Time-Varying Multi-Seasonal AR Models

Abstract: We propose a seasonal AR model with time-varying parameter processes in both the regular and seasonal parameters. The model is parameterized to guarantee stability at every time point and can accommodate multiple seasonal periods. The time evolution is modeled by dynamic shrinkage processes to allow for long periods of essentially constant parameters, periods of rapid change as well as abrupt jumps. A Gibbs sampler is developed with a particle Gibbs update step for the AR parameter trajectories. The near-degeneracy of the model, caused by the dynamic shrinkage processes, is shown to pose a challenge for particle methods. To address this, a more robust, faster and accurate approximate sampler based on the extended Kalman filter is proposed. The model and the numerical effectiveness of the Gibbs sampler are investigated on simulated and real data. An application to more than a century of monthly US industrial production data shows interesting clear changes in seasonality over time, particularly during the Great Depression and the recent Covid-19 pandemic. Keywords: Bayesian inference; Extended Kalman filter; Locally stationary processes; Particle MCMC; Seasonality.