samlet årgang - Økonomisk Institut - Københavns Universitet

THE EFFECT OF LABOUR MARKET CONDITIONS ON HIGHER EDUCATION COMPLETION 87
Assumption 1 [Proportional Hazard]. The hazard function conditional on both
observed and unobserved heterogeneities (X n , U n ) is of the following proportional
hazard (PH) type,
h(t|X n (t), U n )=(t)e Xn(t) e Un = (t)e Xn(t) + U n (4.1)
Here (t) is the baseline hazard function that characterizes the basic shape of the
hazard function and is common across all individuals. Let (t) = ∫
(s)ds denote the in-
0
tegrated baseline hazard.
The expression exp{Xn (t) } is a customary function used to characterize the effect
of observed covariates Xn (t) on the hazard up to a finite set of unknown regression
coefficients . The effect of the unobserved heterogeneity U enters multiplicatively
via the exponential functional form. In this form U has a natural interpretation as an
omitted variable.
Assumption 2 [Heterogeneity]. The random variable Un is independent of Xn , satisfying
a normalization condition E[Un ] = ∫Ωu dG(u) =0 , where Ω is the distribution
support of Un . 5 This set-up allows unobserved heterogeneity to be absent. In that case,
the distribution G(u) will be degenerate with a unit point mass at 0, that is, P(U=0) =1.
Assumption 3 [Grouping]. The data on the duration variable T is grouped into intervals
bounded by integers. For each individual in the sample, instead of directly observing
T n in continuous time, we observe a positive integer K n such that it is known that
either T n falls in the interval (K n , K n +1) (for a completed spell), or T n is at least K n (for
a right-hand censored spell).
Assumption 4 [Covariates]. The covariate vector X(t)
(a) is weakly exogenous to the model parameters, and
(b) is either time-invariant or piecewise constant within each time interval.
In the most general case, neither the baseline hazard function (t) nor the heterogeneity
distribution G(u) is specified. The only parametric part of the model is the
observed covariates effect .
5. Thus, we adopt the customary assumption in the duration modelling literature that the unobserved heterogeneity
is independent from the observed heterogeneity. This assumption on the one hand is more general
then assuming that heterogeneity is absent and on the other hand is a restriction in itself. Because we work
with several spell-specific regressors and several person-spell-year-specific time-varying variables, modelling
dependency between observed regressors and unobserved heterogeneity, as suggested by one referee,
would make the model overly parameterized.