Asymptotic Properties Of Parameter's Bayesian Estimation For ARFIMA Models

In a stationary ARMA process, the autocorrelation decays fast to 0 at an exponential rate. This process is called short-memory process. Yet in many empirical time series, the autocorrelation decays to 0 slowly and the dependence between distant observations is very strong. This behavior is named as long-memory. Granger, Joyeaux and Hosking proposed autoregressive fractionally integrated moving-average (ARFIMA) models, which exhibit long-memory and short-memory behavior. Many researchers have found application of the ARFIMA models in several areas such as economics, finance, geology and hydrology etc. In the last decades, a number of parameter estimation procedures have been proposed, such as the maximum likelihood estimation and Bayesian methods. The Bayesian estimation, which takes the prior information of parameters into consideration, is more efficient than other estimation techniques. Furthermore, the development of Markov chain Monte Carlo methods and Gibbs sampling algorithm have made Bayesian method computationally feasible and more efficient, which has been illustrated by many researchers via applying to real-life examples. But there is no any research about the asymptotic properties of the Bayesian estimation for the ARFIMA models because of the complexity of the parameter's estimation. In this paper, the marginal posterior distribution of the parameter is presented by Bayes theorem and we choose the mode of the marginal posterior distribution as the estimator. Then, followed the analysis of the asympo-totic properties of maximum likelihood estimation for the seasonal ARFIMA models, we prove the consistency, efficiency and asymptotic normality of the Bayesian estimator. The proof technique is based on an approximation of the spectral density proposed by Hannan. Finally, large sample performance of the Bayesian estimates is examined by simulations. It is shown that the estimates behave well when the sample size is large enough.