On the Noisy Four-Parameter Fisher‟s Z-Distribution of Bayesian Mixture Autoregressive (FZBMAR) Process via Mode as a Stable Location Parameter

This paper aims at providing in-depth refinement to switching time-variant autoregressive processes via the mode as a stable location parameter in adopted noisy Fisher’s z-distribution that was impelled in a Bayesian setting. Explicitly, a four-parameter Fisher’s z-distribution of Bayesian Mixture Autoregressive (FZBMAR) process was proposed to congruous k-mixture components of Fisher’s z-switching mixture autoregressive processes that was based on shifting number of modes in the marginal density of any switching time-variant series of interest. The proposed FZBMAR process was not only used to seize what is term “most likely mode value” of the present conditional modal distribution given the immediate past but was also used to capture the conditional modal distribution of the observations given the immediate past that can either be perceived as an asymmetric or symmetric distributed varieties. The proposed FZBMAR process was compared with the existing Student-t Mixture Autoregressive (StMAR) and Gaussian Mixture Autoregressive (GMAR) processes with the demonstration of monthly average share prices (stock prices) of sixteen (16) swaying European economies. Based on the findings, the FZBMAR process outperformed the existing StMAR and GMAR processes in explaining the sixteen (16) swaying European economies share prices via a minimum Pareto-Smoothed Important Sampling Leave-One-Out Cross-Validation (PSIS-LOO) error process performance in comparison with AIC, HQIC by the latters. The same singly truncated student-t prior distribution was adopted for the noisy adoption of Fisher’s z hyper-parameters and the embedded autoregressive coefficients in the proposed FZBMAR process; such that their resulting posterior distributions gave the same singly truncated student-t distribution (conjugate) with an embedded Gamma variate.