This thesis contains six essays on tests for structural change under long memory with applications to economic time series. Chapter 1 introduces structural change tests and the concept of long memory. The main problem examined in Chapters 2 to 5 is that standard change-in-mean tests are invalid in long-memory time series.Chapter 2 reviews the literature on long-memory robust extensions of standard testing principles for a change in mean. Apart from giving a systematic review, an extensive Monte Carlo study is conducted to compare the relative performance of the introduced methods. Special attention is paid to the interaction the test results have with the estimation of the long-memory parameter. Furthermore, it is shown that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.Chapter 3 introduces a simple test on structural change in long-memory time series. In contrast to the testing principles introduced in the previous chapter, it is much easier to implement in statistical software and has a limiting distribution that does not depend on the degree of memory. The test is based on the idea that the test statistic of the standard CUSUM test retains its asymptotic distribution if it is applied to fractionally differenced data. It is proven that the approach is asymptotically valid if the memory is estimated consistently under the null hypothesis. Therefore, the well-known CUSUM test can be used on the differenced data without any further modification. In simulations, the proposed CUSUM test on the differenced series is compared with a CUSUM test on structural change that is specifically constructed for long-memory time series. It is observed that the new approach performs reasonably well.The two chapters described previously find that self-normalized tests on change in mean are robust against size distortions in finite samples and that the CUSUM testing principle tends to be the most powerful. Based on these results, Chapter 4 proposes a new family of self-normalized CUSUM tests for structural change under long memory. The test statistics apply non-parametric kernel-based long-run variance estimators and have well-defined limiting distributions that only depend on the long-memory parameter. A Monte Carlo simulation shows that these tests provide finite sample size control while outperforming the competing procedures, which are presented in the previously described chapters, in terms of power.Chapter 5 presents the memochange package which offers implementations of all methods reviewed and proposed in Chapters 2 to 5 in the programming language R. The package is complemented with implementations of the most prominent change-in-persistence tests and estimation methods.Conversely to standard structural change tests which are invalid in long-memory time series, inference on the memory order is also biased (upwards) by the presence of structural change and other so called 'low-frequency contaminations'. Since the presence of long memory invalidates standard inference about structural breaks and vice versa, it is not clear for many economic time series - such as asset volatilities - what their actual degree of memory is.In Chapter 6, a comprehensive analysis of the memory in volatilities of international stock indices and exchange rates is provided. On the one hand, it is found that the volatility of exchange rates is subject to spurious long memory and the true memory parameter is in the higher stationary range. Stock index volatilities, on the other hand, are free of low-frequency contaminations and the memory is in the lower non-stationary range. These results are obtained using state-of-the-art local Whittle methods that allow consistent estimation in presence of perturbations or low-frequency contaminations.In Chapter 7 standard time series models are combined with search query data to examine whether they can be helpful in predicting sales. Search volume of company as well as product related keywords provided by Google Trends are included as new predictors in models to forecast sales on a product level. Using weekly data from January 2015 to December 2016 of two products of the audio company Sennheiser, evidence is found that using Google Trends data can enhance the prediction performance of conventional models.