Topics in Nonstationary Time Series Analysis

Several interesting applications in areas such as neuroscience, economics, finance and seismology have led to the collection nonstationary time series data wherein the statistical properties of the observed process change across time. The analysis of nonstationary time series data is an important and challenging task with useful applications. In comparison to stationarity, modeling temporal dependence in nonstationary time series is more non-trivial, and numerous methods have been proposed to tackle this problem. Stationarity in time series is more coveted than nonstationarity and many of the existing techniques attempt to transform the problem of nonstationarity to a stationary time series setting. Change point detection is one such method that attempts to find time points wherein the statistical properties of the time series changed. We develop a nonparametric method to detect multiple change points in multivariate piecewise stationary processes when the locations and number of change points are unknown. Based on a test statistic that measures differences in the spectral density matrices through the L₂ norm, we sequentially identify points of local maxima in the test statistic and test for the significance of each of them being change points. In addition, the components responsible for the change in the covariance structure at each detected change point are identified. The asymptotic properties of the test for significant change points under the null and alternative hypothesis are derived. Another related method for handling nonstationarity is the recent technique of stationary subspace analysis (SSA) that aims at finding linear transformations of nonstationary processes that are stationary. We propose an SSA procedure for general multivariate second-order nonstationary processes. It relies on the asymptotic uncorrelatedness of the discrete Fourier transform of a stationary time series to define a measure of departure from stationarity; it is then minimized to find the stationary subspace. The dimension of the subspace is estimated using a sequential testing procedure and its asymptotic properties are discussed. We illustrate the broader applicability and better performance of our method in comparison to existing SSA methods through simulations and discuss an application in neuroeconomics. Here we apply our method to filter out noise in EEG brain signals from an economic choice task experiment. This improves prediction performance and more importantly reduces the number of trials needed from individuals in neuroeconomic experiments thereby aligning with the principle of simple and controlled designs in experimental and behavioral economics