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Band-limited spectral estimation of autoregressive-moving-average processes

Published online by Cambridge University Press:  14 July 2016

P. J. Thomson
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Abstract

Consider an autoregressive-moving-average process of given order where it is known that a number of moving-average roots are of unit modulus. Such a situation might arise, for example, when a time series has been differenced to induce stationarity by removing a non-stationary polynomial or seasonal trend. A band-limited spectral estimation procedure is proposed for estimating the coefficients of such a process and the asymptotic properties of the estimators investigated. The asymptotic theory is illustrated with reference to simulated and real data. A preliminary investigation of the use of Akaike's AIC criterion and this procedure to determine the number of roots of unit modulus (in the case where this is unknown) is also carried out by means of simulation.

The proposed band-limited spectral estimation procedure can also be used to take account of other possible effects met in practice. These include, for example, the band-limited response of a recording device or trend-contaminated low-frequency components.

Keywords

ARMA MODELTIME SERIES INFERENCEFOURIER METHODSFILTERINGASYMPTOTIC RESULTS
Type
Part 2—Estimation for Time Series
Information
Journal of Applied Probability , Volume 23 , Issue A: Essays in Time Series and Allied Processes , 1986 , pp. 143 - 155
Copyright
Copyright © 1986 Applied Probability Trust 

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References

Akaike, H. (1973) Information theory and an extension of the maximum likelihood principle. 2nd Internat. Symp. Information Theory , ed. Petrov, B. N. and Csaki, F., Akademiai Kiado, Budapest, 267–281.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1976) Time Series Analysis: Forecasting and Control (Revised edn). Holden-Day, San Francisco.Google Scholar
Brillinger, D. R. (1974) Fourier analysis of stationary processes. Proc. IEEE 62, 1628–1643.Google Scholar
Brillinger, D. R. (1975) Time Series Data Analysis and Theory. Holt, Rinehart and Winston, New York.Google Scholar
Davies, R. ?. (1983) Optimal inference in the frequency domain. In Handbook of Statistics , Vol. 3, ed. Brillinger, D. R. and Krishnaiah, P. R., North-Holland, Amsterdam, 73–92.Google Scholar
Dzhaparidze, K. O. (1977) Estimation of parameters of a spectral density with fixed zeroes. Theory Prob. Appl. 22, 708–729.Google Scholar
Hannan, E. J. (1969) The estimation of mixed autoregressive moving average models. Biometrika 56, 579–593.CrossRefGoogle Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle Scholar
Hannan, E. J. (1973) The asymptotic theory of linear time-series models. J. Appl. Prob. 10, 130–145.Google Scholar
Hannan, E. J. (1974) Time series analysis. IEEE Trans. Autom. Control AC-19, 706–715.CrossRefGoogle Scholar
Hannan, E. J. (1976) The asymptotic distribution of serial covariances. Ann. Statist. 4, 396–399.CrossRefGoogle Scholar
Hannan, E. J. and Thomson, P. J. (1973) Estimating group delay. Biometrika 60, 241–253.Google Scholar
Pham-Dinh, T. (1978) Estimation of parameters in the ARMA model when the characteristic polynomial of the MA operator has a unit zero. Ann. Statist. 6, 1369–1389.Google Scholar
Robinson, P. M. (1977) The construction and estimation of continuous time models and discrete approximations in econometrics. J. Econometrics 6, 173–197.Google Scholar