Abstract
In this paper, we first establish the limiting distribution of the statistical test under the persistence condition of volatility in Generalized Auto-Regressive Conditional Heteroskedasticity GARCH(1, 1) model. Then, we test the presence of volatility persistence in stock return time series by using S&P500 data and CAC40 data. The performed analysis confirms that the hypothesis of persistence in variance is rejected by the statistical test for both SP&500 and CAC40 with a critical p-value 0.05.