Reparametrization Aspects of Numerical Bayesian Methodology for Autoregressive Moving‐average Models

Abstract

Within the context of likelihood and Bayes approaches to inference in autoregressive moving-average (ARMA) time series models, previous ideas on parameter transformation and numerical integration for implementing Bayesian procedures are reviewed. Some novel transformation ideas are introduced and their role in an efficient numerical integration approach is examined. Some comparisons of the effectivesness of different numerical integration strategies are made.

Bayesian Comparison of ARIMA and Stationary ARMA Models

“…The same principles as in the previous section apply to the prior for 4, while, as we saw there, either a noninformative or conjugate prior can be employed for a. We would propose a noninformative prior for y-the autoregressive and moving average parameters of (3.l)-following, for example, Box & Jenkins (1970), Monahan (1983), and Marriott & Smith (1992). This seems reasonable in practice, as typically one would expect the analyst to have little genuine prior information about these parameters.…”

Bayesian Comparison of ARIMA and Stationary ARMA Models

“…We would propose a noninformative prior for y-the autoregressive and moving average parameters of (3.l)-following, for example, Box & Jenkins (1970), Monahan (1983), and Marriott & Smith (1992). This seems reasonable in practice, as typically one would expect the analyst to have little genuine prior information about these parameters.…”

“…Finally, we note that the general form of the ARMA likelihood is so that analytic integration over 0 in (3.3) is possible. The numerical integration over y can easily be accomplished using the BAYES4 integration rules employed by Marriott & Smith (1992).…”

Robust Bayesian Estimation of Autoregressive‐‐moving‐average Models

“…Zellner (1971, Ch. 7), Box and Jenkins (1976, p. 250), Monahan (1984) and Marriott and Smith (1992). Both maximum likelihood and Bayesian parameter estimates can be badly affected by outliers.…”

“…By extending the ideas in McCulloch and Tsay (1994), Barnett et al (1996) show how to simultaneously choose the model orders of the regular and seasonal polynomials, impose stationarity, robustify against outliers and estimate missing observations. Marriott et al (1996) and Chib and Greenberg (1994) propose Markov chain Monte Carlo samplers for autoregressive-moving-average models which enforce stationarity and invertibility. Their approaches are not made robust to outliers and they do not consider model selection or model averaging.…”