Resuscitating the co-fractional model of Granger (1986)

We study the theoretical properties of the model for fractional cointegration proposed by Granger (1986), namely the FVECM d,b. First, we show that the stability of any discrete time stochastic system of the type II(L)Yt = Et can be assessed by means of the argument principle under mild regularity condition on II (L) where L is the lag operator. Second, we prove that, under stability, the FVECMd,b allows for a representation of the solution that demonstrates the fractional and co-fractional properties and we find a closed-form expression for the impulse response functions. Third, we prove that the model is identifed for any combination of number of lags and cointegration rank, while still being able to generate polynomial co-fractionality. Finally, 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).