The name is absent

.5 [S( k^s ,k^s| θ^g, θ^d )+S( k^s +1,k^s+1| θ^g, θ^d) -S( k^s ,k^s+1| θ^g, θ^d )-S( k^s+1,k^s| θ^g, θ^d )]

is an adjustment for the fact that the data are discrete time.

Let B^w be defined as

B^w (k^f, k^s | θ^f ,θ^g,θ^d,) = P^f ( k^f | θ^f ) P^w( k^s | θ^g,θ^d ), for w=g, d, and sc,

B^w( k^f, k^s| θ^f ,θ^g,θ^d ) = P^f ( k^f | θ^f ), for w = fc,

where k^f denotes the waiting duration to first college enrollment and k^s denotes the duration of college
enrollment.

Summing over the N different triplets of location parameters ( θ^f ,θ^g, θ^d) , the unconditional
probabilities can be written as follows.

Bw(kf,k^s) = ∑ ∕>nl!^w(k^f,k^sθ^f,θ^g,θ^d),
n=1

where

N

∑ p_n = 1.
n=1

Let q^fci be an indicator variable which equals one if individual i did not enroll in any college by the end of
survey year; q^gi is an indicator variable that equals one if this individual graduates, and q^di is an indicator
variable that equals one if individual i drops out from the higher education system, and q^fci is an indicator
variable that equals one if individual i enrolled but the enrollment spell is incomplete or right-censored, i=
1, * ,∙m.

The log-likelihood function is then given by

M
logL=∑ ∑qi^wlog(B^w(ki^f,ki^s))
i= 1 w∈W

and W={fc, d, g, sc}.

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