A Fractional Dickey–Fuller Test for Unit Roots

Juan J. Dolado, Jesús Gonzalo, Laura Mayoral

Econometrica · 2002 · Volume 70 · Issue 5

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 theI(1) versusI(0) case, to the more general setup ofFI(d0) versusFI(d1), withd1<d0. Whend0=1, the proposed test statistics are based on the OLS estimator, or itst–ratio, of the coefficient on Δd1yt−1in a regression of Δyton Δd1yt−1and, possibly, some lags of Δyt. Whend1is not taken to be known a priori, a pre–estimation ofd1is needed to implement the test. We show that the choice of anyT1/2–consistent estimator ofd1∈[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.

Keywords

ARFIMA, Dickey–Fuller test, fractional processes, long memory, unit roots

Enable JavaScript for citation tools and additional paper interactions.