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 discretetime stochastic system of the type Pi(L)Y_t = e_t can be assessed by means of the argument principle under mild regularity condition on Pi(L), where L is the lag operator. Second, we prove that, under stability, the FVECM_{d,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 identified for any combination of number of lags and cointegration rank, while still being able to generate polynomial co-fractionality. In light of these properties, we show that the asymptotic properties of the maximum likelihood estimator reconcile with those of the FCVAR_{d,b} model studied in Johansen and Nielsen (2012). Finally, an empirical illustration is provided.