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The aim of this study is to compare volatility persistence with daily volatility and to analyze the asymmetry effect of volatilities in stock markets of emerging economies. Using daily observations of stock market indices of selected major emerging countries during the period of January 1, 2002 to December 31, 2018, the authors estimate the persistence, the half-life measure of volatility and the daily volatility of the return series using the GARCH model application. The authors also examine the leverage effect on stock market returns using the EGARCH model estimation. In addition, the authors investigate the impact of the 2008 global financial crisis on various volatility measures and the leverage effect of emerging stock market returns. The authors then examine and compare the different speeds of mean reversion, volatility persistence and leverage effects in the national stock market indices during the pre-crisis, crisis, and post-crisis periods. The authors hereby present evidence that the effects of negative shocks are significantly larger than those of positive shocks in emerging stock markets throughout their different sample periods.