lnYi =DilnY1i+(1-Di)Y0i
= Di[Xi(β1 -β0)+u1i -u0i]+β0Xi +u0i
(16)
= γiDi + β0 Xi + u0i
where γi =Xi(β1 -β0)+u1i-u0i is the heterogeneous return to legal status for the ith foreign
farm worker (i.e. the effect varies across all farm workers). If the heterogeneous effect were
from a differential between the βj terms only, this would be ‘observed heterogeneity’; if it were
to arise as a consequence of differences between the uji terms, it would be ‘unobserved
heterogeneity.’ In either case, this parameter would imply different wage effects of legalization
across foreign workers in the farm workforce even if they have identical observable
characteristics. Note that for individuals who gain legal status (D=1), γi captures the benefit of
legal status.
Treatment effect parameters
The literature on marginal treatment effects (MTE) spearheaded by Heckman provides
several interpretations of the MTE which are equivalent under certain assumptions that apply in
this analysis (see Heckman and Vytlacil (2007b) and references therein). One interpretation of
the MTE presents it as a measurement of the marginal return to individuals who are indifferent
between foregoing (D=0) or accepting treatment (D=1 ) when their mean utility (μd(Z)) is
equivalent to Ud . If lnYj are defined as value outcomes, it may be interpreted as a ‘willingness
to pay’ measure for individuals with certain observable characteristics (X) and unobserved
heterogeneity (Ud ) at a specified margin of indifference (Heckman and Li, 2004; Heckman,
Urzua and Vytlacil, 2006a). The other treatment effect estimators - the average treatment effect
(ATE), the average treatment effect on the treated (ATET) and the average treatment effect on
10
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