Bayesian time series analysis of periodic behaviour and spectral structure

Analysis, model selection and forecasting in univariate time series models can be routinely carried out for models in which the model order is relatively small. Under an ARMA assumption, classical estimation, model selection and forecasting can be routinely implemented with the Box–Jenkins time domain representation. However, this approach becomes at best prohibitive and at worst impossible when the model order is high. In particular, the standard assumption of stationarity imposes constraints on the parameter space that are increasingly complex. One solution within the pure AR domain is the latent root factorization in which the characteristic polynomial of the AR model is factorized in the complex domain, and where inference questions of interest and their solution are expressed in terms of the implied (reciprocal) complex roots; by allowing for unit roots, this factorization can identify any sustained periodic components. In this paper, as an alternative to identifying periodic behaviour, we concentrate on frequency domain inference and parameterize the spectrum in terms of the reciprocal roots, and, in addition, incorporate Gegenbauer components. We discuss a Bayesian solution to the various inference problems associated with model selection involving a Markov chain Monte Carlo (MCMC) analysis. One key development presented is a new approach to forecasting that utilizes a Metropolis step to obtain predictions in the time domain even though inference is being carried out in the frequency domain. This approach provides a more complete Bayesian solution to forecasting for ARMA models than the traditional approach that truncates the infinite AR representation, and extends naturally to Gegenbauer ARMA and fractionally differenced models.