A note on calculating autocovariances of long-memory processes. (English) Zbl 1062.62164

This paper deals with a splitting method for calculating the autocovariances of fractional integrated processes ARFIMA and generalized integrated processes GARMA. Autocovariances of the ARFIMA\((0,d,0)\) process and of the GARMA(0,0) process are defined as \[ \gamma^*(k)=\sigma^2\Gamma(k+1)\Gamma(1-2d)(\Gamma(k+1-d)\Gamma(1-d)\Gamma(d))^{-1}, \]
\[ \gamma^*=(\sigma^2/2\sqrt{\pi})\Gamma(1-2\lambda)(2\sin\nu)^{1/2-2\lambda}[P^{2\lambda-1/2}_{k-1/2}(\eta)+ (-1)^{k}P^{2\lambda-1/2}_{k-1/2}(-\eta)], \] where \(\nu=\cos^{-1}(\eta)\) and \(P_{a}^{b}(c)\) is the associate Legendre function of the first kind, \(\Gamma(x)\) is the Gamma function, and \(\sigma^2\) is the variance of the noise. The authors consider a group of ARFIMA and GARMA models with \(p\neq0\) and/or \(q\neq0\). The long memory models are studied numerically, considering parameters \(d\) and \(\lambda\) ranging in \((0,1/2)\). In the case of \(d\) and \(\lambda\in(-1/2,0)\) results are similar. The considered methods generate close theoretical values but splitting is faster and more precise in the case of GARMA\((p,q)\) processes.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
65C60 Computational problems in statistics (MSC2010)
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[1] DOI: 10.1016/0304-4076(95)01732-1 · Zbl 0854.62099
[2] 2J. A. BERAN(1994 )Statistics for long memory processes, New York: Chapman & Hall.
[3] 3P. J. BROCKWELL, and R. DAVIS(1991 )Time series: Theory and methods. NY. Springer-Verlag.
[4] DOI: 10.1016/0165-1765(94)90026-4 · Zbl 0825.62672
[5] Journal of Time Series Analysis 17 pp 111– (1994)
[6] DOI: 10.1016/0304-4076(95)01739-9 · Zbl 0854.62083
[7] GRANGER C. W. J., Journal of Time Series Analysis 1 pp 15– (1980) · Zbl 0541.62106
[8] GRAY H. L., Journal of Time Series Analysis 10 pp 233– (1988)
[9] - and -, Journal of Time Series Analysis 15 pp 561– (1994)
[10] 10J. R. M. HOSKING(1981 )Fractional, differencing,Biometrika, 68 , 165 -76 .
[11] Water Resource Research 20 pp 1898– (1984)
[12] DOI: 10.1016/0304-4076(92)90084-5 · Zbl 04508734
[13] DOI: 10.1016/0304-3932(92)90016-U
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