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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.

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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