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A REPRESENTATION THEORY FOR A CLASS OF VECTOR AUTOREGRESSIVE MODELS FOR FRACTIONAL PROCESSES

Published online by Cambridge University Press:  22 January 2008

SØren Johansen
SØren Johansen*
Affiliation:
University of Copenhagen
*
Address correspondence to Søren Johansen, Department of Economics, University of Copenhagen, Studiestræde 6, DK-1455 Copenhagen K, Denmark; e-mail: sjo@math.ku.dk.
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Abstract

Based on an idea of Granger (1986, Oxford Bulletin of Economics and Statistics 48, 213–228), we analyze a new vector autoregressive model defined from the fractional lag operator 1 − (1 − L)d. We first derive conditions in terms of the coefficients for the model to generate processes that are fractional of order zero. We then show that if there is a unit root, the model generates a fractional process Xt of order d, d > 0, for which there are vectors β so that β‼Xt is fractional of order d − b, 0 < b ≤ d. We find a representation of the solution that demonstrates the fractional properties. Finally we suggest a model that allows for a polynomial fractional vector, that is, the process Xt is fractional of order d, β‼Xt is fractional of order d − b, and a linear combination of β‼Xt and ΔbXt is fractional of order d − 2b. The representations and conditions are analogous to the well-known conditions for I(0), I(1), and I(2) variables.

Information

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 3 , June 2008 , pp. 651 - 676
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Beran, J. (1994) Statistics for Long-Memory Processes. Chapman ’ Hall.Google Scholar
Breitung, J. ’ Hassler, U. (2002) Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics 110, 167–185.CrossRefGoogle Scholar
Brockwell, P.J. ’ Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed.Springer.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.Google Scholar
Engle, R.F. ’ Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Engsted, T. ’ Johansen, S. (1999) Granger's representation theorem and multicointegration. In Engle, R.F. ’ White, H. (eds.), Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger, pp. 200–211. Oxford University Press.CrossRefGoogle Scholar
Gradshteyn, I.S. ’ Ryzhik, I.M. (1971) Tables of Integrals, Sums, Series and Products, 6th ed., Jeffrey, A. ’ Zwillinger, D. (eds.). Academic Press.Google Scholar
Granger, C.W.J. (1986) Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, 213–228.Google Scholar
Granger, C.W.J. ’ Joyeux, R. (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29.CrossRefGoogle Scholar
Grenander, U. ’ Rosenblatt, M. (1956) Statistical Analysis of Stationary Time Series. Almqvist and Wiksell.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.Google Scholar
Johansen, S. (1996) Likelihood Based Inference on Cointegration in the Vector Autoregressive Model, 2nd ed.Oxford University Press.Google Scholar
Johansen, S. (1997) Likelihood analysis of the I(2) model. Scandinavian Journal of Statistics 24, 433–462.Google Scholar
Johansen, S., (in press) Representation of cointegrated autoregressive processes with application to fractional processes. Econometric Review.Google Scholar
Lasak, K. (2005) Likelihood Based Testing for Fractional Cointegration. Discussion paper, University of Barcelona.Google Scholar
Lyhagen, J. (1998) Maximum Likelihood Estimation of the Multivariate Fractional Cointegrating Model. Working paper, Stockholm School of Economics.Google Scholar
Phillips, E.G. (1958) Functions of a Complex Variable with Applications. Oliver and Boyd.Google Scholar
Robinson, P.M. ’ Marinucci, D. (2001) Semiparametric fractional cointegration analysis. Journal of Econometrics 105, 225–247.Google Scholar