ARFIMA Process : Tests and Applications at a White Noise Process, A Random Walk Process and the Stock Exchange Index CAC 40

The assumption of linearity is implicitly accepted in the process which generates a time series condition submitted to a ARIMA. That is why, in this paper, we shall discuss the research of long memory in the processes: the fractional ARIMA models, denoted as ARFIMA, where d and D, the degree of differentiation of the filters is not integer. After presenting the characteristics of the ARFIMA process, we shall discuss the long-memory tests (statistics rescaled Range Lo and R/S* Moody and Wu). Finally three examples and tests on a white noise process, a random walk model and the stock index of Paris Stock Exchange (CAC40) will illustrate the method.