12 - Duration Models

Summary

In the last chapter we presented various models suitable for describing duration related data. These are used to analyse phenomena such as: how long a person remains unemployed, the length and size of bank overdrafts, the delay between successive purchases of a certain good, the life expectancy of certain types of vehicles depending on their characteristics, or the manner in which an employee rises through the ranks of a corporate hierarchy. The principal mathematical characteristics of these data is that they assume a series of positive values. Specification of the distribution of these processes is based upon the theory of renewal processes, the simplest example of which is the Poisson process. While it is beyond the scope of this chapter to develop this theory exhaustively, we shall present some of its aspects. This will allow us to establish the connection between duration models and some models developed in previous chapters: reservation wages (chapter 7), labour-market disequilibria (chapter 8), panel data models (chapter 6), and Poisson models (chapter 11).

The Basic Models

Describing the Distribution of a Positive Real Variable

Let ζ be a duration variable distributed continuously over ℝ⁺. In practice, this variable represents the time elapsed in a particular state (i.e. unemployment, position in a hierarchy, etc.), or separating two events (change of job, births, purchases, etc.)