The Timing-of-Events (ToE) approach concerns the identification of the effect of a treatment given while being in an initial state on the time spent in the same state, allowing both treatments and exits to occur at any point in continuous time. In their seminal paper, Abbring and van den Berg (2003) specify a Mixed Proportional Hazard model and establish conditions under which all parts of the model are non-parametrically identified.
One key feature is that the model allows both the exit rate from the initial state and the treatment rate to be affected by related unobserved determinants, making it an attractive approach in settings where quasi-experimental approaches are unfeasible.
In applications, the ToE model is typically implemented by using a nonparametric maximum likelihood estimator (NPMLE) and approximating the unknown bivariate unobserved heterogeneity by means of a discrete distribution. In practice, however, this could be implemented in different ways. One could pre-specify a low number of support points and increase their number until computational problems arise. Alternatively, information criteria can allow to select the number of support points.
Other specification issues concern the parameterization of the baseline hazard capturing duration dependence. Most often a piecewise constant baseline hazard is used, but this raises additional specification issues concerning how to specify and select the number locally constant parts. Another potentially important aspect when implementing the ToE model is sample size, as the highly non-linear MPH model might be difficult to identify with a small sample.
Data aggregation, if the data truly is continuous or if the data is aggregated at the monthly or the yearly level, could also be important.
In this project, we use a simulation design based on actual data to evaluate these relevant specifications issues. To this end, we modify the so-called Empirical Monte Carlo design (Huber et al., 2013). The key idea is to use information from actually treated observations to simulate placebo treatment for non-treated ones. The attractive feature of this approach is that this simulation design uses actual data instead of a data generating process entirely chosen by the researcher, diminishing the degree of arbitrariness and making it more relevant for real applications.
To this aim, we modify the Empirical Monte Carlo simulation design in a novel way to deal with duration data. The observed time to treatment of the actually treated individuals are used to simulate a placebo time to treatment for each non-participant by using a proportional hazard model. This ensures that true treatment effect is zero and that the treatment assignment process is known.
We then introduce unobserved heterogeneity by excluding subsets of the covariates that were used when generating the placebo treatments. Since these excluded covariates are also found to affect the exit duration (re-employment rate), we have a bivariate duration model with correlated unobserved determinants. With this generated data we are then able to examine the importance and proposed solutions to various specification issues when implementing the ToE model using NPMLE.