Dissertation, Otto-Friedrich-Universität Bamberg, 2025

Since many decades, a lot of methods have been proposed which are capable of investigating (strictly) stationary time series. However, it is expectable that many time series of practical interest (like those that underlie stock market data) are non-stationary. This motivates to consider locally stationary processes, which allow to model not just a stationary behaviour but also gradual distribution changes over a regarded time period.
In this thesis, some new tools for exploring a quite general class of locally stationary processes, which fulfil pretty weak moment conditions, are introduced and applied to log returns of several stock prices. Concretely, instruments are proposed that detect deviations between the distributions of the stationary approximations belonging to a locally stationary process and that also measure how large such deviations are. In addition, a new consistent level-alpha test for independence is introduced, which aims to reveal whether dependences between the stationary approximations of two locally stationary processes exist. Moreover, it is shown that the present methods also allow to formulate approximate statements about deviations between the distributions of the random variables contained in a locally stationary process and about dependences between two locally stationary processes.
In detail, at first, the class of locally stationary Bernoulli shift processes (which underlies the present thesis) and belonging characteristic functions as well as their estimators are introduced. Next, two L^2-distance-based measures that quantify deviations between the distributions of the stationary approximations belonging to a locally stationary process are proposed, which are based on the previously defined characteristic functions. However, these characteristic functions are commonly unknown in practise, such that confidence intervals for both measures are estimated in order to quantify the intensity of deviations between distributions in applications. Therefor, in a first step, empirical versions of both measures are proposed, which are constructed by using estimators for the underlying characteristic functions. Next, in a second step, for each measure, the asymptotic distribution of the (with an appropriate rate of convergence scaled) difference between this measure and its empirical version is stated. Since these limiting distributions depend on parameters which are commonly unknown in practise, dependent wild bootstrap procedures are introduced in a third step that approximate them. Overall, combining these steps yields suitable estimators for the confidence intervals for both measures. Furthermore, based on the empirical measures, consistent level-alpha tests are constructed that aim to detect whether deviations between the distributions of the stationary approximations belonging to a locally stationary process exist, whereby the associated p-values are estimated by an appropriate dependent wild bootstrap procedure. In addition, the first change point in the distributions of the stationary approximations is estimated by using these empirical measures.
Subsequently, a consistent level-alpha test is proposed that allows to reveal whether the stationary approximations of two locally stationary time series dependent on each other within arbitrary but fixed time periods. The belonging test statistic is constructed by using empirical characteristic functions in combination with an L^2-distance and the associated p-values are estimated by a dependent wild bootstrap procedure.
Furthermore, the finite sample behaviour of the instruments introduced in this thesis is evaluated by simulation studies and these tools are applied to log returns of several stocks, whereby the results are interpreted from an economic perspective.