Fisher’s z Distribution-Based Mixture Autoregressive Model

Heri Kuswanto,Kartika Fithriasari,Nur Iriawan,Arifatus Solikhah

doi:10.3390/econometrics9030027

Heri Kuswanto, Kartika Fithriasari + Show 2 more

Open Access

https://doi.org/10.3390/econometrics9030027

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Abstract

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.

Highlights

Many time series indicate non-Gaussian characteristics, such as outliers, flat stretches, bursts of activity, and change points (Le et al 1996)
Several methods have been proposed to deal with the presence of bursts and outliers such as applying robust or resistant estimation procedures (Martin and Yohai 1986) or omitting the outliers based on the use of diagnostics (Bruce and Martin 1989)
The Gaussian Mixture Transition Distribution (GMTD), which is a special form of MTD, was generalized to a Gaussian Mixture Autoregressive (GMAR) model by Wong and Li (2000)

Summary

Introduction

Many time series indicate non-Gaussian characteristics, such as outliers, flat stretches, bursts of activity, and change points (Le et al 1996). The use of the Gaussian distribution in the GMAR model still leaves problems, because it is able to capture only short-tailed data patterns. Fisher’s z uses the errors in each component of the MAR model to capture the ‘most likely’ mode value—(not the mean, median, or quantile) of the conditional distribution Yt given the past information. This research applies the Bayesian method to estimate the parameters of the ZMAR model, using MCMC with the NUTS algorithm, as well as simulation studies to examine different scenarios in order to evaluate whether the proposed mixture model outperforms its counterparts. We used cross-validation Leave-One-Out (LOO) coupled with the Pareto-smoothed important sampling (PSIS), namely PSIS-LOO This approach has very efficient computation and was stronger than the Widely Applicable Information.

Four-Parameter Fisher’s z Distribution

Model Specification

Bayesian Approach for ZMAR Model

Simulation Studies

IBM Stock Prices

Brent Crude Oil Prices

Monthly average

Conclusions