The modelling of stock market volatility is considered to be important for practitioners and academics in finance due to its use in forecasting aspects of future returns. The GARCH class models have now firmly established themselves as one of the foremost techniques for modelling volatility in financial markets. The application of GARCH class models in developed and emerging markets (including the Egyptian Stock Market) provides evidence of GARCH effects in stock returns. However, most of the studies conducted on modelling the volatility of stock returns are based on the aggregated market index. This thesis argues that this will not reflect significant differences of variation in the pattern of volatility associated with different stocks. However, in order to examine the similarities and differences between the conditional variance structures of stocks from the same or different industries in the same equity market, this thesis estimates pooled-panel models. These novel models are used to test for similarities and differences in the conditional variance equation in panels of time series within a general to specific framework of nested tests. This is done using panel samples of sector indices and stocks from the Egyptian Stock Market covering the period from 1997 to 2002. The results suggest that there are similarities in the temporal volatility structures of stocks from the same sector or industry, but there are significant differences in the temporal volatility structures of stocks from different sectors or industries. This suggests that using indices alone for modelling the volatility of an equity market, which is the method used in the majority of studies cited in the literature, may not be appropriate. The thesis concludes with a discussion of some of the implications of these results and suggestions for further research. / Doctor of Philosophy (PhD)