r= 1, 2, 3, ∙ .4 Further, it is assumed that θf is distributed independently of the vector of
predictor variables z .
In the competing risks model, the joint survivor function for college completion
and college dropout, conditional on the two unobserved variables θg and θd, the vector of
predictor variables, x, and the waiting time until enrollment Tf is assumed to have the
following form:
kg
S(kg,kd∖x,θg,θd,Tf ) = exp[-θg∑ exp(γrg + (βg )'x + αT )
r =1
kd
- θ ∑ exp(γd + (βd )'x + αd Tf )](2)
r=1
where the parameters γsr are the baseline hazard parameters and the vector βsr measures
the (possibly time-varying) effects of the regressors, and αsr measures the (possibly time-
varying) effect of the waiting time until college enrollment, s = g, d, r=1, 2, 3,∙. Further,
it is assumed that θg and θd are distributed independently of the observed predictor
variables. The joint distribution of θf, θg, θd is denoted by G( θf, θg, θd). If θf is correlated
with θg and/or θd, then Tf will be correlated with θg and/or θd. However, we will jointly
estimate equations (1) and (2) using maximum likelihood estimation to explicitly account
4For simplicity of notation, the regressors are assumed to be time-constant. The model is easily extended
to the case of time-varying regressors although the notation is cumbersome.
10
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