samlet årgang - Økonomisk Institut - Københavns Universitet

THE EFFECT OF LABOUR MARKET CONDITIONS ON HIGHER EDUCATION COMPLETION 89
maximum likelihood estimator for the extended parameter vector ( , 1 , 2 , …, m )
maximizes the sample log likelihood subject to 0 ≤ 1 ≤ 2 ≤ ... ≤ m ≤ .
To get around these inequality constraints, it is customary to adopt an alternative
parametrization. Let 1 = log( 1 ) and j = log( j – j-1 ) for j = 2,3,…, m. With the new
parametrization the log likelihood function can be maximized in terms of = ( , 1 ,..,
m ) .
(C) Incorporating Heterogeneity Terms
In the presence of unobserved heterogeneity, since U is not observed, the contribution
to the likelihood of an observation with a completed spell is of the following integral
form:
P(T n [K n ,K n +1) |X n )=E G [S(K n |U)]–E G [S(K n +1|X n , U)]
= ∫ exp(–e Xn (Kn )e u )dG(u) – ∫ exp(–e Xn (Kn + 1)e u )dG(u) (4.6)
Ω Ω
There are three alternatives proposed in the literature in dealing with this integration
problem:
(i) Make a transformation of the random variable from U to W = e U . Assume L(w) to
be the distribution function of W. In terms of W, (4.6) becomes
P(T n [K n ,K n +1) |X n )=E L [S(K n |X n , W)]–E L [S(K n +1|X n , W)]
= ∫ exp(–e Xn (Kn )w)dL(w) – ∫ exp(–e Xn (Kn + 1)w)dL(w) (4.7)
Each of the two integrals is exactly the moment generating function (m.g.f.) of the
distribution function L. Due to this m.g.f. representation, parametric distributions that
have tractable analytical m.g.f.’s are convenient choices for the heterogeneity distribution.
A particular example is the (unit-mean) gamma density for W:
dL(w) / dw = (w;)= w w-1 e - ,w≥ 0 (4.8)
()
whose m.g.f. is m L (t) = (1 – t/) - . Adopting this gamma distribution, the sample log
likelihood function has analytical form in terms of (, ). Maximization of this likeli-