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A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes

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  • Johansen, SØren

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 ) null . 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 X null of order d , d > 0, for which there are vectors β so that β‼ X null 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 X null is fractional of order d , β‼ X null is fractional of order d − b , and a linear combination of β‼ X null and Δ null X null is fractional of order d − 2 b . The representations and conditions are analogous to the well-known conditions for I (0), I (1), and I (2) variables.

Suggested Citation

  • Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(03), pages 651-676, June.
  • Handle: RePEc:cup:etheor:v:24:y:2008:i:03:p:651-676_08
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