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Spectral Factorization of Periodically Correlated MA(1) Processes

Part of: Convergence and divergence of infinite limiting processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mohamed Bentarzi  and
Marc Hallin
Mohamed Bentarzi*
Affiliation:
Université des Sciences et Techniques d'Alger
Marc Hallin*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Institut de Mathématiques, Université des Sciences et Techniques d'Alger, B.P. 9 Dar El Beida, Alger, Algeria
∗∗Postal address: Institut de Statistique, Université Libre de Bruxelles, cp 210, B1050 Bruxelles, Belgium
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Abstract

The spectral factorization problem, i.e. the problem of obtaining all possible MA representations of a process with given autocovariance function, is considered for univariate, d-periodic MA(1) (equivalently, 1-dependent in the second-order sense) processes. The solutions are provided explicitly, and their invertibility properties are investigated. A characterization, in terms of their autocovariance functions, of non-invertible d-periodic 1-dependent processes, extending to the periodic case the traditional unit root condition, is provided.

Keywords

Periodic modelsmoving average modelsinvertibility

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc. 40A15: Convergence and divergence of continued fractions
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 46 - 54
Copyright
Copyright © Applied Probability Trust 1998 

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References

Adams, G.J., and Goodwin, G.C. (1995). Parameter estimation for periodic ARMA models. J. Time Series Anal. 16, 127–145.CrossRefGoogle Scholar
Anderson, P.L., and Vecchia, A.V. (1993). Asymptotic results for periodic autoregressive moving-average processes. J. Time Series Anal. 14, 1–18.CrossRefGoogle Scholar
Bentarzi, M., and Hallin, M. (1994). On the invertibility of periodic moving-average models. J. Time Series Anal. 15, 263–268.CrossRefGoogle Scholar
Cipra, T. (1985). Periodic moving average processes. Aplikace Matematik. 30, 218–229.Google Scholar
Cramér, H. (1961). On a class of nonstationary stochastic processes. Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability 2, 57–78.Google Scholar
Franses, P.H. (1993). A multivariate approach to modeling univariate seasonal time series. J. Econometrics 63, 133–151.CrossRefGoogle Scholar
Ghysels, E., and Hall, A. (1992). Periodically correlated random sequences. Discussion paper. Université de Montréal.Google Scholar
Gladyshev, E.G. (1961). Periodically correlated random sequences. Soviet Math. 2, 385–388.Google Scholar
Hallin, M. (1984). Spectral factorization of nonstationary moving average processes. Ann. Statist. 12, 172–192.CrossRefGoogle Scholar
Hallin, M. (1986). Nonstationary q-dependent processes and time-varying moving average models. Adv. Appl. Prob. 18, 170–210.CrossRefGoogle Scholar
Jones, R.H., and Brelsford, W.M. (1967). Time series with periodic structure. Biometrika 54, 403–408.CrossRefGoogle ScholarPubMed
Lütkepohl, H. (1991). Introduction to Multiple Time Series Analysis. Springer, Berlin.CrossRefGoogle Scholar
McLeod, A.I. (1993). Parsimony, model adequacy and periodic correlation in time series analysis forecasting. Int. Statist. Rev. 61, 387–393.CrossRefGoogle Scholar
Pagano, M. (1978). On periodic and multiple autoregressions. Ann. Statist. 6, 1310–1317.Google Scholar
Tiao, G.C., and Grupe, M.R. (1980). Hidden periodic autoregressive-moving average models in time series data. Biometrika 67, 365–373.Google Scholar
Troutman, B.M. (1979). Some results in periodic autoregression. Biometrika 66, 219–228.CrossRefGoogle Scholar