Bayesian model selection and parameter estimation for possibly asymmetric and non-stationary time series using a reversible jump Markov chain Monte Carlo approach

Abstract

A Markov chain Monte Carlo (MCMC) approach, called a reversible jump MCMC, is employed in model selection and parameter estimation for possibly non-stationary and non-linear time series data. The non-linear structure is modelled by the asymmetric momentum threshold autoregressive process (MTAR) of Enders & Granger (1998) or by the asymmetric self-exciting threshold autoregressive process (SETAR) of Tong (1990). The non-stationary and non-linear feature is represented by the MTAR (or SETAR) model in which one ( 𝜌 1 ) of the AR coefficients is greater than one, and the other ( 𝜌 2 ) is smaller than one. The other non-stationary and linear, stationary and nonlinear, and stationary and linear features, represented respectively by ( 𝜌 1 = 𝜌 2 = 1 ), ( 𝜌 1 p 𝜌 2 < 1 ) and ( 𝜌 1 = 𝜌 2 < 1 ), are also considered as possible models. The reversible jump MCMC provides estimates of posterior probabilities for these four different models as well as estimates of the AR coefficients 𝜌 1 and 𝜌 2 . The proposed method is illustrated by analysing six series of US interest rates in terms of model selection, parameter estimation, and forecasting.