As in Part I, Markov models are very attractive for categorical processes, because of the ease of interpreting the model, making likelihood inferences (see Remark B.2.1.2 in the appendix) and making forecasts. However, without further restrictions concerning the conditional distributions, the number of model parameters becomes quite large. Therefore, Section 7.1 presents approaches for defining parsimoniously parametrized Markov models. One of these approaches is linked to a family of discrete ARMA models, which exhibit an ARMA-like serial dependence structure and allow also for non-Markovian forms of dependence; see Section 7.2. Note that the count data version of this model was discussed in Section 5.3. Two other approaches from Chapter 5, namely hidden-Markov models and regression models, can also be adapted to the categorical case, as described in Sections 7.3 and 7.4, respectively.
7.1 Parsimoniously Parametrized Markov Models
Perhaps the most obvious approach to model a categorical process with state space is to use a (homogeneous) pth-order Markov model; see (B.1) for the definition:
The idea of having a limited memory is often ...