Structural Change in (Economic) Time Series

Methods for detecting structural changes, or change points, in time series data are widely used in many fields of science and engineering. This chapter sketches some basic methods for the analysis of structural changes in time series data. The exposition is confined to retrospective methods for univariate time series. Several recent methods for dating structural changes are compared using a time series of oil prices spanning more than 60 years. The methods broadly agree for the first part of the series up to the mid-1980s, for which changes are associated with major historical events, but provide somewhat different solutions thereafter, reflecting a gradual increase in oil prices that is not well described by a step function. As a further illustration, 1990s data on the volatility of the Hang Seng stock market index are reanalyzed.Comment: 12 pages, 6 figure

Novel Methods for Efficient Changepoint Detection

This thesis introduces several novel computationally efficient methods for offline and online changepoint detection. The first part of the thesis considers the challenge of detecting abrupt changes in scenarios where there is some autocorrelated noise or where the mean fluctuates locally between the changes. In such situations, existing implementations can lead to substantial overestimation of the number of changes. In response to this challenge, we introduce DeCAFS, an efficient dynamic programming algorithm to deal with such scenarios. DeCAFS models local fluctuations as a random walk process and autocorrelated noise as an AR(1) process. Through theory and empirical studies we demonstrate that this approach has greater power at detecting abrupt changes than existing approaches. The second part of the thesis considers a practical, computational challenge that can arise with online changepoint detection within the real-time domain. We introduce a new procedure, called FOCuS, a fast online changepoint detection algorithm based on the simple Page-CUSUM sequential likelihood ratio test. FOCuS enables the online changepoint detection problem to be solved sequentially in time, through an efficient dynamic programming recursion. In particular, we establish that FOCuS outperforms current state-of-the-art algorithms both in terms of efficiency and statistical power, and can be readily extended to more general scenarios. The final part of the thesis extends ideas from the nonparametric changepoint detection literature to the online setting. Specifically, a novel algorithm, NUNC, is introduced to perform an online detection for changes in the distribution of real-time data. We explore the properties of two variants of this algorithm using both simulated and real data examples

Novel Methods for the Detection of Emergent Phenomena in Streaming Data

In the fast paced and data rich world of today there is an increased demand for methods that analyse a stream of data in real time. In particular, there is a desire for methods that can identify phenomena in the data stream as they are emerging. These emergent phenomena can be viewed as observations being received that are surprising when compared to the history of the data. Motivated by challenges in the telecommunications sector, we develop methods that operate when the stream does not follow classical assumptions. This includes when the data are not independent or identically distributed, or when the phenomena occur gradually over time. This thesis makes three contributions to the field of anomaly detection for streaming data. The first, Non-Parametric Unbounded Change (NUNC), provides a non-parametric method for identifying changes in the distribution of a data stream. The second, Functional Anomaly Sequential Test (FAST), provides a method for identifying deviations from an expected shape in a stream of partially observed functional data. The third, mvFAST, extends FAST to the multivariate functional data setting

Sequential Cross-Validated Bandwidth Selection Under Dependence and Anscombe-Type Extensions to Random Time Horizons

To detect changes in the mean of a time series, one may use previsible detection procedures based on nonparametric kernel prediction smoothers which cover various classic detection statistics as special cases. Bandwidth selection, particularly in a data-adaptive way, is a serious issue and not well studied for detection problems. To ensure data adaptation, we select the bandwidth by cross-validation, but in a sequential way leading to a functional estimation approach. This article provides the asymptotic theory for the method under fairly weak assumptions on the dependence structure of the error terms, which cover, e.g., GARCH($p,q$) processes, by establishing (sequential) functional central limit theorems for the cross-validation objective function and the associated bandwidth selector. It turns out that the proof can be based in a neat way on \cite{KurtzProtter1996}'s results on the weak convergence of \ito integrals and a diagonal argument. Our gradual change-point model covers multiple change-points in that it allows for a nonlinear regression function after the first change-point possibly with further jumps and Lipschitz continuous between those discontinuities. In applications, the time horizon where monitoring stops latest is often determined by a random experiment, e.g. a first-exit stopping time applied to a cumulated cost process or a risk measure, possibly stochastically dependent from the monitored time series. Thus, we also study that case and establish related limit theorems in the spirit of \citet{Anscombe1952}'s result. The result has various applications including statistical parameter estimation and monitoring financial investment strategies with risk-controlled early termination, which are briefly discussed