In this article, we have modeled the log of the US and the UK real oil prices in terms of fractionally integrated processes with a mean shift. We used different versions of the tests of Robinson (1994), which have standard null and local limit distributions. The results indicated that if we model the series without a mean shift, then they are both non-stationary I(1). However, by including a mean shift component during the oil crises, they become fractionally integrated with an order of integration smaller than one and, thus, showing mean reverting behavior.