We generalize the Gaussian Mixture Autoregressive (GMAR) model to the Fisher’s z Mixture Autoregressive (ZMAR) model for modeling nonlinear time series. The model consists of a mixture of K‐component Fisher’s z autoregressive models with the mixing proportions changing over time. This model can capture time series with both heteroskedasticity and multimodal conditional distribution, using Fisher’s z distribution as an innovation in the MAR model. The ZMAR model is classified as nonlinearity in the level (or mode) model because the mode of the Fisher’s z distribution is stable in its location parameter, whether symmetric or asymmetric. Using the Markov Chain Monte Carlo (MCMC) algorithm, e.g., the No‐U‐Turn Sampler (NUTS), we conducted a simulation study to investigate the model performance compared to the GMAR model and Student t Mixture Autoregressive (TMAR) model. The models are applied to the daily IBM stock prices and the monthly Brent crude oil prices. The results show that the proposed model outperforms the ex-isting ones, as indicated by the Pareto‐Smoothed Important Sampling Leave‐One‐Out cross‐valida-tion (PSIS‐LOO) minimum criterion.

Original languageEnglish
Article number27
Issue number3
Publication statusPublished - Sept 2021


  • Bayesian analysis
  • Fisher’s z distribution
  • Mixture autoregressive model
  • No‐U‐turn sampler
  • Stan program
  • The Brent crude oil prices
  • The IBM stock prices


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