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A STATE SPACE CANONICAL FORM FOR UNIT ROOT PROCESSES

Published online by Cambridge University Press:  21 May 2012

Dietmar Bauer  and
Martin Wagner
Dietmar Bauer
Affiliation:
Austrian Institute of Technology
Martin Wagner*
Affiliation:
Institute for Advanced Studies and Frisch Centre for Economic Research
*
*Address correspondence to Martin Wagner, Institute for Advanced Studies, Department of Economics and Finance, Stumpergasse 56, A-1060 Vienna, Austria; e-mail: [email protected].
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Abstract

In this paper we develop a canonical state space representation of autoregressive moving average (ARMA) processes with unit roots with integer integration orders at arbitrary unit root frequencies. The developed representation utilizes a state process with a particularly simple dynamic structure, which in turn renders this representation highly suitable for unit root, cointegration, and polynomial cointegration analysis. We also propose a new definition of polynomial cointegration that overcomes limitations of existing definitions and extends the definition of multicointegration for I(2) processes of Granger and Lee (1989a, Journal of Applied Econometrics4, 145–159). A major purpose of the canonical representation for statistical analysis is the development of parameterizations of the sets of all state space systems of a given system order with specified unit root frequencies and integration orders. This is, e.g., useful for pseudo maximum likelihood estimation. In this respect an advantage of the state space representation, compared to ARMA representations, is that it easily allows one to put in place restrictions on the (co)integration properties. The results of the paper are exemplified for the cases of largest interest in applied work.

Type
Articles
Information
Econometric Theory , Volume 28 , Issue 6 , December 2012 , pp. 1313 - 1349
Copyright
Copyright © Cambridge University Press 2012 

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References

REFERENCES

Ahn, S.K. & Reinsel, G.C. (1994) Estimation of partially nonstationary vector autoregressive models with seasonal behavior. Journal of Econometrics 62, 317–350.CrossRefGoogle Scholar
Aoki, M. (1987) State Space Modeling of Time Series. Springer.Google Scholar
Aoki, M. & Havenner, A. (1989) A method for approximate representation of vector valued time series and its relation to two alternatives. Journal of Econometrics 42, 181–199.CrossRefGoogle Scholar
Aoki, M. & Havenner, A. (1991), State space modeling of multiple time series. Econometric Reviews 10(1), 1–59.CrossRefGoogle Scholar
Bauer, D. & Wagner, M. (2006) Asymptotic Theory of Pseudo Maximum Likelihood Estimators for Multiple Frequency I(1) Processes. Mimeo.Google Scholar
Bauer, D. & Wagner, M. (2007) Autoregressive Approximations of Multiple Frequency I(1) Processes. Mimeo.Google Scholar
Gantmacher, F. (1966) Matrizentheorie. Springer.Google Scholar
Granger, C.W.J. & Lee, T. (1989a) Investigation of production, sales and inventory relationship using multicointegration and non-symmetric error correction models. Journal of Applied Econometrics 4, 145–159.CrossRefGoogle Scholar
Granger, C.W.J. & Lee, T. (1989b) Multicointegration. In Rhodes, G. & Fomby, T. (eds.), Advances in Econometrics: Cointegration, Spurios Regression and Unit Roots, pp. 71–84. JAI Press.Google Scholar
Gregoir, S. (1999a) Multivariate time series with various hidden unit roots, part I. Econometric Theory 15, 435–468.CrossRefGoogle Scholar
Gregoir, S. (1999b) Multivariate time series with various hidden unit roots, part II. Econometric Theory 15, 469–518.CrossRefGoogle Scholar
Gregoir, S. & Laroque, G. (1994) Polynomial cointegration: Estimation and test. Journal of Econometrics 63, 183–214.Google Scholar
Hannan, E.J. & Deistler, M. (1988) The Statistical Theory of Linear Systems. Wiley.Google Scholar
Hansen, P.R. (2005) Granger’s representation theorem: A closed form expression for I(1) processes. The Econometrics Journal 8, 23–38.CrossRefGoogle Scholar
Harvey, A. (1989) Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.Google Scholar
Hylleberg, S., Engle, R.F., Granger, C.W.J., & Yoo, B. (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215–238.CrossRefGoogle Scholar
Johansen, S. (1995) Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models. Oxford University Press.CrossRefGoogle Scholar
Johansen, S. (1997) Likelihood analysis of the I(2) model. Scandinavian Journal of Statistics 24, 433–462.CrossRefGoogle Scholar
Johansen, S. & Schaumburg, E. (1999) Likelihood analysis of seasonal cointegration. Journal of Econometrics 88, 301–339.CrossRefGoogle Scholar
Kailath, T. (1980) Linear Systems. Prentice-Hall.Google Scholar
Lee, H. (1992) Maximum likelihood inference on cointegration and seasonal cointegration. Journal of Econometrics 54, 1–47.CrossRefGoogle Scholar
Ober, R. (1996) Balanced canonical forms. In Bittanti, S. and Picci, G. (eds.), Identification, Adaptation, Learning: The Science of Learning Models from Data, pp. 120–183. Springer.Google Scholar
Phillips, P.C.B. (1998) Impulse response and forecast error variance asymptotics in nonstationary VARs. Journal of Econometrics 83, 21–56.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.C. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.Google Scholar
Sims, C.A., Stock, J.H., & Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113–144.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.CrossRefGoogle Scholar
Yoo, B. (1986) Multi-cointegrated time series and generalized error correction models. Technical report, University of California San Diego.Google Scholar