Topics on unobserved component detection for time series

The Impact of Vintage on the Persistence of Gross Domestic Product Shocks. The first chapter of the thesis aims to demonstrate that the data revision process affects the persistence of gross domestic product shocks. The analysis is based on two alternative models, the Fractional Unit Root and the Linear Trend, and it benefits from new semiparametric procedures. The analysis of results seems to suggest that changes in the definition of the output significantly affects the performance of the models which are typically used to study the GDP series. - Seasonality in HighFrequency Data. The second chapter of the thesis aims to study the intradaily seasonal pattern of the Dow Jones volatility. An unobserved component analysis, following the socalled ModelBased approach, is examined in order to separate the seasonal pattern from the remaining long and short term components. The major novelty of the work is to explore the seasonal behavior of the volatility as a stochastic component which evolves over time according to a specific ARMA model. In more detail, high frequency data are used to recover 30minute realized volatility measures. Particular attention has been devoted to checking whether estimates were robust to market microstructure, jumps and outliers. The analysis of results emphasizes that the volatility of the Dow Jones is characterized by a stochastic seasonal component which recalls the “Ushape” pattern. - Bayesian Unobserved Components in Time Series. The third chapter of the thesis aims to present a full Bayesian framework to identify, extract and forecast unobserved components in time series. The major novelty of the approach is the definition of 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 components are recovered by exploiting some recent developments in the WienerKolmogorov and circular process literature. An empirical application shows how to capture the seasonal component in the volatility of financial high frequency data. The posterior forecasting distributions are finally recovered by exploiting a relationship between spectral densities and linear processes. An empirical application shows how to forecast seasonal adjusted financial time series. Finally, a generalization of the BernsteinDirichlet prior distribution is proposed in order to implement a frequencypass spectral density estimator. - Objective Priors for AR(p) models. The fourth chapter of the thesis aims to derive objective prior distributions for general autoregressive models. The core of the paper is based on the study of the Jeffreys' principle and on a particular adaptation of the “reference” algorithm for dependent data. The analysis of stationarity turns out to be rather complicated for general autoregressive processes. Therefore, two alternative parameterizations which respectively depend on the partial autocorrelation functions (PACFs) and the roots are considered. For the causal and stationary parameter subspace, the PACFs and the roots are always defined within the unit circle. This simplifies the derivation of the prior distributions. Two main results are obtained. The first is a general formula which depends on the PACFs and represents the Jeffreys' prior distribution for the causal subspace. The second result depends on the roots of the process and represents a particular reference prior distribution which is asymptotically independent from the initial conditions and covers the noncausal subspace. Ultimately, simulation results are presented for the autoregressive process of order two.