Semi-Parametric Duration Models: The Cox Model

The piecewise-constant exponential model is a semi-parametric continuous time duration model.1 The model is semi-parametric in the sense that the shape of the hazard rate with respect to time is not specified a priori but is determined by the data. The basic idea underlying the PCE model is that the duration time can be divided up into discrete units in each of which the hazard rate is assumed to be constant across time. In other words, the hazard rate is allowed to differ in different time periods but is assumed to be constant within any given time period. The advantage of the PCE approach is that the overall shape of the hazard rate does not have to be imposed in advance as with the parametric models. One use of the PCE model is to see how the hazard rate varies with time and to use this information to choose an appropriate parametric model. For example, if the PCE reveals that the hazard rate increases monotonically with time as in Figure 1, we could safely adopt the Weibull parametric model. In this sense, the PCE model is an alternative method for adjudicating between different parametric models.