THE NONSTATIONARY FRACTIONAL UNIT ROOT

This paper deals with a scalar l(d) process {yj}, where the inlegratiun order d is any real number. Under this setting, we first explore asymptotie properties of various slatislics associated with {yj}, assuming that d is known and is greater than or equal lo 1/2. Note that {yj} becomes stationary when d <1/2, whose case is not our concern here. It turns out that the case of d = 1/2 needs a separate treatment from d >1/2. We then consider, under the normality assumption, testing and estimation fur d, allowing for any value of d. The tests suggested here are asymplotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptolic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymplolic results hold for lest statistics and estimators, where d need not he restricted lo d >Simulation experiments are conducted lo examine the finite sample performance of both the tests and estimators.