Bayesian analysis of autoregressive fractionally integrated moving‐average processes

Abstract

For the autoregressive fractionally integrated moving-average (ARFIMA) processes which characterize both long-memory and short-memory behavior in time series, we formulate Bayesian inference using Markov chain Monte Carlo methods. We derive a form for the joint posterior distribution of the parameters that is computationally feasible for repetitive evaluation within a modi®ed Gibbs sampling algorithm that we employ. We illustrate our approach through two examples.

1/f noise in human cognition: Is it ubiquitous, and what does it mean?

“…Thornton and Gilden (2005) have demonstrated the use of maximum likelihood and Bayesian estimation in the spectral classifier; others have demonstrated the use of exact maximum likelihood (Hauser, 1999;Sowell, 1992), approximate maximum likelihood (Haslett & Raftery, 1989), prequential (Wagenmakers, Grünwald, & Steyvers, 2006, and Bayesian (Hsu & Breidt, 2003;Pai & Ravishanker, 1998) ARFIMA modeling. Table 1 also lists some differences between the two approaches.…”

Efficient Bayesian inference for natural time series using ARFIMA processes

“…Our method can be succinctly described as a modernization of the blocked MCMC method of Pai and Ravishanker (1998). Isolating parameters by blocking provides significant scope for modularization,…”

“…We follow earlier work (Koop et al, 1997;Pai and Ravishanker, 1998) and assume a priori independence for components of ψ. Each component will leverage familiar prior forms with diffuse versions as limiting cases.…”

“…Very few Bayesian methods have been studied, most probably due to computational difficulties. The earliest works are parametric and include Koop et al (1997), Pai and Ravishanker (1998), and Hsu and Breidt (2003). If computational challenges could be mitigated, the Bayesian paradigm would offer advantages over classical methods including flexibility in specification of priors (i.e., physical expertise could be used to elicit an informative prior).…”

“…Our sampling scheme can be best described as a modernization of a blocked MCMC scheme proposed by Pai and Ravishanker (1998) -adapting it to the approximate likelihood and extending it to handle a richer form of (known) short-memory effects. We then further extend the analysis to the case where the short-memory form is unknown, which requires trans-dimensional MCMC, in which the model order (the p and q parameters in the ARFIMA model) varies and, thus, so does the dimension of the problem.…”