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On the relationship between Levinson recursion and the R and S arrays for ARMA model identification. (English) Zbl 0594.62102

Summary: Motivated by the papers of W. A. Woodward and H. L. Gray [On the relationship between the R and S arrays and the Box-Jenkins method of ARMA model identification. Tech. Rep. 134, Dpt. Stat., ONR Contract, SMU, Dallas, Texas (1979)] and H. L. Gray, G. D. Kelly and D. D. McIntire [Commun. Stat., Simulation Comput. B 7, 79-96 (1978; Zbl 0385.62063)] on the R and S array approach to ARMA modeling, the authors show that the R and S array algorithm is completely equivalent to Levinson recursion. Since entries in the R and S array can be computed by either algorithm, the equivalence provides greater insight into the R and S methodology as well as its links to Akaike’s AIC or FPE.
Numerical simulations serve to highlight the differences between the various approaches as well as illustrate the problems associated with exact methods. The R and S array approach is shown to be an effective procedure for determining ARMA model orders.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Citations:

Zbl 0385.62063
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Full Text: DOI

References:

[1] Akaike H., System Identfication: Advances and Case Studies (1976)
[2] Akaike H., Ann.Inst.Statist.Math 21 pp 243– (1969) · Zbl 0202.17301 · doi:10.1007/BF02532251
[3] Akaike H., Ann.Inst.Statist.Math 22 pp 203– (1970) · Zbl 0259.62076 · doi:10.1007/BF02506337
[4] Akaike H., Computer Science Monographs #11 (1978)
[5] Bednar J.B., IEEE Trans. on ASSP 22 (1982)
[6] Box G.E.P., Time Series Analysis:Forecasting and Control (1976)
[7] de Ballore M., Bull.Soc.Math pp 28– (1902)
[8] Durbin J., Rev.Inst.Int.Stat. 28 pp 233– (1960) · Zbl 0101.35604 · doi:10.2307/1401322
[9] Friedlander B., Linear Algebra and Its Applications 27 pp 31– (1979) · Zbl 0414.15005 · doi:10.1016/0024-3795(79)90030-2
[10] Gray H.L., Comm.of Stat.–Sirnula. and Comput 7 (1) pp 1– (1978) · doi:10.1080/03610917808812057
[11] Levinson N., J.Math.Phy. 25 (1) pp 261– (1947)
[12] Luenberger D.G., Optimization by Vector Space Methods (1969) · Zbl 0176.12701
[13] Makridakis S., Interactive Forecasting (1978)
[14] Parzen E., IEEE Trans. on Automatic Control 19 pp 723– (1974) · Zbl 0317.62063 · doi:10.1109/TAC.1974.1100733
[15] Pye W.D., SIAM J.Numerical Analysis 10 pp 1– (1973) · Zbl 0257.65109 · doi:10.1137/0710001
[16] Trench W.F., SIAM 7 (3) pp 515– (1964)
[17] Woodward W.A., On the relationship between the R and S arrays and the Box-Jenkins method of ARMA model identification (1979)
[18] Yule G.U., Phil.Trans 226 (3) pp 267– (1927) · JFM 53.0509.02 · doi:10.1098/rsta.1927.0007
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