Bayesian Identification, Extraction and Forecasting of Unobserved Components for Time Series in the Frequency Domains

This work aims to present a full Bayesian framework to identify, extract and forecast unobserved components in time series. The major novelty is to present a probabilistic framework to analyze the identification conditions. More precisely, informative prior distributions are assigned to the spectral densities of the unobserved components. This entails a interesting feature: the possibility to analyze more than one decomposition at once by studying the posterior distributions of the unobserved spectra. Particular attention is given to an empirical application where the canonical decomposition of sunspot data is compared with some alternative decompositions. The posterior distributions of the unobserved spectra are implemented to sample from the posterior distributions of the unobserved components; in doing so, some recent developments in the Wiener-Kolmogorov and circular process literature are exploited. An empirical application shows how to capture the seasonal component in the volatility of financial high frequency data. The posterior distributions of the unobserved spectra are finally implemented in order to sample from the posterior forecasting distributions of the unobserved components; this is obtained by exploiting the relationship between spectral densities and linear processes. An empirical application shows how to forecast seasonal adjusted financial time series. Finally, a generalization of the Bernstein-Dirichlet prior distribution is proposed in order to implement a frequency-pass spectral density estimator.