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A REPRESENTATION THEORY FOR POLYNOMIAL COFRACTIONALITY IN VECTOR AUTOREGRESSIVE MODELS

Published online by Cambridge University Press:  04 November 2009

Massimo Franchi
Massimo Franchi*
Affiliation:
Università di Roma “La Sapienza”
*
*Address correspondence to Massimo Franchi, Dipartimento di Statistica, Probabilità e Statistiche Applicate, Università di Roma “La Sapienza,” P. le Aldo Moro 5, 00185 Rome, Italy; e-mail: massimo.franchi@uniroma1.it.
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Abstract

We extend the representation theory of the autoregressive model in the fractional lag operator of Johansen (2008, Econometric Theory 24, 651–676). A recursive algorithm for the characterization of cofractional relations and the corresponding adjustment coefficients is given, and it is shown under which condition the solution of the model is fractional of order d and displays cofractional relations of order d − b and polynomial cofractional relations of order d − 2b,…, d − cb ≥ 0 for integer c; the cofractional relations and the corresponding moving average representation are characterized in terms of the autoregressive coefficients by the same algorithm. For c = 1 and c = 2 we find the results of Johansen (2008).

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Type
ARTICLES
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Econometric Theory , Volume 26 , Issue 4 , August 2010 , pp. 1201 - 1217
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Baillie, R. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5–59.CrossRefGoogle Scholar
Baillie, R. & Bollerslev, T. (1994) Cointegration, fractional cointegration, and exchange rate dynamics. Journal of Finance 49, 737–745.CrossRefGoogle Scholar
Bauer, D. & Wagner, M. (2007) A Canonical Form for Unit Root Processes in the State Space Framework. Mimeo, Institute for Advanced Studies, Wien.Google Scholar
Breitung, J. & Hassler, U. (2002) Inference on the cointegrating rank in fractionally integrated processes. Journal of Econometrics 110, 167–185.CrossRefGoogle Scholar
Cheung, Y.-E. & Lai, K. (1993) A fractional cointegration analysis of purchasing power parity. Journal of Business & Economic Statistics 11, 103–122.CrossRefGoogle Scholar
Davidson, J. (2002) A model of fractional cointegration, and tests for cointegration using the bootstrap. Journal of Econometrics 110, 187–212.CrossRefGoogle Scholar
Dittmann, I. (2004) Error correction models for fractionally cointegrated time series. Journal of Time Series Analysis 25, 27–32.CrossRefGoogle Scholar
Dolado, J. & Marmol, F. (2004) Asymptotic inference results for multivariate long memory processes. Econometrics Journal 7, 160–190.CrossRefGoogle Scholar
Dueker, M. & Startz, R. (1998) Maximum-likelihood estimation of fractional cointegration with an application to U.S. and Canadian bond rates. Review of Economics and Statistics 83, 420–426.CrossRefGoogle Scholar
Engle, R. & Granger, C. (1987) Co-integration and error correction: Representation, estimation, and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Franchi, M. (2007) The integration order of vector autoregressive processes. Econometric Theory 23, 546–553.CrossRefGoogle Scholar
Franchi, M. & Paruolo, P. (2008) A Representation of Vector Autoregressive Processes with Common Cyclical Features. Mimeo, Centro Interuniversitario di Econometria (CIdE) and Associazione Carlo Giannini, Università di Pavia.Google Scholar
Granger, C. (1986) Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, 213–228.CrossRefGoogle Scholar
Granger, C. & Joyeux, R. (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29.CrossRefGoogle Scholar
Gregoir, S. (1999) Multivariate time series with various hidden unit roots, part I: Integral operator algebra and representation theorem. Econometric Theory 15, 435–468.CrossRefGoogle Scholar
Gregoir, S. & Laroque, G. (1993) Multivariate integrated time series: A polynomial error correction representation theorem. Econometric Theory 9, 329–342.CrossRefGoogle Scholar
Henry, M. & Zaffaroni, P. (2003) The long range dependence paradigm for macroeconomics and finance. In Doukhan, P., Oppenheim, G., & Taqqu, M.S. (eds.), Theory and Applications of Long-Range Dependence, pp. 419–438. Birkhäuser.Google Scholar
Hualde, J. (2006) Unbalanced cointegration. Econometric Theory 22, 765–814.CrossRefGoogle Scholar
Hualde, J. (2007) Estimation of Long-Run Parameters in Unbalanced Cointegration. Working paper, Universidad de Navarra.Google Scholar
Johansen, S. (1988) The mathematical structure of error correction models. Contemporary Mathematics 80, 359–386.CrossRefGoogle Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.Google Scholar
Johansen, S. (2008) A representation theory for a class of vector autoregressive models for fractional processes. Econometric Theory 24, 651–676.CrossRefGoogle Scholar
Johansen, S. & Nielsen, M.O. (2010) Likelihood inference for a nonstationary fractional autoregressive model. Journal of Econometrics, forthcoming.CrossRefGoogle Scholar
Kim, C. & Phillips, P. (2001) Fully Modified Estimation of Fractional Cointegration Models. Preprint, Yale University.Google Scholar
la Cour, L. (1998) A parametric characterization of integrated vector autoregressive (VAR) processes. Econometric Theory 14, 187–199.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P. (2001) Semiparametric fractional cointegration analysis. Journal of Econometrics 105, 225–247.CrossRefGoogle Scholar
Nielsen, M. (2008) Nonparametric Cointegration Analysis of Fractional Systems with Unknown Integration Orders. QED Working paper 1174, Queen’s University.CrossRefGoogle Scholar
Stock, J. & Watson, M. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.CrossRefGoogle Scholar
Velasco, C. (2003) Gaussian semi-parametric estimation of fractional cointegration. Journal of Time Series Analysis 24, 345–378.CrossRefGoogle Scholar