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 existing ones, as indicated by the Pareto-Smoothed Important Sampling Leave-One-Out cross-validation (PSIS-LOO) minimum criterion.

中文翻译：

---

Fisher 基于 z 分布的混合自回归模型

我们将高斯混合自回归 (GMAR) 模型推广到用于建模非线性时间序列的 Fisher z混合自回归 (ZMAR) 模型。该模型由K分量 Fisher z自回归模型的混合组成，混合比例随时间变化。该模型可以捕获具有异方差和多峰条件分布的时间序列，使用 Fisher 的z分布作为 MAR 模型的创新。ZMAR 模型在水平（或模式）模型中被归类为非线性，因为 Fisher 的z分布在其位置参数上是稳定的，无论是对称的还是非对称的。使用马尔可夫链蒙特卡罗 (MCMC) 算法，例如 No-U-Turn Sampler (NUTS)，我们进行了模拟研究，以研究与 GMAR 模型和 Student t Mixture Autoregressive (TMAR) 模型相比的模型性能。这些模型适用于每日 IBM 股票价格和每月布伦特原油价格。结果表明，所提出的模型优于现有模型，如帕累托平滑重要抽样留一交叉验证 (PSIS-LOO) 最小标准所示。