# zbMATH — the first resource for mathematics

Two-dimensional long memory models. (English) Zbl 0649.62086
The paper investigates the two-dimensional model $$(I-AB)^{\delta}X_ t=\epsilon_ t$$, $$0<\delta <$$, where B is the back-shift operator and A is a matrix with eigenvalues $$| \lambda_ j| \leq 1$$. Exact formulae for the spectral density and the covariance function R(t) are derived and the corresponding AR($$\infty)$$ and MA($$\infty)$$ representations are given. The cases of long memory dependence $$\sum_{t}\sum_{j,k}| R_{jk}(t)| =\infty$$ are described.
Reviewer: I.G.Zhurbenko
##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text:
##### References:
 [1] J. Anděl: Long memory time series models. Kybernetika 22 (1986), 105-123. · Zbl 0607.62111 [2] W. Dunsmuir, E. J. Hannan: Vector linear time series models. J. Appl. Probab. 10 (1973), 130-145. · Zbl 0261.62073 [3] G. M. Fichtengolc: Kurs differencialnogo i integralnogo isčislenija III. Gos. izd. fiz.-mat. lit., Moskva 1960. [4] J. Geweke, S. Porter-Hudak: The estimation and application of long memory time series models. J. Time Ser. Anal. 4 (1983), 221-238. · Zbl 0534.62062 [5] C. W. Granger, R. Joyeux: An introduction to long memory time series models and fractional differencing. J. Time Ser. Anal. 1 (1980), 15-29. · Zbl 0503.62079 [6] J. R. M. Hosking: Fractional differencing. Biometrika 68 (1981), 165-176. · Zbl 0464.62088 [7] A. I. McLeod, K. W. Hipel: Preservation of the rescaled adjusted range. 1. A reassessment of the Hurst phenomenon. Water Resour. Res. 14 (1978), 491-508.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.