Duration Models: Specification, Identification, and Multiple Durations

Since the early 1980s, the econometric analysis of duration variables has become widespread. This chapter provides an overview of duration analysis, with an emphasis on the specification and identification of duration models, and with special attention to models for multiple durations. Most of the chapter deals with so-called reduced-form duration models, notably the popular Mixed Proportional Hazard (MPH) model and its multivariate extensions. The MPH model is often used to describe the relation between the empirical exit rate and ``background variables'' in a concise way. However, since the applications usually interpret the results in terms of some economic-theoretical model, we examine to what extent the deep structural parameters of some important theoretical models can be related to reduced-form parameters. We subsequently examine the specification and identification of the MPH model in great detail, we provide intuition on what drives identification, and we infer to what extent biases may occur because of misspecifications. This examination is carried out separately for the case of single-spell data and the case of multi-spell data. We also compare different functional forms for the unobserved heterogeneity distribution. Next, we examine models for multiple durations. In the applied econometric literature on the estimation of multiple-duration models, the range of different models is actually not very large. Typically, the models allow for dependence between the duration variables by way of their unobserved determinants, with each single duration following its own MPH model. In addition to this, the model may allow for an interesting ``causal'' effect of one duration on the other, as motivated by an underlying economic theory. For all these models we examine the conditions for identification. Some of these are intimately linked to particular estimation strategies. The multiple-duration model where the marginal duration distributions each satisfy an MPH specification, and the durations can only be dependent by way of their unobserved determinants, is called the Multivariate Mixed Proportional Hazard (MMPH) model. For this model, we address the issue of the dimensionality of the heterogeneity distribution and we compare the flexibility of different parametric heterogeneity distributions. On a number of occasions, we incorporate recent insights from the biostatistical literature on duration analysis, and we contrast points of view in this literature to those in the econometric literature. Finally, throughout the chapter, we discuss the importance of the possible collection of additional data.