Markov Chain and Duration Models
We can use the term time series models in a broad sense to mean statistical models that specify how the distribution of random variables observed over time depends on their past observations. Thus defined, Markov chain models and duration models, as well as the models discussed in Chapter S, are special cases of time series models. However, time series models in a narrow sense refer to the models of Chapter 5, in which random variables take on continuous values and are observed at discrete times. Thus we may characterize the models of Chapter 5 as continuous-state, discrete-time models. Continuous – state, continuous-time models also constitute an important class of models, although we have not discussed them. In contrast, Markov chain models (or, more simply, Markov models) may be characterized as discrete-state, discrete-time models, aruiduration models (or, survival models) as discrete-state, continuous-time models. In this chapter we shall take up these two models in turn.
The reader who wishes to pursue either of these topics in greater detail than is presented here should consult the textbooks by Bartholomew (1982) for Markov models and by Kalbfleisch and Prentice (1980) or Miller (1981) for duration models. For recent results on duration models with econometric applications, see Heckman and Singer (1984b).