How can we Define the Concept of Long Memory? An Econometric Survey

Abstract

In this paper we discuss different aspects of long memory behavior and applicable parametric models. We discuss the confusion that can arise when the empirical autocorrelation function decreases in a hyperbolic way.Estimation theory, Long memory, Returns, Spectral domain, Switching,

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

“…Among three main definitions of memory in the case of finite variance in the mentioned paper [12], there is one that uses the so-called Allan variance. It turns out that this notion can be extended to the case of infinite variance.…”

“…Three-four decades ago, the notion of weak dependence, measured by different quantities such as Rosenblat or Ibragimov mixing coefficients, was very popular and was well suited for theoretical purposes, but later it was realized that it was very difficult to estimate these quantities, and the notions of long-and short-range dependencies became more popular since they were based on covariances, the quantities that can be estimated quite easily. However, it is necessary to note that the notions of long-and short-range dependencies can be based not only on covariances; there are several ways to define long-range dependence; see, for example, important survey paper [34] or [12], where even eight definitions are provided, three of them are the main definitions (via the covariance function; via the spectral density; and via the growth of partial sums, the so-called Allan variance), and the other five are only modifications.…”

Memory parameter estimation for long range dependent random fields

“…Long range spatiotemporal correlation is often involved in the study of deterministic chaos in the spatiotemporal pattern of dynamic systems (Anh and Lunney, 1995). For more details about analysis and applications of long memory time series and long memory random fields in general, we refer the readers to Dobrushin and Major (1979), Ivanov and Leonenko (1989), Beran (1994), Leonenko (1999), Doukhan et al (2003), Guégan (2005), Lavancier (2006), Palma (2007) and the references therein.…”

Forecasting electricity spot market prices with a k-factor GIGARCH process

“…The autocorrelation function does not converge exponentially to zero in covariance sense, (Guégan, 2005, for discussions on this concept), thus we assume that the series has a long range dependence behavior. Moreover, the spectral density represented by the periodogram (Figure 5), is unbounded at three frequencies corresponding respectively, to the daily, weekly and half daily seasonalities.…”