Standard unit root tests and cointegration tests are sensitive to atypical events such as outliers and structural breaks. This paper uses outlier robust estimation techniques to reduce the impact of these events on cointegration analysis. As a byproduct of computing the robust estimator, we obtain weights for all observations in the sample. These weights can be used to identify the approximate dates of the atypical events. We evaluate our method using some illustrative simulated data. Furthermore, since our robust approach involves a few additional decisions on the values of key parameters, we investigate the sensitivity of our method through extensive Monte-Carlo simulations. Finally, we present an empirical example based on real-life data to show that OLS-based cointegration tests can spuriously indicate stationarity.