The paper tests four hypotheses at the same time using an autoregressive return-generating process and an asymmetric conditional variance specification, both also including deterministic day of the week dummies. The daily stock index returns from 19 countries are employed to test: (H1) predictable time variation in conditional volatility; (H2) asymmetry in volatility and leverage effect; (H3) effects of estimated volatility on returns; and (H4) day of the week effects on both returns and their volatility. Evidence is provided for predictable time varying daily volatility in all markets among which eight also exhibit a significant leverage effect. There is a significantly positive relationship between returns and their conditional volatility in only three countries. The nature of the day of the week effects on returns and their conditional volatility differs greatly among countries and across days. Thirteen countries exhibit seasonality in either mean returns (seven countries) or volatility (eight countries) or both (two countries). Each day is at least once reported to exhibit significant positive and negative effects in both mean and volatility with the exception that there is no negative effect on mean returns and no positive effect in volatility on Wednesdays. © 2001 Taylor & Francis Ltd.