Unified theoretical framework for unit root and fractional unit root

To distinguish between purely fractionally integrated (FI) processes, we propose in this article an appropriate fractional Dickey-Fuller test (F-DF). This extends the familiar Dickey-Fuller (1979) type tests for unit root (I(1) against I(0)) by embedding the case d=0 and d=1 in continuum of memory properties. The new F-DF test is easy to implement and is based on two time domains properties of FI(d) processes. First, if time series follows FI(d) process, then the (-1+d)th differenced series follows an I(1) process. Second, the purely FI(d) processes are non explosive for d be reel number.