Bayesian Inference for Double Seasonal Moving Average Models: A Gibbs Sampling Approach

Ayman Amin

doi:10.18187/pjsor.v13i3.1647

Abstract

In this paper we use the Gibbs sampling algorithm to develop a Bayesian inference for multiplicative double seasonal moving average (DSMA) models. Assuming the model errors are normally distributed and using natural conjugate priors, we show that the conditional posterior distribution of the model parameters and variance are multivariate normal and inverse gamma respectively, and then we apply the Gibbs sampling to approximate empirically the marginal posterior distributions. The proposed Bayesian methodology is illustrated using simulation study.

Highlights

High frequency time series that are observed at small time units may be characterized by exhibiting multiple seasonal patterns
Seasonal autoregressive moving average (SARMA) models being widely applied to analyze time series with single seasonal pattern need to be modified and extended to accommodate multiple seasonalities, see for example Box et al (1994) and Taylor (2003)
The conditional posterior distribution for each one of the double seasonal moving average (DSMA) parameters is obtained from the joint posterior distribution (11) by first grouping together terms in the joint posterior that depend on this parameter, and finding the appropriate normalizing constant to form a proper and closed-form density

Summary

Introduction

High frequency time series that are observed at small time units may be characterized by exhibiting multiple seasonal patterns. Seasonal autoregressive moving average (SARMA) models being widely applied to analyze time series with single seasonal pattern need to be modified and extended to accommodate multiple seasonalities, see for example Box et al (1994) and Taylor (2003). Ismail (2003a&b) used Gibbs sampling algorithm to achieve Bayesian analysis for multiplicative seasonal moving average (SMA) and seasonal autoregressive (SAR) models. Amin and Ismail (2015) have used Gibbs sampling algorithm to develop a Bayesian analysis to multiplicative double SAR models. We extend this work to develop a Bayesian analysis to multiplicative DSMA models based on Gibbs sampling algorithm. For more details about the properties of SARMA models see Box et al (1994)

Likelihood Function

Prior Specification

Full Conditional Posterior Distributions

The conditional posterior of ε0

The Proposed Gibbs Sampler

Simulate

Simulation Study

Findings

Conclusion