Linear factor models have attracted considerable interest over recent years especially in the econometrics literature. The intuitively appealing idea to explain a panel of economic variables by a few common factors is one of the reasons for their popularity. From a statistical viewpoint, the need to reduce the cross-section dimension to a much smaller factor space dimension is obvious considering the large data sets available in economics and finance. One of the characteristics of the traditional factor model is that the process is stationary in the time dimension. This appears restrictive, given the fact that over long time periods it is unlikely that e.g. factor loadings remain constant. For example, in the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), typical empirical results show that factor loadings are time-varying, which in the CAPM is caused by time-varying second moments. In this thesis we generalize the tools of factor analysis for the study of stochastic processes whose behavior evolves over time. In particular, we introduce a new class of factor models with loadings that are allowed to be smooth functions of time. To estimate the resulting nonstationary factor model we generalize the properties of the principal components technique to the time-varying framework. We mainly consider separately two classes of Evolutionary Factor Models: Evolutionary Static Factor Models (Chapter 2) and Evolutionary Dynamic Factor Models (Chapter 3). In Chapter 2 we propose a new approximate factor model where the common components are static but nonstationary. The nonstationarity is introduced by the time-varying factor loadings, that are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators of the time varying covariance matrix, the loadings and the factors. This paper generalizes to the locally stationary case the results given by Bai (2003) in the stationary framework. A simulation study illustrates the performance of these estimators. The estimators proposed in Chapter 2 are based on a nonparametric estimator of the covariance matrix whose entries are computed with the same moothing parameter. This approach has the advantage of guaranteeing a positive definite estimator but it does not adapt to the different degree of smoothness of the different entries of the covariance matrix. In Chapter 5 we give an additional theoretical result which explains how to construct a positive definite estimate of the covariance matrix while while permitting different smoothing parameters. This estimator is based on the Cholesky decomposition of a pre-estimator of the covariance matrix. In Chapter 3 we introduce the dynamics in our modeling. This model generalizes the dynamic (but stationary) factor model of Forni et al. (2000), as well as the nonstationary (but static) factor model of Chapter 2. In the stationary (dynamic) case, Forni et al. (2000) show that the common components are estimated by the eigenvectors of a consistent estimator of the spectral density matrix, which is a matrix depending only on the frequency. In the evolutionary framework the dynamics of the model is explained by a time-varying spectral density matrix. This operator is a function of time as well as of the frequency. In this chapter we show that the common components of a locally stationary dynamic factor model can be estimated consistently by the eigenvectors of a consistent estimator of the time-varying spectral density matrix. In Chapter 4 we apply our theoretical results to real data and compare the performance of our approach with that based on standard techniques. Chapter 6 concludes and mention the main questions for future research.

The goal in this thesis is to develop a novel statistical approach to identity functional interactions between regions in a brain network. Wavelets are effective for capturing time varying properties of non-stationary signals because they have compact support that can be compressed or stretched according to the dynamic properties of the signal. Wavelets provide a multi-scale decomposition of signals and thus can be few for exploring potential cross-scale interactions between signals. To achieve this, we propose the scale-specific sub-processes of a multivariate locally stationary wavelet stochastic process. Under this proposed framework, a novel cross-scale dependence measurement is developed, which provides a measure for dependence structure of components at different scales of multivariate time series. Extensive simulation experiments are conducted to demonstrate that the theoretical properties hold in practice. The developed cross-scale analysis is performed on the electroencephalogram (EEG) data to study alterations in the functional connectivity structure in children diagnosed with attention deficit hyperactivity disorder (ADHD). Our approach identified novel interesting cross-scale interactions between channels in the brain network. The proposed framework can be extended to other signals, which can also capture the statistical association between the stocks at different time scales.

No contexto dos modelos fatoriais existem diferentes metodologias para abordar a modelagem de séries temporais multivariadas que exibem uma estrutura não estacionária de segunda ordem, co- movimentos e transições no tempo. Modelos com mudanças estruturais abruptas e restrições rigorosas (muitas vezes irreais) nas cargas fatoriais, quando elas são funções determinísticas no tempo, foram propostos na literatura para lidar com séries multivariadas que possuem essas características. Neste trabalho, apresentamos um modelo fatorial com cargas variando continuamente no tempo para modelar séries temporais não estacionárias e um procedimento para sua estimação que consiste em dois estágios. No primeiro, os fatores latentes são estimados empregando os componentes principais das séries observadas. Em um segundo estágio, tratamos estes componentes principais como co-variáveis e as cargas funcionais são estimadas através de funções de ondaletas e mínimos quadrados generalizados. Propriedades assintóticas dos estimadores de componentes principais e de mínimos quadrados dos coeficientes de ondaletas são apresentados. O desempenho da metodologia é ilustrado através de estudos de simulação. Uma aplicação do modelo proposto no mercado spot de energia do Nord Pool é apresentado. / In the context of the factor models there are different methodologies to modeling multivariate time series that exhibit a second order non-stationary structure, co-movements and transitions over time. Models with abrupt structural changes and strict restrictions (often unrealistic) in factor loadings, when they are deterministic functions of time, have been proposed in the literature to deal with multivariate series that have these characteristics. In this work, we present a factor model with time-varying loadings continuously to modeling non-stationary time series and a procedure for its estimation that consists of two stages. First, latent factors are estimated using the principal components of the observed series. Second, we treat principal components obtained in first stage as covariate and the functional loadings are estimated by wavelet functions and generalized least squares. Asymptotic properties of the principal components estimators and least squares estimators of the wavelet coefficients are presented. The per- formance of the methodology is illustrated by simulations. An application to the model proposed in the energy spot market of the Nord Pool is presented.

No contexto dos modelos fatoriais existem diferentes metodologias para abordar a modelagem de séries temporais multivariadas que exibem uma estrutura não estacionária de segunda ordem, co- movimentos e transições no tempo. Modelos com mudanças estruturais abruptas e restrições rigorosas (muitas vezes irreais) nas cargas fatoriais, quando elas são funções determinísticas no tempo, foram propostos na literatura para lidar com séries multivariadas que possuem essas características. Neste trabalho, apresentamos um modelo fatorial com cargas variando continuamente no tempo para modelar séries temporais não estacionárias e um procedimento para sua estimação que consiste em dois estágios. No primeiro, os fatores latentes são estimados empregando os componentes principais das séries observadas. Em um segundo estágio, tratamos estes componentes principais como co-variáveis e as cargas funcionais são estimadas através de funções de ondaletas e mínimos quadrados generalizados. Propriedades assintóticas dos estimadores de componentes principais e de mínimos quadrados dos coeficientes de ondaletas são apresentados. O desempenho da metodologia é ilustrado através de estudos de simulação. Uma aplicação do modelo proposto no mercado spot de energia do Nord Pool é apresentado. / In the context of the factor models there are different methodologies to modeling multivariate time series that exhibit a second order non-stationary structure, co-movements and transitions over time. Models with abrupt structural changes and strict restrictions (often unrealistic) in factor loadings, when they are deterministic functions of time, have been proposed in the literature to deal with multivariate series that have these characteristics. In this work, we present a factor model with time-varying loadings continuously to modeling non-stationary time series and a procedure for its estimation that consists of two stages. First, latent factors are estimated using the principal components of the observed series. Second, we treat principal components obtained in first stage as covariate and the functional loadings are estimated by wavelet functions and generalized least squares. Asymptotic properties of the principal components estimators and least squares estimators of the wavelet coefficients are presented. The per- formance of the methodology is illustrated by simulations. An application to the model proposed in the energy spot market of the Nord Pool is presented.

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D). In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed. In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided. In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity. In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.