Empirical research indicates that the volatility of stock return time series has long memory. However, it has been demonstrated that short memory processes contaminated by random level shifts can often be confused with long memory, a feature often referred to as spurious long memory. This paper represents an empirical study of the random level shift (RLS) model for the volatility of daily stock return data for five Latin American countries. This model consists of the sum of a short term memory component and a level shift component that is governed by a Bernoulli process with a shift probability α. The results suggest that level shifts in the volatility of daily stock return data are infrequent but when taken into account, the long memory characteristic and GARCH effects disappear. An out-of-sample forecasting exercise is also provided.