Modelling stock return behaviour provides an important tool for analysing investment decisions and choices. In this paper, the assumption of linearity is tested for the returns of three North African markets—Egypt, Morocco and Tunisia. I find that the assumption of independently and identically distributed (iid) innovations is grossly violated in the North African data. Consequently, models of the GARCH family were employed to uncover the dynamics of the first two moments. The evidence indicates that volatility is persistent in our sample. Further non-linearity provides a starting point for understanding the dynamics of African stock returns. However, evidence to reject weak form efficiency requires more information than assumed in the literature due to the joint hypothesis problem inherent in all tests of market efficiency.