Smoothing Hazard Functions and Time-Varying Effects in Discrete Duration and Competing Risks Models

State space or dynamic approaches to discrete or grouped duration
data with competing risks or multiple terminating events allow
simultaneous modelling and smooth estimation of hazard functions
and time-varying effects in a flexible way. Full Bayesian or
posterior mean estimation, using numerical integration techniques
or Monte Carlo methods, can become computationally rather demanding
or even infeasible for higher dimensions and larger data sets.
Therefore, based on previous work on filtering and smoothing for
multicategorical time series and longitudinal data, our approach
uses posterior mode estimation. Thus we have to maximize posterior
densities or, equivalently, a penalized likelihood, which enforces
smoothness of hazard functions and time-varying effects by a
roughness penalty. Dropping the Bayesian smoothness prior and
adopting a nonparametric viewpoint, one might also start directly
from maximizing this penalized likelihood. We show how Fisher
scoring smoothing iterations can be carried out efficiently by
iteratively applying linear Kalman filtering and smoothing to a
working model. This algorithm can be combined with an EM-type
procedure to estimate unknown smoothing- or hyperparameters. The
methods are applied to a larger set of unemployment duration data
with one and, in a further analysis, multiple terminating events
from the German socio-economic panel GSOEP.