Testing for no cointegration in vector autoregressions with estimated degree of fractional integration

Confirming the existence of long-run relations between persistent variables is an important empirical goal. Fractional (co)integration allows for flexible modelling of such variables. The persistence of many fractional VAR processes is characterized by the integration parameter d and, under cointegration, by the persistence of the equilibrium deviations, where both are typically unknown in practice. To deal with unknown persistence of equilibrium deviations, we linearize here the involved fractional difference filter. When d is known, this results in test statistics possessing standard (normal or chi(2)) asymptotic distributions. For unknown d, conditions are provided, under which plugging in consistent estimators of d does not affect the standard asymptotics; the key insight is that letting the number of lags increase as the sample size goes to infinity accounts for the estimation error of d. The proposed approach is illustrated in Monte Carlo experiments and a fractional cointegration analysis of short-term U.S. T-bills.