Nonlinear GARCH-type models for ordinal time series

Despite their relevance in various areas of application, only few stochastic models for ordinal time series are discussed in the literature. To allow for a flexible serial dependence structure, different ordinal GARCH-type models are proposed, which can handle nonlinear dependence as well as kinds of an intensified memory. The (logistic) ordinal GARCH model accounts for the natural order among the categories by relying on the conditional cumulative distributions. As an alternative, a conditionally multinomial model is developed which uses the softmax response function. The resulting softmax GARCH model incorporates the ordinal information by considering the past (expected) categories. It is shown that this latter model is easily combined with an artificial neural network response function. This introduces great flexibility into the resulting neural softmax GARCH model, which turns out to be beneficial in three real-world time series applications (air quality levels, fear states, cloud coverage).

Automated summary

Despite being relevant in various areas, there are few stochastic models for ordinal time series discussed in the literature. This study proposes different ordinal GARCH-type models to accommodate flexible serial dependence structure and handle nonlinear dependence and intensified memory. The logistic ordinal GARCH model considers the natural order by relying on conditional cumulative distributions, while the conditionally multinomial model utilizes softmax response function to incorporate ordinal information by considering past categories. The study shows that the resulting neural softmax GARCH model, which combines the latter model with artificial neural network response function, offers great flexibility and brings benefits in real-world applications.