Outlier detection has become an important part of time series analysis. This paper studies the problem of detecting and correcting outliers in time series data, and proposes a method based on the Gumbel distribution as a limiting distribution for outliers. Outlier detection influences modelling, testing and inference, because outliers can lead to model misspecification, biased parameter estimation, poor forecasts and inappropriate decomposition of series. We develop an algorithm for determining when an observation can be classified as an outlier. The method is then applied to some residuals of autoregressive integrated moving average (ARIMA) models fitted to Zimbabwe Stock Exchange Indices and if any outliers are detected a correction procedure is then applied to rid the data of the outliers. A new model is then fitted to the corrected data series and some analyses are performed. The results show that the method proposed is effective in detecting outliers, and the correction procedure ensures that the correct model for the data is specified and the parameter estimates are unbiased.