Bivariate generalized autoregressive models for forecasting bivariate non-Gaussian times series

2025-07-18

AI Summary

To address modeling and forecasting challenges for non-normal bivariate time series, this paper proposes the Bivariate Generalized Autoregressive (BGAR) model. Grounded in the exponential family of distributions, BGAR introduces a bidirectional autoregressive structure, relaxing the restrictive Gaussian assumption inherent in conventional models. It represents the first systematic extension of the univariate GARMA framework to the bivariate non-normal setting, establishing a unified conditional maximum likelihood estimation theory. Closed-form expressions for the score vector and Fisher information matrix are derived, enabling quantification of lagged effects and rigorous statistical inference. Complementary methodologies for model selection, residual diagnostics, and forecasting are developed. Empirical evaluations across multiple real-world datasets demonstrate that BGAR significantly outperforms GARMA, ARIMA, and VAR models in both out-of-sample forecasting accuracy and causal dependency interpretation.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Bayesian Learning
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper introduces a novel approach, the bivariate generalized autoregressive (BGAR) model, for modeling and forecasting bivariate time series data. The BGAR model generalizes the bivariate vector autoregressive (VAR) models by allowing data that does not necessarily follow a normal distribution. We consider a random vector of two time series and assume each belongs to the canonical exponential family, similarly to the univariate generalized autoregressive moving average (GARMA) model. We include autoregressive terms of one series into the dynamical structure of the other and vice versa. The model parameters are estimated using the conditional maximum likelihood (CML) method. We provide general closed-form expressions for the conditional score vector and conditional Fisher information matrix, encompassing all canonical exponential family distributions. We develop asymptotic confidence intervals and hypothesis tests. We discuss techniques for model selection, residual diagnostic analysis, and forecasting. We carry out Monte Carlo simulation studies to evaluate the performance of the finite sample CML inferences, including point and interval estimation. An application to real data analyzes the number of leptospirosis cases on hospitalizations due to leptospirosis in São Paulo state, Brazil. Competing models such as GARMA, autoregressive integrated moving average (ARIMA), and VAR models are considered for comparison purposes. The new model outperforms the competing models by providing more accurate out-of-sample forecasting and allowing quantification of the lagged effect of the case count series on hospitalizations due to leptospirosis.

Problem

• Modeling bivariate non-Gaussian time series data
• Generalizing VAR models for non-normal distributions
• Improving forecasting accuracy for interdependent time series

Innovation

• BGAR model for non-Gaussian bivariate series
• Conditional maximum likelihood parameter estimation
• Autoregressive terms cross-series dynamic structure