This thesis contains four essays on persistence change tests and non-stationarity tests.Persistence change tests are analysed under non-standard conditions and a new familyof tests to detect changes in persistence and unit roots is proposed that is based on theCUSUM testing principle. These can be applied in economic and financial time series.Chapter 1 introduces the existence and implications of persistence in time series andstructural changes. Furthermore, the impact of asymmetric volatility and different typesof outliers is discussed. A new testing principle based on the concept of squared CUSUMof residuals is developed.Chapter 2 reviews the literature on different methods for persistence change testsincluding parametric and non-parametric modifications. A family GARCH model is presentedto consider different asymmetric conditional volatility models within the persistencechange model. The Wild bootstrap approach is introduced and bootstrap analoguesof the persistence change tests are derived. The bootstrap procedure is conducted in acomprehensive Monte Carlo study to analyse the behaviour of the tests under asymmetricvolatility. The results show that the tests suffer from severe size distortions, while thebootstrap method provides reasonable results in small samples. In an application to theU.S. stock market, asymmetric volatility models are estimated on the return series, wherethe persistence change tests and the bootstrap analogues are conducted. The main findingis that the tests falsely detect a change in persistence under asymmetric volatility, whilethe bootstrap analogues assume stationary behaviour.In chapter 3 the effect of outliers on inference in models with changing persistenceis under consideration. We introduce the additive and innovative outlier with differentoutlier detection and removal methods. In a Monte Carlo study, the performance of thetests is investigated and compared in uncontaminated, outlier contaminated and adjustedseries. The main finding is that innovative outliers do not affect the size, while additiveoutliers deteriorate the performance of the tests if the series exhibits a high degree ofpersistence. We present a modified outlier detection and removal method which is appliedin a simulation study. In an empirical application to inflation data of the G7 countriesthe tests and the new method are conducted.Chapter 4 introduces a new approach to test for a unit root based on squared CUSUMsof residuals. The procedure is based on the squared sum of all different consecutive observationsof the time series. The limiting distributions of the tests are derived and consistencycan be shown. A comprehensive simulation study in ARMA models suggests that the newmethod provides better properties. In stationary processes the tests show higher powerthan commonly used unit root tests, while the size is closer to the nominal significancelevel, when a unit root is present in the data. The empirical application to the historicalNelson-Plosser data provides slightly different results compared to the findings in theliterature.In chapter 5 the same procedure as in chapter 4 is used to develop tests for a changein persistence based on squared CUSUMs of sub-sample residuals. We construct one testfor the null hypothesis of a stationary process and another test for the non-stationarityhypothesis. Similar to previous testing procedures a maximum and a ratio based test isconstructed for the alternative of a change in persistence in an unknown direction. Whilecommon persistence change tests weight the residuals in the partial sums differently orignore cross-dependencies of the residuals, the presented tests provide squared partialsums of equally weighted observations and exploit the cross-dependencies. The limitingdistributions of the tests are derived and consistency against a change in persistencecan be shown. The simulation study provides better size and power properties for bothdeveloped tests.