A REPRESENTATION THEORY FOR A CLASS OF VECTOR AUTOREGRESSIVE MODELS FOR FRACTIONAL PROCESSES
Based on an idea of Granger (1986, <italic>Oxford Bulletin of Economics and Statistics</italic> 48, 213–228), we analyze a new vector autoregressive model defined from the fractional lag operator 1 − (1 − <italic>L</italic>)<sup>null</sup>. 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 <italic>X</italic><sub>null</sub> of order <italic>d</italic>, <italic>d</italic> > 0, for which there are vectors β so that β‼<italic>X</italic><sub>null</sub> is fractional of order <italic>d</italic> − <italic>b</italic>, 0 < <italic>b</italic> ≤ <italic>d</italic>. 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 <italic>X</italic><sub>null</sub> is fractional of order <italic>d</italic>, β‼<italic>X</italic><sub>null</sub> is fractional of order <italic>d</italic> − <italic>b</italic>, and a linear combination of β‼<italic>X</italic><sub>null</sub> and Δ<sup>null</sup><italic>X</italic><sub>null</sub> is fractional of order <italic>d</italic> − 2<italic>b</italic>. The representations and conditions are analogous to the well-known conditions for <italic>I</italic>(0), <italic>I</italic>(1), and <italic>I</italic>(2) variables.
