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Computation of vector ARMA autocovariances. (English) Zbl 1417.62253
Summary: This note describes an algorithm for computing the autocovariance sequence of a VARMA process, without requiring the intermediary step of determining the Wold representation. Although the recursive formula for the autocovariances is well-known, the initialization of this recursion in standard treatments (such as [P. J. Brockwell and R. A. Davis, Time series: theory and methods. 2nd ed. Berlin etc.: Springer-Verlag (1991; Zbl 0709.62080); H. Lütkepohl, New introduction to multiple time series analysis. Corrected 2nd printing. Berlin: Springer (2006; Zbl 1141.62071)] is slightly nuanced; we provide explicit formulas and algorithms for the initial autocovariances.
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
65C50 Other computational problems in probability (MSC2010)
Full Text: DOI
[1] Barone, P., A method for generating independent realizations of a multivariate normal stationary and invertible ARMA(\(p, q\)) process, J. Time Series Anal., 8, 125-130, (1987) · Zbl 0608.62109
[2] Brockwell, P.; Davis, R., Time series: theory and methods, (1991), Springer New York
[3] Lütkepohl, H., New introduction to multiple time series analysis, (2007), Springer-Verlag Berlin
[4] Mittnik, S., Computation of the theoretical autocovariance matrices of multivariate autoregressive moving average time series, J. R. Stat. Soc. Ser. B Stat. Methodol., 52, 151-155, (1990) · Zbl 0699.62085
[5] Mittnik, S., Computing theoretical autocovariances of multivariate autoregressive moving average models by using a block Levinson method, J. R. Stat. Soc. Ser. B Stat. Methodol., 55, 435-440, (1993) · Zbl 0797.62074
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