Categorical---or qualitative---time series data with random time-dependent covariates are frequently encountered in diverse applications as the list of examples shows. As with "ordinary'' time series, the data analyst is faced with the same problems of modeling, estimation, model checking, diagnostics and prediction. The present work shows that these questions can be attacked by means of regression theory for categorical time series whose foundation is based on generalized linear models and partial likelihood inference. A variety of models are provided to illustrate the selection of the link function and recent large sample results are reviewed. The theory is developed without resorting to the Markov assumption and to the notion of stationarity. Moreover, regression methods for categorical time series allow for parsimonious modeling and incorporation of random time-dependent covariates as opposed to other procedures. In particular, nominal and ordinal time series are analyzed and compared empirically to Markov chains and mixture transition distribution models.
"Regression Theory for Categorical Time Series." Statist. Sci. 18 (3) 357 - 376, August 2003. https://doi.org/10.1214/ss/1076102425