Abstract
The serial dependence of categorical data is commonly described using Markovian models. Such models are very flexible, but they can suffer from a huge number of parameters if the state space or the model order becomes large. To address the problem of a large number of model parameters, the class of (new) discrete autoregressive moving-average (NDARMA) models has been proposed as a parsimonious alternative to Markov models. However, NDARMA models do not allow any negative model parameters, which might be a severe drawback in practical applications. In particular, this model class cannot capture any negative serial correlation. For the special case of binary data, we propose an extension of the NDARMA model class that allows for negative model parameters, and, hence, autocorrelations leading to the considerably larger and more flexible model class of generalized binary ARMA (gbARMA) processes. We provide stationary conditions, give the stationary solution, and derive stochastic properties of gbARMA processes. For the purely autoregressive case, classical Yule–Walker equations hold that facilitate parameter estimation of gbAR models. Yule–Walker type equations are also derived for gbARMA processes.
Highlights
Categorical time series data are collected in many fields of applications and the statistical research focusing on such data structures evolved considerably over the last years
Give the stationary solution, and derive stochastic properties of generalized binary ARMA (gbARMA) processes
Yule–Walker type equations are derived for gbARMA processes
Summary
Categorical time series data are collected in many fields of applications and the statistical research focusing on such data structures evolved considerably over the last years. To address this lacking flexibility of the NDARMA model class, we propose a simple and straightforward extension of the original idea of Jacobs and Lewis (1983) that allows negative serial dependence. We define generalized binary AR(p) (gbAR(p)) models for binary data based on the notation of NDAR(p) models by adopting the idea of replacing Xt−1 by 1 − Xt−1 for a negative parameter α as in Equations (5) and (6) separately for all or some of the lagged values Xt−1 , . Is that only one random variable is multiplied with the lagged value Xt−i , whereas at is an additional random variable that accounts for the switching that leads to negative model coefficients
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