A fractional Dickey-Fuller test for unit roots

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well-known Dickey-Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d(0)) versus FI(d(1)), with d(1) < d(0). When d(0) = 1, the proposed test statistics are based on the OLS estimator, or its t-ratio, of the coefficient on Delta(d1)y(t-1) in a regression of Deltay(t) on Delta(d1)y(t-1) and, possibly, some lags of Deltay(t). When d, is not taken to be known a priori, a pre-estimation of d, is needed to implement the test. We show that the choice of any T-1/2-consistent estimator of d(1) epsilon [0, 1) suffices to make the test feasible, while achieving asymptotic normality. Monte-Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.