This paper analyses the long‐memory properties of US and European stock indices, as well as their linkages, using fractional integration and fractional cointegration techniques. These methods are more general and have higher power than the standard ones usually employed in the literature. The empirical evidence based on them suggests the presence of unit roots in both the Standard and Poor's 500 Index and the Euro Stoxx 50 Index. Also, fractional cointegration appears to hold at least for the subsample from December 1996 to March 2009 ending when the global financial crisis was still severe; subsequently, the US and European stock markets diverged and followed different recovery paths, possibly as a result of various factors such as diverging growth and monetary policy. Establishing whether the degree of cointegration has changed over time is important because past literature has shown that diversification benefits arise when markets are not cointegrated. Copyright © 2015 John Wiley & Sons, Ltd.