Nonlinear GARCH-type models for ordinal time series

• Published in: Stochastic environmental research and risk assessment 2024-02, Vol.38 (2), p.637-649
• Main Authors: Jahn, Malte, Weiß, Christian H.
• Format: Article
• Language: English
• Subjects: Air quality; Aquatic Pollution; Artificial neural networks; Autoregressive models; Binomial distribution; Chemistry and Earth Sciences; Computational Intelligence; Computer Science; Earth and Environmental Science; Earth Sciences; Environment; Environmental research; Generalized linear models; Math. Appl. in Environmental Science; Neural networks; Original Paper; Physics; Probability Theory and Stochastic Processes; Random variables; Response functions; Statistics for Engineering; Stochastic models; Stochasticity; Time series; Waste Water Technology; Water Management; Water Pollution Control
• ISSN: 1436-3240; 1436-3259
• DOI: 10.1007/s00477-023-02591-1

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).