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

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 X-t of order d, d > 0, for which there are vectors 6 so that beta'X-t 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-t is fractional of order d, beta'X-t is fractional of order d - b, and a linear combination of beta'X-t and Delta X-b(t) 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.