), which are assumed to be independent of the observable characteristics (Z and
X). Based on this assumption, the expectations on the errors of the outcome equations reflect the
differences in legal status choice (D=1 if legalized/treated, D=0 if not legalized/untreated):
3 Information on the MTE is available at http://jenni.uchicago.edu/underiv/ (Cited as Heckman, Urzua and Vytlacil,
2006c in reference list). Also, see Heckman, Urzua and Vytlacil (2006a; 2006b).
4 Parameters μ(Z) and ε are assumed to be additively separable as is the predominant specification in the
literature.