Bayesian Analysis Influences Autoregressive Models

Abstract

The models, principles and steps of Bayesian time series analysis and forecasting have been established extensively during the past fifty years. In order to estimate parameters of an autoregressive (AR) model we develop Markov chain Monte Carlo (MCMC) schemes for inference of AR model. It is our interest to propose a new prior distribution placed directly on the AR parameters of the model. Thus, we revisit the stationarity conditions to determine a flexible prior for AR model parameters. A MCMC procedure is proposed to estimate coefficients of AR(p) model. In order to set Bayesian steps, we determined prior distribution with the purpose of applying MCMC. We advocate the use of prior distribution placed directly on parameters. We have proposed a set of sufficient stationarity conditions for autoregressive models of any lag order. In this thesis, a set of new stationarity conditions have been proposed for the AR model. We motivated the new methodology by considering the autoregressive model of AR(2) and AR(3). Additionally, through simulation we studied sufficiency and necessity of the proposed conditions of stationarity. The researcher, additionally draw parameter space of AR(3) model for stationary region of Barndorff-Nielsen and Schou (1973) and our new suggested condition. A new prior distribution has been proposed placed directly on the parameters of the AR(p) model. This is motivated by priors proposed for the AR(1), AR(2),..., AR(6), which take advantage of the range of the AR parameters. We then develop a Metropolis step within Gibbs sampling for estimation. This scheme is illustrated using simulated data, for the AR(2), AR(3) and AR(4) models and extended to models with higher lag order. The thesis compared the new proposed prior distribution with the prior distributions obtained from the correspondence relationship between partial autocorrelations and parameters discussed by Barndorff-Nielsen and Schou (1973). Keywords— Bayesian Analysis, Autoregressive Models, Time series, Stationarity.