Time series data, or mixed time series and cross section data, are often analysed with the theory of processes. This procedure may, for example, be applied to linear models with serial correlation or with lagged endogenous variables – in the latter case the values assumed by the endogenous variables are described by a Markov process. Similar models can be built when the endogenous variable is qualitative; these derive from the theory of Markov chains (however, cf. section 2.8.3 for a different approach.)

It is beyond the scope of this chapter to develop a full treatment of this theory. Rather, we shall restrict our analysis to showing how it can be used to build qualitative models of time series and to studying estimation problems associated with this type of model.

Definition of a Markov Chain

Consider a qualitative variable y assuming J values, j = 0, …, J − 1, for which we have observations over a period of time, t = 0, …, T. These observations (y0, y1, …, yt, …, yT) have a joint distribution which can be characterized in several ways.

One approach is to postulate a priori the marginal distribution of y0, the conditional distribution of y1 given y0, the conditional distribution of y2 given (y1, y0), etc.