A representation theory for a class of vector autoregressive models for fractional processes

Based on an idea of Granger (1986), we analyze a new vector autoregressive model de…ned from the fractional lag operator 1 (1 L)d: We …rst derive conditions in terms of the coe¢ cients for the model to generate processes which 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 so that Xt is fractional of order d b; 0 < b d: We …nd a representation of the solution which 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; Xt is fractional of order d b and a linear combination of Xt and Xt 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. 1 Introduction and motivation Since the de…nition of fractional processes by Granger and Joyeux (1980) and Hosking (1981), attention has been given to the estimation of the fractional order of a time series, see for instance Beran (1994), with minimal assumptions on the processes. The problem considered in this paper is di¤erent. We propose a vector autoregressive model that allows for a fractional process as solution, but also allows for modelling cofractional relations and adjustments to these, so that economic hypotheses can be formulated within a model, that can be I would like to thank the Danish Social Sciences Research Council for continuing support, and the members of the ESF-EMM network for discussions. As usual the insight and help of Søren Tolver Jensen is greatly appreciated in connection with Lemma 10. I am grateful for comments from an anonymous referee which helped clarify some explanations, and to Bent Nielsen who helped with the proof of Lemma 11. Finally Massimo Franchi had some good comments to the almost …nal version.