Following Xu and Perron (2014), I applied the extended RLS model to the daily stock market returns of Argentina, Brazil, Chile, Mexico and Peru. This model replaces the constant probability of level shifts for the entire sample with varying probabilities that record periods with extremely negative returns. Furthermore, it incorporates a mean reversion mechanism with which the magnitude and the sign of the level shift component vary in accordance with past level shifts that deviate from the long-term mean. Therefore, four RLS models are estimated: the Basic RLS, the RLS with varying probabilities, the RLS with mean reversion, and a combined RLS model with mean reversion and varying probabilities. The results show that the estimated parameters are highly significant, especially that of the mean reversion model. An analysis of ARFIMA and GARCH models is also performed in the presence of level shifts, which shows that once these shifts are taken into account in the modeling, the long memory characteristics and GARCH effects disappear. Also, I find that the performance prediction of the RLS models is superior to the classic models involving long memory as the ARFIMA(p,d,q) models, the GARCH and the FIGARCH models. The evidence indicates that except in rare exceptions, the RLS models (in all its variants) are showing the best performance or belong to the 10% of the Model Confidence Set (MCS). On rare occasions the GARCH and the ARFIMA models appear to dominate but they are rare exceptions. When the volatility is measured by the squared returns, the great exception is Argentina where a dominance of GARCH and FIGARCH models is appreciated.