A REPRESENTATION THEORY FOR A CLASS OF VECTOR AUTOREGRESSIVE MODELS FOR FRACTIONAL PROCESSES
基于Granger的思想，分析由分数滞后算子定义的新向量自回归模型，推导生成分数过程的系数条件，并给出单位根情形下的解表示，适用于研究分数阶协整关系。
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.