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The asymptotic theory of linear time-series models

Published online by Cambridge University Press:  14 July 2016

E. J. Hannan
E. J. Hannan*
Affiliation:
The Australian National University
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Abstract

A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

Keywords

TIME SERIES INFERENCEFOURIER METHODSMARTINGALE CENTRAL LIMIT THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 130 - 145
Copyright
Copyright © Applied Probability Trust 1973 

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References

Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis. Holden-Day, San Francisco.Google Scholar
Brown, B. M. (1971) Martingale central limit theorems. Ann. Math. Statist. 42, 59–66.Google Scholar
Eicker, F. (1967) Limit theorems for regression with unequal and dependent errors. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 1, 59–82. Univ. California Press.Google Scholar
Grenander, U. and Szego, G. (1958) Toeplitz Forms and their Application. Univ. California Press.CrossRefGoogle Scholar
Hannan, E. J. (1970) Multiple Time Series. John Wiley, New York.CrossRefGoogle Scholar
Hannan, E. J. (1971) Non-linear time series regression. J. Appl. Prob. 8, 767–780.CrossRefGoogle Scholar
Hannan, E. J. and Heyde, C. C. (1972) On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 44, 2058–2066.Google Scholar
Hoffman, K. (1967) Banach Spaces of Analytic Functions. Prentice Hall, New Jersey.Google Scholar
Scott, D. J. (1973) Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Adv. Appl. Prob. 5, 118–136.CrossRefGoogle Scholar
Walker, A. M. (1964) Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series. J. Aust. Math. Soc. 4, 363–384.CrossRefGoogle Scholar
Whittle, P. (1951) Hypothesis Testing in Time Series Analysis. , Uppsala University. Almqvist and Wiksell, Uppsala; Hafner, New York.Google Scholar
Whittle, P. (1962) Gaussian estimation in stationary time series. Bull. Indian Stand. Inst. 39, 105–129.Google Scholar
Zygmund, A. (1959) Trigonometric Series. Cambridge University Press.Google Scholar