In this paper we examine the statistical properties of several stock market indices in Europe, the US and Asia by means of determining the degree of dependence in both the level and the volatility of the processes. In the latter case, we use the squared returns as a proxy for the volatility. We also investigate the cyclical pattern observed in the data and in particular, if the degree of dependence changes depending on whether there is a bull or a bear period. We use fractional integration and GARCH specifications. The results indicate that the indices are all nonstationary I(1) processes with the squared returns displaying a degree of long memory behaviour. With respect to the bull and bear periods, we do not observe a systematic pattern in terms of the degree of persistence though for some of the indices (FTSE, Dax, Hang Seng and STI) there is a higher degree of dependence in both the level and the volatility during the bull periods.