Time-Varying Multi-Seasonal ARMA Models

2026-08-11

AI Summary

This study addresses the limitations of traditional ARMA models in simultaneously capturing multiple seasonalities, time-varying parameters, and heterogeneous noise. To overcome these challenges, we propose a novel multi-seasonal ARMA framework that incorporates time-varying autoregressive and moving average coefficients along with stochastic volatility. A key innovation is the integration of dynamic shrinkage priors, enabling model parameters to adaptively switch among constant, smoothly evolving, and abruptly changing regimes while preserving stationarity and invertibility at each time point. For efficient inference, we develop an exact-likelihood-based Gibbs sampler augmented with an extended Kalman filter to rapidly update latent states and initial histories. Empirical evaluations on both simulated and real-world data demonstrate superior performance; notably, our analysis of U.S. monthly air passenger traffic from 1990 to 2024 reveals that the COVID-19 pandemic substantially reshaped seasonal patterns.

Abstract

We propose an ARMA model that allows for multiple seasonal periods and time varying parameters in both regular and seasonal components, building upon previous work for pure AR processes and the conditional likelihood. The model is parameterized to ensure stability and invertibility at each time point. The parameter evolution is governed by dynamic shrinkage processes, enabling extended periods of essentially constant parameters, gradual changes, and abrupt shifts. The model includes a stochastic volatility component to account for potentially heterogeneous noise, also modeled by a dynamic shrinkage process. A Gibbs sampler is developed using the exact likelihood, with separate updating steps for the latent errors and the unobserved pre-sample history of the process. The time-varying AR and MA parameters are sampled jointly using a fast posterior sampler based on the extended Kalman filter. The model and the efficiency of the Gibbs sampler are evaluated using simulated and real data. A case study on monthly air passenger data in the US during 1990-2024 reveals significant changes in seasonality during the Covid-19 pandemic.

Problem

• time-varying parameters
• multi-seasonal
• ARMA
• stochastic volatility
• seasonality

Innovation

• time-varying ARMA
• multi-seasonal
• dynamic shrinkage
• stochastic volatility
• Gibbs sampler