E(u1 | X =
'
x, D = 1, P ( Z ) = p ) = P1
φ(α'Z∕ T
ʌ σεε J
P (Z )
J
(12)
E (u o| X = x, D = 0, P (Z ) = p ) = Po
φα'% 1
I /— J
1 - P (z )
I J
(13)
where ρ1 = —; ρ0 = — are the correlations between the disturbances of the respective
σε σε
outcome equations and the choice equation, and φ(.) denotes the standard normal density
function (Heckman, Urzua and Vytlacil, 2006b).
The probability of becoming legalized is defined as:
Pr(z) = Pr(D = 1∖Z = z) = Pr(α'Z > ε) = Φε{α'Z) (14)
where Φ(.)is the cumulative distribution ofε. Heckman, Urzua and Vytlacil (2006a) refer to
this function as a propensity score, taken as a monotonic function of the mean utility of treatment
(legal status). This is reflected in the acceptance decision:
D = 1[Φ εμ( Z ))> Φ ε(ε )]= 1[P (Z )> Ud ] (15)
where Ud denotes the unobserved characteristics of individuals. The algorithm estimates the
propensity score using a probit model, from which the predicted values for the treated and
untreated groups are used to define values over which the marginal treatment effect (MTE) of
legalization may be identified (Heckman, Urzua and Vytlacil, 2006b).
Since it is impossible to observe an individual in the treated and untreated states
simultaneously, the actual outcome to be estimated:
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