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This thesis proposes novel methods to address specific challenges in analyzing the frequency- and time-domain properties of nonstationary time series data motivated by the study of electrophysiological signals. A new method is proposed for the simultaneous and automatic analysis of the association between the time-varying power spectrum and covariates. The procedure adaptively partitions the grid of time and covariate values into an unknown number of approximately stationary blocks and nonparametrically estimates local spectra within blocks through penalized splines. The approach is formulated in a fully Bayesian framework, in which the number and locations of partition points are random, and fit using reversible jump Markov chain Monte Carlo techniques. Estimation and inference averaged over the distribution of partitions allows for the accurate analysis of spectra with both smooth and abrupt changes. The new methodology is used to analyze the association between the time-varying spectrum of heart rate variability and self-reported sleep quality in a study of older adults serving as the primary caregiver for their ill spouse. Another method proposed in this dissertation develops a unique framework for automatically identifying bands of frequencies exhibiting similar nonstationary behavior. This proposal provides a standardized, unifying approach to constructing customized frequency bands for different signals under study across different settings. A frequency-domain, iterative cumulative sum procedure is formulated to identify frequency bands that exhibit similar nonstationary patterns in the power spectrum through time. A formal hypothesis testing procedure is also developed to test which, if any, frequency bands remain stationary. This method is shown to consistently estimate the number of frequency bands and the location of the upper and lower bounds defining each frequency band. This method is used to estimate frequency bands useful in summarizing nonstationary behavior of full night heart rate variability data. / Statistics

<p>This dissertation is a collection of four essays on nonstationary time series econometrics, which are grouped into four chapters. The first chapter investigates the inference in mildly explosive autoregressions under unconditional heteroskedasticity. The second chapter develops a new approach to forecasting a highly persistent time series that employs feasible generalized least squares (FGLS) estimation of the deterministic components in conjunction with Mallows model averaging. The third chapter proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by nonstationary volatility. The last chapter studies the problem of testing partial parameter stability in cointegrated regression models.</p>

Die vorliegende Arbeit befaßt sich mit instationären, uni- oder multivariaten Zeitreihen, die bei der Beobachtung komplexer nichtlinearer dynamischer Systeme entstehen und sich der Modellierung durch ein globales Modell entziehen. In vielen natürlichen oder gesellschaftlichen Prozessen kann man jedoch wiederkehrende Phänomene beobachten, die von deren Rhythmen beeinflußt sind; ebenso lassen sich in technischen Prozessen beispielsweise aufgrund einer bedarfsorientierten Steuerung wiederholte, aber nicht periodische Verhaltensweisen ausmachen. Für solche Systeme und Zeitreihen wird deshalb vorgeschlagen, eine partielle Modellierung durch mehrere lokale Modelle vorzunehmen, die wiederkehrende Phänomene in Form zeitlich begrenzter Muster beschreiben. Um den Unwägbarkeiten dieser und sich anschließender Aufgabenstellungen Rechnung zu tragen, werden in dieser Arbeit durchgehend unscharfe Ansätze zur Modellierung von Mustern und ihrer Weiterverarbeitung gewählt und ausgearbeitet. Die Aufgabenstellung der Erkennung von Mustern in fortlaufenden Zeitreihen wird dahingehend verallgemeinert, daß unvollständige, sich noch in Entwicklung befindliche Musterinstanzen erkannt werden können. Basierend auf ebendieser frühzeitigen Erkennung kann der Verlauf der Zeitreihe -- und damit das weitere Systemverhalten -- lokal prognostiziert werden. Auf Besonderheiten und Schwierigkeiten, die sich aus der neuartigen Aufgabe der Online-Erkennung von Mustern ergeben, wird jeweils vermittels geeigneter Beispiele eingegangen, ebenso die praktische Verwendbarkeit des musterbasierten Vorhersageprinzips anhand realer Daten dokumentiert. / This dissertation focuses on non-stationary multivariate time series stemming from the observation of complex nonlinear dynamical systems. While one global model for such systems and time series may not always be feasible, we may observe recurring phenomena (patterns) in some of these time series. These phenomena might, for example, be caused by the rhythms of natural or societal processes, or a demand-oriented control of technical processes. For such systems and time series a partial modelling by means of multiple local models is being proposed. To cope with the intrinsic uncertainties of this task, fuzzy methods and models are being used throughout this work. Means are introduced for modelling and recognition of patterns in multivariate time series. Based on a novel method for the early recognition of incomplete patterns in streaming time series, a short-time prediction becomes feasible. Peculiarities and intrinsic difficulties of an online recognition of incomplete patterns are being discussed with the help of suitable examples. The usability of the pattern-based prediction approach is being demonstrated by means of real-world data.

Die vorliegende Arbeit befaßt sich mit instationären, uni- oder multivariaten Zeitreihen, die bei der Beobachtung komplexer nichtlinearer dynamischer Systeme entstehen und sich der Modellierung durch ein globales Modell entziehen. In vielen natürlichen oder gesellschaftlichen Prozessen kann man jedoch wiederkehrende Phänomene beobachten, die von deren Rhythmen beeinflußt sind; ebenso lassen sich in technischen Prozessen beispielsweise aufgrund einer bedarfsorientierten Steuerung wiederholte, aber nicht periodische Verhaltensweisen ausmachen. Für solche Systeme und Zeitreihen wird deshalb vorgeschlagen, eine partielle Modellierung durch mehrere lokale Modelle vorzunehmen, die wiederkehrende Phänomene in Form zeitlich begrenzter Muster beschreiben. Um den Unwägbarkeiten dieser und sich anschließender Aufgabenstellungen Rechnung zu tragen, werden in dieser Arbeit durchgehend unscharfe Ansätze zur Modellierung von Mustern und ihrer Weiterverarbeitung gewählt und ausgearbeitet. Die Aufgabenstellung der Erkennung von Mustern in fortlaufenden Zeitreihen wird dahingehend verallgemeinert, daß unvollständige, sich noch in Entwicklung befindliche Musterinstanzen erkannt werden können. Basierend auf ebendieser frühzeitigen Erkennung kann der Verlauf der Zeitreihe -- und damit das weitere Systemverhalten -- lokal prognostiziert werden. Auf Besonderheiten und Schwierigkeiten, die sich aus der neuartigen Aufgabe der Online-Erkennung von Mustern ergeben, wird jeweils vermittels geeigneter Beispiele eingegangen, ebenso die praktische Verwendbarkeit des musterbasierten Vorhersageprinzips anhand realer Daten dokumentiert. / This dissertation focuses on non-stationary multivariate time series stemming from the observation of complex nonlinear dynamical systems. While one global model for such systems and time series may not always be feasible, we may observe recurring phenomena (patterns) in some of these time series. These phenomena might, for example, be caused by the rhythms of natural or societal processes, or a demand-oriented control of technical processes. For such systems and time series a partial modelling by means of multiple local models is being proposed. To cope with the intrinsic uncertainties of this task, fuzzy methods and models are being used throughout this work. Means are introduced for modelling and recognition of patterns in multivariate time series. Based on a novel method for the early recognition of incomplete patterns in streaming time series, a short-time prediction becomes feasible. Peculiarities and intrinsic difficulties of an online recognition of incomplete patterns are being discussed with the help of suitable examples. The usability of the pattern-based prediction approach is being demonstrated by means of real-world data.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal. / In this thesis, we deal with three particular issues in the literature on nonstationary time series. The first essay deals with various unit root tests in the context of structural change. The second paper studies some residual based tests in order to identify cointegration. Finally, in the third essay, we analyze several tests in order to identify additive outliers in nonstationary time series. The first paper analyzes the hypothesis that some time series can be characterized as stationary with a broken trend. We extend the class of M-tests and ADF test for a unit root to the case where a change in the trend function is allowed to occur at an unknown time. These tests (MGLS, ADFGLS) adopt the Generalized Least Squares (GLS) detrending approach to eliminate the set of deterministic components present in the model. We consider two models in the context of the structural change literature. The first model allows for a change in slope and the other for a change in slope as well as intercept. We derive the asymptotic distribution of the tests as well as that of the feasible point optimal test (PF-Ls) which allows us to find the power envelope. The asymptotic critical values of the tests are tabulated and we compute the non-centrality parameter used for the local GLS detrending that permits the tests to have 50% asymptotic power at that value. Two methods to select the break point are analyzed. A first method estimates the break point that yields the minimal value of the statistic. In the second method, the break point is selected such that the absolute value of the t-statistic on the change in slope is maximized. We show that the MGLS and PTGLS tests have an asymptotic power function close to the power envelope. An extensive simulation study analyzes the size and power of the tests in finite samples under various methods to select the truncation lag for the autoregressive spectral density estimator. In an empirical application, we consider two U.S. macroeconomic annual series widely used in the unit root literature: real wages and common stock prices. Our results suggest a rejection of the unit root hypothesis. In other words, we find that these series can be considered as trend stationary with a broken trend. Given the fact that using the GLS detrending approach allows us to attain gains in the power of the unit root tests, a natural extension is to propose this approach to the context of tests based on residuals to identify cointegration. This is the objective of the second paper in the thesis. In fact, we propose residual based tests for cointegration using local GLS detrending to eliminate separately the deterministic components in the series. We consider two cases, one where only a constant is included and one where a constant and a time trend are included. The limiting distributions of various residuals based tests are derived for a general quasi-differencing parameter and critical values are tabulated for values of c = 0 irrespective of the nature of the deterministic components and also for other values as proposed in the unit root literature. Simulations show that GLS detrending yields tests with higher power. Furthermore, using c = -7.0 or c = -13.5 as the quasi-differencing parameter, based on the two cases analyzed, is preferable. The third paper is an extension of a recently proposed method to detect outliers which explicitly imposes the null hypothesis of a unit root. it works in an iterative fashion to select multiple outliers in a given series. We show, via simulation, that under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that this iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first-differenced data that has considerably more power. The issues are illustrated using two US/Finland real exchange rate series.

The idea of time series forecasting techniques is that the past has certain information about future. So, the question of how the information is encoded in the past can be interpreted and later used to extrapolate events of future constitute the crux of time series analysis and forecasting. Several methods such as qualitative techniques (e.g., Delphi method), causal techniques (e.g., least squares regression), quantitative techniques (e.g., smoothing method, time series models) have been developed in the past in which the concept lies in establishing a model either theoretically or mathematically from past observations and estimate future from it. Of all the models, time series methods such as autoregressive moving average (ARMA) process have gained popularity because of their simplicity in implementation and accuracy in obtaining forecasts. But, these models were formulated based on certain properties that a time series is assumed to possess. Classical decomposition techniques were developed to supplement the requirements of time series models. These methods try to define a time series in terms of simple patterns called trend, cyclical and seasonal patterns along with noise. So, the idea of decomposing a time series into component patterns, later modeling each component using forecasting processes and finally combining the component forecasts to obtain actual time series predictions yielded superior performance over standard forecasting techniques. All these methods involve basic principle of moving average computation. But, the developed classical decomposition methods are disadvantageous in terms of containing fixed number of components for any time series, data independent decompositions. During moving average computation, edges of time series might not get modeled properly which affects long range forecasting. So, these issues are to be addressed by more efficient and advanced decomposition techniques such as Wavelets and Empirical Mode Decomposition (EMD). Wavelets and EMD are some of the most innovative concepts considered in time series analysis and are focused on processing nonlinear and nonstationary time series. Hence, this research has been undertaken to ascertain the predictability of nonstationary time series using wavelet and Empirical Mode Decomposition (EMD) based ARMA models. The development of wavelets has been made based on concepts of Fourier analysis and Window Fourier Transform. In accordance with this, initially, the necessity of involving the advent of wavelets has been presented. This is followed by the discussion regarding the advantages that are provided by wavelets. Primarily, the wavelets were defined in the sense of continuous time series. Later, in order to match the real world requirements, wavelets analysis has been defined in discrete scenario which is called as Discrete Wavelet Transform (DWT). The current thesis utilized DWT for performing time series decomposition. The detailed discussion regarding the theory behind time series decomposition is presented in the thesis. This is followed by description regarding mathematical viewpoint of time series decomposition using DWT, which involves decomposition algorithm. EMD also comes under same class as wavelets in the consequence of time series decomposition. EMD is developed out of the fact that most of the time series in nature contain multiple frequencies leading to existence of different scales simultaneously. This method, when compared to standard Fourier analysis and wavelet algorithms, has greater scope of adaptation in processing various nonstationary time series. The method involves decomposing any complicated time series into a very small number of finite empirical modes (IMFs-Intrinsic Mode Functions), where each mode contains information of the original time series. The algorithm of time series decomposition using EMD is presented post conceptual elucidation in the current thesis. Later, the proposed time series forecasting algorithm that couples EMD and ARMA model is presented that even considers the number of time steps ahead of which forecasting needs to be performed. In order to test the methodologies of wavelet and EMD based algorithms for prediction of time series with non stationarity, series of streamflow data from USA and rainfall data from India are used in the study. Four non-stationary streamflow sites (USGS data resources) of monthly total volumes and two non-stationary gridded rainfall sites (IMD) of monthly total rainfall are considered for the study. The predictability by the proposed algorithm is checked in two scenarios, first being six months ahead forecast and the second being twelve months ahead forecast. Normalized Root Mean Square Error (NRMSE) and Nash Sutcliffe Efficiency Index (Ef) are considered to evaluate the performance of the proposed techniques. Based on the performance measures, the results indicate that wavelet based analyses generate good variations in the case of six months ahead forecast maintaining harmony with the observed values at most of the sites. Although the methods are observed to capture the minima of the time series effectively both in the case of six and twelve months ahead predictions, better forecasts are obtained with wavelet based method over EMD based method in the case of twelve months ahead predictions. It is therefore inferred that wavelet based method has better prediction capabilities over EMD based method despite some of the limitations of time series methods and the manner in which decomposition takes place. Finally, the study concludes that the wavelet based time series algorithm could be used to model events such as droughts with reasonable accuracy. Also, some modifications that could be made in the model have been suggested which can extend the scope of applicability to other areas in the field of hydrology.