Analysis Of Stationary And Non-stationary Long Memory Processes: Estimation, Applications And Forecast

In this thesis, we consider two classes of long memory processes:the stationary long memory processes and the non-stationary long memory processes. We are devoted to the study of their probabilistic properties, estimation methods, forecast methods and the statistical tests.Stationary long memory processes have been extensively studied over the past decades. It has been shown that some long memory processes have the properties of self-similarity, which are important for parameter estimation. We review the self-similar properties of continuous-time and discrete-time long memory processes. We establish the propositions that stationary long memory process is asymptotically second-order self-similar, while stationary short memory process is not asymptotically second-order self-similar. Then we extend the results to specific long memory processes such as k-factor GARMA processes and k-factor GIGARCH processes. We also investigate the self-similar properties of some heteroscedastic models and the processes with switches and jumps.We make a review for the stationary long memory processes' parameter estimation methods, including the parametric methods (for example, maximum likelihood estimation, approximate maximum likelihood estimation) and the semiparametric methods (for example, GPH method, Whittle method, Robinson method). The consistency and asymptotic normality behaviors are also investigated for the estimators.Testing the fractionally integrated order of seasonal and non-seasonal unit roots of the stochastic stationary long memory process is quite important for the economic and financial time series modeling. The widely used Robinson test (1994) is applied to various well-known long memory models. Via Monte Carlo experiments, we study and compare the performances of this test using several sample sizes, which provide a good reference for the practitioners who want to apply Robinson's test.In practice, seasonality and time-varying long-range dependence can often be observed and thus some kind of non-stationarity exists inside the economic and financial data sets. To take into account this kind of phenomena, we review the existing non-stationary processes and we propose a new class of non-stationary stochastic process:the locally stationary k-factor Gegenbauer process. We describe a procedure of estimating consistently the time-vary ing parameters with the help of the discrete wavelet packet transform (DWPT). The consistency and asymptotic normality of the estimates are proved. The robustness of the algorithm is investigated through simulation study.We also propose the forecast method for this new non-stationary long memory processes. Applications and forecasts based on the error correction term in the error correction model of the Nikkei Stock Average 225 (NSA 225) index and theWest Texas Intermediate (WTI) crude oil price are followed.