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 β so that β‼ 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 , β‼ X t is fractional of order d − b , and a linear combination of β‼ X t and Δ b X t 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.