Beyond the co-fractional model of Granger (1986)

We illustrate a number of appealing properties of a new model for fractional cointegration, which extends the model proposed by Granger (1986), by letting the short-term dynamics to be driven by powers of the fractional lag operator L_c=1-(1-L)^c where c>0. We show that the non-explosiveness of the system can be checked through the argument principle and we demonstrate that the model admits a Granger representation. We also prove that the model is identified for any combination of number of lags and cointegration rank if c is fixed to a constant, c*, satisfying a mild lower-bound requirement. Furthermore, we show that the asymptotic properties of the maximum likelihood estimator reconcile with those of the FCVARd,b model studied in Johansen and Nielsen (2012). Finally, we provide an empirical illustration analyzing the equilibrium relation between realized variance and VIX.