Robust Bayesian Estimation of Autoregressive‐‐moving‐average Models

Abstract

A Bayesian approach is presented for modeling a time series by an autoregressive-moving-average model. The treatment is robust to innovation and additive outliers and identifies such outliers. It enforces stationarity on the autoregressive parameters and invertibility on the moving-average parameters, and takes account of uncertainty about the correct model by averaging the parameter estimates and forecasts of future observations over the set of permissible models. Posterior moments and densities of unknown parameters and observations are obtained by Markov chain Monte Carlo in O(n) operations, where n is the sample size. The methodology is illustrated by applying it to a data set previously analyzed by Martin, Samarov and Vandaele (Robust methods for ARIMA models.

Bayesian model selection for unit root testing with multiple structural breaks

“…, 1 are implicitly assumed to have non-zero coefficients. However due to the special form of the ADF-type test regression (1), more efficient approaches as in Chen (1999), Gerlach et al (2000) and Barnett et al (1996Barnett et al ( , 1997 cannot be applied here directly. The presented approach is most similar to that of Troughton and Godsill (1997a), who propose a sampling scheme for a lag order selection in autoregressive models, see also Godsill (2001) and Ehlers and Brooks (2002).…”

“…So far many approaches to model selection in time series models have been proposed in the Bayesian literature. Among the many works that focus on lag order determination in ARMA models are Barnett et al (1996Barnett et al ( , 1997, Huerta and West (1999), Chen (1999), Gerlach et al (2000), Vermaak et al (2004), Ehlers and Brooks (2004) and Philippe (2006), inter alia. Many of the existing works that deal with the detection of change points treat the selection of the number of breaks as a successive problem, which is solved by using information criteria or Bayes factors, but do not treat the number of change points together with the number of lags as additional unknown parameters in their sampling schemes, see for example Chib (1998), Zivot and Wang (2000) and Koop and Potter (2004).…”

Bayesian Enhancement of Speech and Audio Signals which can be Modelled as ARMA Processes

“…We use a form of ARMA model which leads to Gaussian conditionals for the AR and MA coefficients and discuss an efficient scheme for detecting both innovational and observational outliers (impulses), which forms an integral part of the MCMC simulation. We discuss the relationship of the methods to existing MCMC techniques for ARMA modelling (Chib & Greenberg, 1994; Marriott, Ravishanker, Gelfand & Pai, 1996; Barnett, Kohn & Sheather, 1996).…”

“…no background component) models used by Godsill (1993) and Godsill & Rayner (1995a) for the processing of audio signals and more recently implemented using MCMC methods (Godsill & Rayner, 1996b). This latter work is closely related to MCMC work in statistical outlier analysis (McCulloch & Tsay, 1994;Barnett et al, 1996).…”

“…This might also be expected to give slightly better convergence since changing the value of a single uI affects the subsequent q + 1 values of the restored signal x,. Also, Barnett et al (1996) prohibit the condition it = j , = 1 on grounds of identifiability and computation. Although only a minor point, this seems to us to be wrong if innovational and observational outlier processes are to be regarded as independent: if there is posterior uncertainty about whether an outlier is innovational or observational (or bpth) then that uncertainty will automatically be accounted for in the posterior moments estimated by the MCMC method.…”