Soft-Clipping Autoregressive Models for Ordinal Time Series

Abstract

The linear autoregressive models are among the most popular models in the practice of time series analysis, which constitutes an incentive to adapt them to ordinal time series as well. Our starting point for modeling ordinal time series data is the latent variable approach to define a generalized linear model. This method, however, typically leads to a non-linear relationship between the past observations and the current conditional cumulative distribution function (cdf). To overcome this problem, we use the soft-clipping link to obtain an approximately linear model structure and propose a wide and flexible class of soft-clipping autoregressive (scAR) models. The constraints imposed on the model parameters allow us to identify relevant special cases of the scAR model family. We study the calculation of transition probabilities as well as approximate formulae for the CDF. Our proposals are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation as well as model selection is analyzed. The novel model family is successfully applied to a real-world ordinal time series from finance.