Jahn, Malte

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).

Existing integer-valued generalised autoregressive conditional heteroskedasticity (INGARCH) models for spatio-temporal counts do not allow for negative parameter and autocorrelation values. Using approximately linear INGARCH models, the unified and flexible spatio-temporal (B)INGARCH framework for modelling unbounded (bounded) counts is proposed. These models combine negative dependencies with kinds of a long memory. They are easily adapted to special marginal features or cross-dependencies: When modelling precipitation data (counts of rainy hours), we account for zero-inflation, while for cloud-coverage data (counts of okta), we deal with missing data and additional cross-correlation. A copula related to the spatial error model shows an appealing performance.