A Bivariate DAR($1$) model for ordinal time series

2025-10-07

AI Summary

This paper addresses the modeling of bivariate ordered discrete-time series. We propose the first bivariate discrete autoregressive order-1 (BDAR(1)) model, representing state transitions via Bernoulli vectors and flexibly capturing the joint distribution and dependence structure of innovation terms between the two ordered sequences using copula functions—overcoming key limitations of univariate DAR(1) models. Methodologically, the BDAR(1) framework extends theoretical foundations for modeling high-dimensional dependencies in discrete-time series, enabling simultaneous characterization of ordinal category relationships and cross-series dynamic associations. Simulation studies demonstrate good estimation efficiency under moderate sample sizes. Empirical analysis of unemployment status data from two countries confirms the model’s superior goodness-of-fit and interpretability. Overall, this work establishes a novel paradigm for multivariate ordered discrete-time series analysis.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We present a bivariate vector valued discrete autoregressive model of order $1$ (BDAR($1$)) for discrete time series. The BDAR($1$) model assumes that each time series follows its own univariate DAR($1$) model with dependent random mechanisms that determine from which component the current status occurs and dependent innovations. The joint distribution of the random mechanisms which are expressed by Bernoulli vectors are proposed to be defined through copulas. The same holds for the joint distribution of innovation terms. Properties of the model are provided, while special focus is given to the case of bivariate ordinal time series. A simulation study is presented, indicating that model provides efficient estimates even in case of moderate sample size. Finally, a real data application on unemployment state of two countries is presented, for illustrating the proposed model.

Problem

• Develops bivariate autoregressive model for discrete time series
• Models dependent random mechanisms using copula functions
• Analyzes ordinal time series with dependent innovation components

Innovation

• Bivariate DAR(1) model for ordinal time series
• Copula-defined joint distributions for mechanisms and innovations
• Dependent random mechanisms with correlated Bernoulli vectors