A Note On Long Memory Time Series
and VIVIEN GUIRAUD
Louis Pasteur de Strasbourg, Faculte
des Sciences Economiques et de
Gestion, Strasbourg Cedex, France. E-mail: firstname.lastname@example.org
Abstract. This note presents the fractional integrated processes which are the main models
used to describe long memory phenomena.
Section 1 brieﬂy deﬁnes the concept of fractional
integration, shows the fundamental properties and provides a short summary of the estimation
methods. Section 2 consists of a survey of their extensions in order to model long-term cycles.
Key words: fractional integration, long memory
1. Fractional Integrated Processes
THE CONCEPT OF FRACTIONAL INTEGRATION
The class of fractional integrated processes is an extension of the class of auto
regressive integrated moving average (ARIMA) processes stemming from
Box and Jenkins methodology. One of the originalities is the explicit
modelling of the long term correlation structure.
According to the values of parameters, these processes will possess the
long term dependence property or long memory introduced by Hurst (1957)
and Mandelbrot (1968).
; t ¼ 1 ...n be a time series and q(k) its autocorrelation function:
. The stationarity property is veriﬁed if
In this case, it is said that x
has the long memory property if
Author for Correspondence: Claude Diebolt, BETA/CNRS, Universite
Louis Pasteur de
des Sciences Economiques et de Gestion, 61 Avenue de la Foreˆ t Noire,
67085 Strasbourg Cedex, France. Tel. 33(0)22.214.171.124.87 (direct line), Fax. 03.90.24.20.70,
Quality & Quantity (2005) 39:827–836 Ó Springer 2005