Unified theoretical framework for unit root test and fractional unit root test

This paper proposes a new testing procedure for the degree of fractional integration of a time series inspired on the unit root test of Dickey-Fuller (1979). The composite null hypothesis is that of d>=d0 against d<d0. The test statistics is the same as in Dickey-Fuller test, exploiting the fact that if the process, under study, is I(d0) then the (-1+d0)th differenced series is I(1) under the null d=d0. If d>=d0, using the generalization of Sowell's results (1990), we propose a test based on the least favorable case d=d0, to control type I error and when d<d0 we show that the usual tests statistics diverges to infinity, providing consistency.