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ON THE IDENTIFICATION AND ESTIMATION OF NONSTATIONARY AND COINTEGRATED ARMAX SYSTEMS

Published online by Cambridge University Press:  03 November 2006

D.S. Poskitt
D.S. Poskitt
Affiliation:
Monash University
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Abstract

This paper extends current theory on the identification and estimation of vector time series models to nonstationary processes. It examines the structure of dynamic simultaneous equations systems or ARMAX processes that start from a given set of initial conditions and evolve over a given, possibly infinite, future time horizon. The analysis proceeds by deriving the echelon canonical form for such processes. The results are obtained by amalgamating ideas from the theory of stochastic difference equations with adaptations of the Kronecker index theory of dynamic systems. An extension of these results to the analysis of unit-root, partially nonstationary (cointegrated) time series models is also presented, leading to straightforward identification conditions for the error correction, echelon canonical form. An innovations algorithm for the evaluation of the exact Gaussian likelihood is given. The asymptotic properties of the approximate Gaussian estimator and the exact maximum likelihood estimator based upon the algorithm are derived for the cointegrated case. Examples illustrating the theory are discussed, and some experimental evidence is also presented.I thank two referees for insightful comments and helpful suggestions on the content and presentation of this paper. I am particularly grateful for the correction of errors in earlier drafts and reference to the work of B. Hanzon. Financial support under ARC grant DP0343811 is gratefully acknowledged.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 6 , December 2006 , pp. 1138 - 1175
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Akaike, H. (1974) Stochastic theory of minimal realization. IEEE Transactions on Automatic Control AC-19, 667–674.CrossRefGoogle Scholar
Banerjee, M., J. Dolado, J. Galbraith, & D.F. Hendry (1993) Co-integration, Error-correction and the Econometric Analysis of Non-stationary Data. Oxford University Press.CrossRefGoogle Scholar
Brockwell, P.J. & R.A. Davis (1988) Simple consistent estimation of the coefficients of a linear filter. Stochastic Processes and Their Applications 22, 47–59.CrossRefGoogle Scholar
Cox, D.R. (1975) Partial likelihood. Biometrika 62, 269–276.CrossRefGoogle Scholar
Deistler, M. (1983) The properties of the parameterisation of ARMAX systems and their relevance for structural estimation and dynamic specification. Econometrica 51, 1187–1206.CrossRefGoogle Scholar
Deistler, M. (1985) The general structure and parameterisation of ARMA and State-Space systems and its relation to statistical problems. In E.J. Hannan, P.R. Krishnaiah, & M.M. Rao (eds.), Handbook of Statistics, vol. 5, Time Series in the Time Domain, pp. 257–278. North-Holland.CrossRefGoogle Scholar
Dhrymes, P.J. (1998) Time Series, Unit Roots, and Cointegration. Academic Press.Google Scholar
Engle, R.F. & C.W.J. Granger (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Engle, R.F., D.F. Hendry, & J.F. Richard (1983) Exogeneity. Econometrica 51, 277–304.CrossRefGoogle Scholar
Gevers, M. (1986) ARMA models, their Kronecker indices and their McMillan degree. International Journal of Control 43, 1745–1761.CrossRefGoogle Scholar
Geweke, J. (1984) Measures of conditional linear independence and feedback between time series. Journal of the American Statistical Association 79, 907–915.CrossRefGoogle Scholar
Granger, C.W.J. (1981) Some properties of time series data and their use in econometric model specification. Journal of Econometrics 16, 121–130.CrossRefGoogle Scholar
Hannan, E.J. (1971) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. (1974) The identification problem for multiple equation systems with moving average errors. Econometrica 39, 751–765.CrossRefGoogle Scholar
Hannan, E.J. (1976) The identification and parameterisation of ARMA and State Space forms. Econometrica 44, 713–723.CrossRefGoogle Scholar
Hannan, E.J. & M. Deistler (1988) The Statistical Theory of Linear Systems. Wiley.Google Scholar
Hanzon, B. (1989) Identifiability, Recursive Identification and Spaces of Linear Dynamical Systems: Part I. CWI Tract 63, Center for Mathematics and Computer Science, Amsterdam.Google Scholar
Hatanaka, M. (1996) Time-Series-Based Econometrics: Unit Roots and Co-integration. Oxford University Press.CrossRefGoogle Scholar
Hosoya, Y. (1977) On the Granger condition for non-causality. Econometrica 45, 1735–1736.CrossRefGoogle Scholar
Hotelling, H. (1936) Relations between two sets of variates. Biometrika 28, 321–377.CrossRefGoogle Scholar
Hsiao, C. (1997) Cointegration and dynamic simultaneous equations model. Econometrica 65, 647–670.CrossRefGoogle Scholar
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.CrossRefGoogle Scholar
Lütkepohl, H. & H. Claessen (1997) Analysis of cointegrated VARMA processes. Journal of Econometrics 80, 223–239.CrossRefGoogle Scholar
Lütkepohl, H. & D. Poskitt (1996) Specification of echelon form VARMA models. Journal of Business & Economic Statistics 14, 69–79.Google Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Poskitt, D.S. (1992) Identification of echelon canonical forms for vector linear processes using least squares. Annals of Statistics 20, 195–215.CrossRefGoogle Scholar
Poskitt, D.S. (2000) Strongly consistent determination of cointegrating rank via canonical correlations. Journal of Business & Economic Statistics 18, 71–90.Google Scholar
Poskitt, D.S. (2003) On the specification of cointegrated autoregressive moving-average forecasting systems. International Journal of Forecasting 19, 503–519.CrossRefGoogle Scholar
Poskitt, D.S. & A.R. Tremayne (1982) Diagnostic tests for multiple time series models. Annals of Statistics 10, 114–120.CrossRefGoogle Scholar
Reinsel, G.C.S. (1993) Elements of Multivariate Time Series Analysis. Springer-Verlag.CrossRefGoogle Scholar
Rissanen, J. & L. Barbosa (1969) A factorization problem and the problem of predicting non-stationary vector-valued stochastic processes. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 12, 255–266.CrossRefGoogle Scholar
Rissanen, J. & P.E. Caines (1979) The strong consistency of maximum likelihood estimators for ARMA processes. Annals of Statistics 7, 297–315.CrossRefGoogle Scholar
Saikkonen, P. (1996) Problems with the asymptotic theory of maximum likelihood estimation in integrated and cointegrated systems. Econometric Theory 11, 888–911.CrossRefGoogle Scholar
Stock, J.H. & M.W. Watson (1988) Testing for common trends. Journal of the American Statistical Association 83, 1097–1107.CrossRefGoogle Scholar
Tanaka, K. (1996) Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley.Google Scholar
Tunnicliffe-Wilson, G. (1972) The factorization of matricial spectral densities. SIAM Journal of Applied Mathematics 23, 420–426.CrossRefGoogle Scholar
Wickens, M.R. (1996) Interpreting cointegrating vectors and common stochastic trends. Journal of Econometrics 74, 255–271.CrossRefGoogle Scholar
Wickens, M.R. & T.S. Breusch (1988) Dynamic specification, the long-run and the estimation of transformed regression models. Economic Journal 98, 189–205.CrossRefGoogle Scholar
Yap, S. & G. Reinsel (1995) Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model. Journal of the American Statistical Association 90, 253–267.CrossRefGoogle Scholar