Analysis of stationary and non-stationary long memory processes: estimates, applications and predictions.

Summary In this thesis, we consider two types of long memory processes: stationary and non-stationary processes. We focus on the study of their statistical properties, estimation methods, prediction methods and statistical tests. Stationary long memory processes have been widely studied in the last decades. It has been shown that long memory processes have self-similarity properties, which are important for parameter estimation. We review the self-similarity properties of long memory processes in continuous and discrete time. We propose two propositions showing that long memory processes are asymptotically second order self-similar, while short memory processes are not asymptotically second order self-similar. Then, we study the self-similarity of specific long memory processes such as k-factor GARMA processes and k-factor GIGARCH processes. We also study the self-similarity properties of heteroscedastic models and processes with jumps. We review the methods for estimating the parameters of long memory processes, by parametric methods (e.g., maximum likelihood estimation and pseudo-maximum likelihood estimation) and semiparametric methods (e.g., GPH method, Whittle method, Robinson method). The consistency and asymptotic normality behaviors are also studied for these estimators. The test on the integrated fractional order of the seasonal and non-seasonal unit root of long memory stationary processes is very important for the modeling of economic and financial series. The Robinson (1994) test is widely used and applied to various well-known long memory models. Using Monte Carlo methods, we study and compare the performance of this test using several sample sizes. This work is important for practitioners who want to use the Robinson test. In practice, when dealing with financial and economic data, seasonality and time dependence can often be observed. Thus a kind of non-stationarity exists in financial data. In order to take into account such phenomena, we review non-stationary processes and propose a new class of stochastic processes: locally stationary k-factor Gegenbauer processes. We propose a procedure for estimating the parameter function using the discrete wavelet packet transformation (DWPT). The robustness of the algorithm is studied by simulations. We also propose prediction methods for this new class of non-stationary long memory processes. We give applications on the error correction term of the fractional cointegration analysis of the Nikkei Stock Average 225 index and we study world crude oil prices.