Searching for fractional evidence using combined unit root tests

Abstract

It has become a fully accepted rule in applied work that rejection of both the difference stationarity and trend stationarity null hypotheses, could imply the possibility that the underlying time series behaves as a fractionally integrated process. In this paper we prove this claim in a rigorous way by showing the consistency of the customary Dickey-Fuller and KPSS tests against fractional alternatives. It is shown that the combined use of both tests only achieves consistency if two-tailed tests are implemented.