The standard macroeconomic view links the equilibrium level of foreign exchange rates to the state of the macroeconomic fundamentals. Any deviation from the equilibrium level is viewed as temporary since there are forces ensuring quickly mean-reverting dynamics. The aim of this article is to investigate whether the empirical observation of the real exchange rate misalignments in five European countries over the period 1979–1999 was consistent with the hypothesis of temporary deviations from the fundamentals, or whether they must be associated with significant persistent dynamics. We depart from the traditional framework of linear cointegration by using fractional cointegration or non-linear cointegration. Therefore, we will try to discriminate between linear long memory dynamics and non-linear short memory dynamics.