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 Xt of order d, d 0, for which there are vectors b so that bʹXt 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 Xt is fractional of order d, bʹXt is fractional of order d b, and a linear combination of bʹXt and DbXt is fractional of order d 2b+ The representations and conditions are analogous to the wellknown conditions for I ~0!, I ~1!, and I ~2! variables+