The Estimation of Continuous Parameter Long-Memory Time Series Models

Abstract

A class of univariate fractional ARIMA models with a continuous time parameter is developed for the purpose of modeling long-memory time series. The spectral density of discretely observed data is derived for both point observations (stock variables) and integral observations (flow variables). A frequency domain maximum likelihood method is proposed for estimating the longmemory parameter and is shown to be consistent and asymptotically normally distributed, and some issues associated with the computation of the spectral density are explored.

On the robustness to small trends of parameter estimation for continuous-time stationary models with memory

“…with some constants C > 0, γ > 0 and β > 1/4, then 10) and hence the asymptotic relations (3.4) and (3.5) …”

“…An example of a continuous-time model that displays the above defined memory structures is the continuous-time autoregressive fractionally integrated moving-average (CARFIMA) process (see, e.g., Chambers [10], Tsai and Chan [29]). …”

Estimation of fractional integration under temporal aggregation

“…For the special case of fractional integration, spectral results have been obtained by Chambers (1998), Hwang (2000, Chan (2005b), andSouza (2005). Further, Chambers (1996) and Tsai and Chan (2005a) cover the related case of discrete-time sampling from a continuous-time long memory process, while Souza (2007Souza ( , 2008 focusses on the effect of temporal aggregation on widely used memory estimators.…”

Fractional random fields associated with stochastic fractional heat equations

“…Recent studies have found that data in many fields of application (including geostatistics, hydrology, turbulence, economics, and finance) display fractal structure (see Adler (1981), Hosking (1981), Chambers (1996), Woyczyński (1998), Hilfer (2000), Christakos (2000), and the references therein).…”

“…Earlier, Granger and Joyeux (1980) and Hosking (1981) constructed long-memory time series in discrete time via fractional differencing, and Chambers (1996) used fractional derivatives to obtain long-memory phenomena for continuous-time stochastic processes. Some other examples of fractional random fields can be found in Anh et al (1999), Anh and Leonenko (2000), (2001), (2002) and Ruiz-Medina et al (2001), (2003), (2004).…”