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Vector linear time series models

Published online by Cambridge University Press:  01 July 2016

W. Dunsmuir  and
E. J. Hannan
W. Dunsmuir
Affiliation:
Australian National University
E. J. Hannan
Affiliation:
Australian National University
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Abstract

This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models.

Keywords

LINEAR PROCESSESVECTOR ARMA MODELSIDENTIFICATIONLIMIT THEOREMSMARTINGALESFOURIER METHODS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 339 - 364
Copyright
Copyright © Applied Probability Trust 1976 

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References

Auslander, L. and Mackenzie, R. E. (1963) Introduction to Differentiable Manifolds. McGraw-Hill, New York.Google Scholar
Billingsley, P. (1961) The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12, 788–792.Google Scholar
Bloomfield, P. (1973) An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226.CrossRefGoogle Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time Series, Forecasting and Control. Holden Day, San Francisco.Google Scholar
Grenander, U. and Szego, G. (1958) Toeplitz Forms and their Applications. University of California Press, Berkeley.Google Scholar
Hannan, E. J. (1969) The identification of vector mixed autoregressive-moving average systems. Biometrika 56, 223–225.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle Scholar
Hannan, E. J. (1971) The identification problem for multiple equation systems with moving average errors. Econometrica 39, 751–765.Google Scholar
Hannan, E. J. (1973) The asymptotic theory of linear time series models. J. Appl. Prob. 10, 130–145.CrossRefGoogle Scholar
Hannan, E. J. (1975) The estimation of ARMA models. Ann. Statistics 3, 975–981.CrossRefGoogle Scholar
Hannan, E. J. (1976) Multivariate ARMA Theory. In Essays in Honour of Professor A. W. Phillips, ed. Peston, M. Wiley, New York.Google Scholar
Hannan, E. J. and Heyde, C. C. (1972) On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 43, 2058–2066.Google Scholar
Kang, K. M. (1973) A comparison of least squares and maximum likelihood estimation for moving average processes. Presented at the Australian Statistical Conference, Perth, 1973.Google Scholar
MacDuffee, C. C. (1956) The Theory of Matrices. Chelsea, New York.Google Scholar
Matsushima, Y. (1972) Differentiable Manifolds. Marcel Dekker, New York.Google Scholar
Pagan, A. R. (1971) Alternative estimates of ARMAX models. Working Paper No. 4 in Economics and Econometrics, Australian National University.Google Scholar
Tunnicliffe Wilson, G. (1973) The estimation of parameters in multivariate time series. J. R. Statist. Soc. B 35, 76–85.Google Scholar
Walker, A. M. (1964) Asymptotic properties of least squares estimates of parameters of the spectrum of a stationary non-deterministic time series. J. Austral. Math. Soc. 4, 363–384.Google Scholar
Ward, L. E. (1972) Topology. Marcel Dekker, New York.Google Scholar
Whittle, P. (1951) Hypothesis Testing in Time Series Analysis. Thesis, Uppsala University. Almqvist and Wiksell, Uppsala; Hafner, New York.Google Scholar
Whittle, P. (1961) Gaussian estimation in stationary time series. Bull. Inst. Int. Statist. 33, 1–26.Google Scholar
Zygmund, A. (1959) Trigonometric Series. Cambridge University Press.Google Scholar