CHECKING STATIONARITY AND INVERTIBILITY IN TIME SERIES MODELS—FINDING THE INVERTIBLE FORM IN THE VECTOR CASE

REFERENCES

• Box , G. E. P. and Jenkins , G. M. 1970 . Time Series Analysis: Forecasting and Control San Francisco : Holden-Day .
   Google Scholar
• Hannan , E. J. 1969 . The identification of vector mixed autoregressive-moving average systems . Biometrika , 56 : 223 – 225 .
   Web of Science ®Google Scholar
• Hannan , E. J. and Deistler , M. 1988 . The Statistical Theory of Linear Systems New York : Wiley .
   Google Scholar
• LÜtkepohl , H. 1993 . Introduction to Multiple Time Series Analysis New York : Springer-Verlag .
   Google Scholar
• Box , G. E. P. and Luceño , A. 1997 . Statistical Control by Monitoring and Feedback Adjustment New York : Wiley .
   Google Scholar
• Ansley , C. F. 1979 . An algorithm for the exact likelihood of a mixed autoregressive-moving average process . Biometrika , 66 : 59 – 65 .
   Web of Science ®Google Scholar
• Brockwell , P. J. and Davis , R. A. 1987 . Time Series: Theory and Methods New York : Springer-Verlag .
   Google Scholar
• Luceño , A. 1994 . A fast algorithm for the exact likelihood of stationary and partially nonstationary vector autoregressive moving average processes . Biometrika , 81 : 555 – 565 .
   Web of Science ®Google Scholar
• Luceño , A. 1997 . Estimation of missing values in possibly partially nonstationary vector time series . Biometrika , 84 : 495 – 499 .
   Web of Science ®Google Scholar
• Luceño , A. 1998 . Detecting possibly nonconsecutive outliers in industrial time series . Journal of the Royal Statistical Society Series B , 60 : 295 – 310 .
   Google Scholar
• Åström , K. J. and Wittenmark , B. 1997 . Computer-controlled Systems. Theory and Design , 3rd Ed. Upper Saddle River, NJ : Prentice Hall .
   Google Scholar
• Ogunnaike , B. A. and Ray , W. H. 1994 . Process Dynamics, Modeling, and Control Oxford : Oxford University Press .
   Google Scholar
• Dahlquist , G. and Björck , Å. 1974 . Numerical Methods Englewood Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Anderson , O. D. 1975 . The recursive nature of the stationarity and invertibility restraints on the parameters of mixed autoregressive-moving average processes . Biometrika , 62 : 704 – 706 .
   Web of Science ®Google Scholar
• Cohn , A. 1922 . Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise . Mathematische Zeitschrift , 14 : 110 – 148 .
   Google Scholar
• Hurwitz , A. 1895 . Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitz . Mathematische Annalen , 46 : 273 – 284 .
   Google Scholar
• Jury , E. I. and Blanchard , J. 1961 . Stability test for linear discrete systems in table form . Proceedings of the Institute of Radio Engineers , 49 : 1947 – 1948 .
   Google Scholar
• Levinson , N. 1946 . The Wiener root mean square error criterion in filter design and prediction . Journal of Mathematical Physics , 25 : 261 – 278 .
   Web of Science ®Google Scholar
• Routh , E. J. 1877 . A Treatise on the Stability of a Given State of Motion London : Macmillan and Co .
   Google Scholar
• Samuelson , P. A. 1941 . Conditions that the roots of a polynomial be less than unity in absolute value . Annals of Mathematical Statistics , 12 : 360 – 364 .
   Google Scholar
• Schur , I. 1917 . Über Potensreihen in Innern des Einhheitskreises . Journal fÜr die Reine und Angewante Mathematik , 147 : 215 – 232 .
   Google Scholar
• Wise , J. 1956 . Stationarity conditions for stochastic processes of the autoregressive moving average type . Biometrika , 43 : 215 – 219 .
   Web of Science ®Google Scholar
• Copson , E. T. 1970 . An Introduction to the Theory of Functions of a Complex Variable Oxford : Clarendon Press .
   Google Scholar
• Lang , S. 1977 . Complex Analysis Reading, Massachusetts : Addison-Wesley .
   Google Scholar
• Barnett , S. and Cameron , R. G. 1985 . Introduction to Mathematical Control Theory , 2nd Ed. Oxford : Clarendon Press .
   Google Scholar
• Press , W. H. , Teukolsky , S. A. , Vetterling , W. T. and Flannery , B. P. 1992 . Numerical Recipes Cambridge : Cambridge University Press .
   Google Scholar
• Stoer , J. and Bulirsch , R. 1993 . Introduction to Numerical Analysis , 2nd Ed. New York : Springer-Verlag .
   Google Scholar
• Wilkinson , J. H. 1965 . The Algebraic Eigenvalue Problem Oxford : Clarendon Press .
   Google Scholar
• Mccullagh , P. 1994 . Does the moment-generating function characterize a distribution? . American Statistician , 48 : 208
   Web of Science ®Google Scholar
• Luceño , A. 1997 . Further evidence supporting the numerical usefulness of characteristic functions . American Statistician , 51 : 233 – 234 .
   Web of Science ®Google Scholar
• Golub , G. H. and Van Loan , C. F. 1987 . Matrix Computations Baltimore : Johns Hopkins University Press .
   Google Scholar
• Ansley , C. F. and Kohn , R. 1985 . On the rate of convergence of the innovation representation of a moving average process . Biometrika , 72 : 325 – 330 .
   Web of Science ®Google Scholar
• Newton , H. J. 1980 . Effcient estimation of multivariate moving average autocovariances . Biometrika , 67 : 227 – 231 .
   Web of Science ®Google Scholar
• Tunnicliffe-Wilson , G. 1972 . The factorization of matricial spectral densities . SIAM Journal on Applied Mathematics , 23 : 420 – 426 .
   Web of Science ®Google Scholar
• Tunnicliffe-Wilson , G. 1978 . A convergence theorem for spectral factorization . Journal of Multivariate Analysis , 8 : 222 – 232 .
   Web of Science ®Google Scholar
• Tunnicliffe-Wilson , G. 1994 . “ Developments in multivariate covariance generation and factorization ” . In Developments in Time Series, a Volume in Honor of Maurice B. Priestly Edited by: Rao , T. Subba . 26 – 36 . London : Chapman Hall .
   Google Scholar