p.1963

A Fractional Dickey–Fuller Test for Unit Roots

Juan J. Dolado
Jesú:s Gonzalo
Laura Mayoral

Abstract

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(d0) versus FI(d1), with d1<d0. When d0=1, the proposed test statistics are based on the OLS estimator, or its t–ratio, of the coefficient on ?^d1yt−1 in a regression of ?yt on ?^d1yt−1 and, possibly, some lags of ?yt. When d1 is not taken to be known a priori, a pre–estimation of d1 is needed to implement the test. We show that the choice of any T^1/2–consistent estimator of d1∈[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.