Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T21:08:22.098Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-stationary q-dependent processes and time-varying moving-average models: invertibility properties and the forecasting problem

Published online by Cambridge University Press:  01 July 2016

Marc Hallin
Marc Hallin*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Institut de Statistique, Université Libre de Bruxelles, C.P. 210, Campus de la Plaine, B 1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

The spectral factorization problem was solved in Hallin (1984) for the class of (non-stationary) m-variate MA(q) stochastic processes, i.e. the class of second-order q-dependent processes. It was shown that such a process generally admits an infinite (mq(mq +1)/2-dimensional) family of possible MA(q) representations. The present paper deals with the invertibility properties and asymptotic behaviour of these MA(q) models, in connection with the problem of producing asymptotically efficient forecasts. Invertible and borderline non-invertible models are characterized (Theorems 3.1 and 3.2). A criterion is provided (Theorem 4.1) by which it can be checked whether a given MA model is a Wold–Cramér decomposition or not; and it is shown (Theorem 4.2) that, under mild conditions, almost every MA model is asymptotically identical with some Wold–Cramér decomposition. The forecasting problem is investigated in detail, and it is established that the relevant invertibility concept, with respect to asymptotic forecasting efficiency, is what we define as Granger-Andersen invertibility rather than the classical invertibility concept (Theorem 5.3). The properties of this new invertibility concept are studied and contrasted with those of its classical counterpart (Theorems 5.2 and 5.4). Numerical examples are also treated (Section 6), illustrating the fact that non-invertible models may provide asymptotically efficient forecasts, whereas invertible models, in some cases, may not. The mathematical tools throughout the paper are linear difference equations (Green's matrices, adjoint operators, dominated solutions, etc.), and a matrix generalization of continued fractions.

Keywords

NON-STATIONARY TIME SERIESSPECTRAL FACTORIZATIONINVERTIBILITYSTOCHASTIC DIFFERENCE EQUATIONSCONTINUED FRACTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 1 , March 1986 , pp. 170 - 210
Copyright
Copyright © Applied Probability Trust 1986 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdrabbo, N. A. and Priestley, M. B. (1967) On the prediction of nonstationary processes. J. R. Statist. Soc. B 29, 570–585.Google Scholar
Anderson, O. D. (1978) On the invertibility conditions for moving average processes. Math. Operationsforsch., Ser. Statist. 9, 545–529.Google Scholar
Anderson, O. D. (1977) The time series concept of invertibility. Math. Operationsforsch., Ser. Statist. 8, 399–406.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1976) Time Series Analysis. Holden-Day, San Francisco.Google Scholar
Cramér, H. (1961) On some classes of nonstationary stochastic processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 57–78.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Davies, N., Pate, M. B. and Frost, M. G. (1974) Maximum autocorrelations for moving average processes. Biometrika 61, 199–200.CrossRefGoogle Scholar
De Bruin, M. G. (1974) Generalized C-fractions and a Multidimensional Padé Table. Ph.D. Dissertation, University of Amsterdam.Google Scholar
De Bruin, M. G. (1977) Convergence along steplines in a generalized Padé table. In Padé and Rational Approximation , Academic Press, New York, 15–22.CrossRefGoogle Scholar
De Bruin, M. G. (1978) Convergence of generalized C-fractions. J. Approximation Theory 24, 177–207.CrossRefGoogle Scholar
Granger, C. W. J. and Andersen, A. (1978) On the invertibility of time series models. Stoch. Proc. Appl. 8, 87–92.CrossRefGoogle Scholar
Granger, C. W. J. and Newbold, P. (1977) Forecasting Economic Time Series. Academic Press, New York.Google Scholar
Hallin, M. (1980) Invertibility and generalized invertibility of time series models. J. R. Statist. Soc. B 42, 210–212; 43, 103.Google Scholar
Hallin, M. (1981) Nonstationary first-order moving average processes: the model-building problem. In Time Series Analysis , ed. Anderson, O. D. and Perryman, M. R., North-Holland, Amsterdam, 189–206.Google Scholar
Hallin, M. (1982a) Nonstationary second-order moving average processes. In Applied Time Series , ed. Anderson, O. D. and Perryman, M. R., North-Holland, Amsterdam, 75–83.Google Scholar
Hallin, M. (1982b) Une propriété des opérateurs moyenne mobile. Cahiers du CERO 24 (Mélanges offerts au Professeur P. P. Gillis à l'occasion de son 70e anniversaire) , 229–236.Google Scholar
Hallin, M. (1983) Nonstationary second-order moving average processes II: model-building and invertibility. In Time Series Analysis, Theory and Practice 4, ed. Anderson, O. D., North-Holland, Amsterdam, 55–64.Google Scholar
Hallin, ?. (1984) Spectral factorization of nonstationary moving average processes. Ann. Statist. 12, 172–192.CrossRefGoogle Scholar
Hallin, M. and Ingenbleek, J.-Fr. (1981) The model-building problem for non-stationary multivariate autoregressive processes. In Time Series Analysis, Theory and Practice 1, ed. Anderson, O. D., North-Holland, Amsterdam, 599–606.Google Scholar
Hallin, M. and Ingenbleek, J.-Fr. (1983) Nonstationary Yule-Walker equations. Statist. Prob. Letters 1, 189–195.CrossRefGoogle Scholar
Jones, R. H. and Brelsford, W. M. (1967) Time series with periodic structure. Biometrika 54, 403–408.CrossRefGoogle ScholarPubMed
Kendall, M. G. (1971) Book review. J. R. Statist. Soc. A134, 450–453.Google Scholar
Levy, H. and Lessman, F. (1961) Finite Difference Equations. Pitman, London.CrossRefGoogle Scholar
Magnus, A. (1977) Fractions continues généralisées et matrices infinies. Bull. Soc. Math. Belg. 29B, 145–159.Google Scholar
Mattheij, R. M. M. (1980) Characterizations of dominant and dominated solutions of linear recursions. Numer. Math. 23, 421–442.CrossRefGoogle Scholar
Melard, G. (1962) The likelihood function of a time-dependent ARMA model. In Applied Time Series Analysis , ed. Anderson, O. D. and Perryman, M. R., North-Holland, Amsterdam, 229–239.Google Scholar
Melard, G. (1984a) A fast algorithm for the exact likelihood of autoregressive-moving average models. J. R. Statist. Soc. C Appl. Statist. 33, 104–114.Google Scholar
Melard, G. (1984b) Analyse des Données chronologiques. Séminaire de Math. Sup., P.U. de Montréal, Montréal.Google Scholar
Miller, K. S. (1968) Linear Difference Equations. Benjamin, New York.CrossRefGoogle Scholar
Miller, K. S. (1969) Nonstationary autoregressive processes. IEEE Trans. Inf. Theory 15, 315–316.CrossRefGoogle Scholar
Milne-Thomson, L. M. (1951) The Calculus of Finite Differences. McMillan, London.Google Scholar
Pagano, M. (1978) On periodic and multiple autoregressions. Ann. Statist. 6, 1310–1317.CrossRefGoogle Scholar
Peiris, S. (1984) Some results on the prediction with nonstationary ARMA models. Statistics Research Report 88, Dept. of Mathematics, Monash University, Australia.Google Scholar
Priestley, M. B. (1965) Evolutionary spectra and nonstationary processes. J. R. Statist. Soc. B 27, 204–237.Google Scholar
Rutishauser, H. (1958) Über eine Verallgemeinerung der Kettenbrüche. Z. Angew. Math. Mech. 39, 278–279.CrossRefGoogle Scholar
Tjöstheim, D. and Tyssedal, J. S. (1982) Autoregressive processes with a time-dependent variance. J. Time Series Anal. 3, 209–217.Google Scholar
Troutman, B. M. (1979) Some results in periodic autoregression. Biometrika 66, 219–228.CrossRefGoogle Scholar
Van Der Cruyssen, P. (1979) Linear difference equations and generalized continued fractions. Computing 22, 269–278.CrossRefGoogle Scholar
Wall, H. S. (1967) Analytic Theory of Continued Fractions. Chelsea, New York.Google Scholar
Wegman, E. J. (1974) Some results on nonstationary first order autoregression. Technometrics 16, 321–322.CrossRefGoogle Scholar
Whittle, P. (1965) Recursive relations for predictors of nonstationary processes. J. R. Statist. Soc. B 27, 523–532.Google Scholar
Wold, H. (1954) A Study in the Analysis of Stationary Time Series , 2nd edn. Almqvist and Wiksell, Stockholm.Google Scholar