In stead of assuming the starting observations as zero, the cointegration test statistic derived in this paper takes the initial conditions into consideration. The statistic helps us understand how the initial conditions affect the test and helps us recognize whether the cointegration relationship exists in the data generation process or not. Beside, with this statistic derived with the concept of discriminant function and residual based test, we can simulate our own critical value table in according to what starting observations we have in hand, what significant level we want and what value of rho we meet under H_1.

This thesis consists of four manuscripts in the area of nonlinear time series econometrics on topics of testing, modeling and forecasting nonlinear common features. The aim of this thesis is to develop new econometric contributions for hypothesis testing and forecasting in these area. Both stationary and nonstationary time series are concerned. A definition of common features is proposed in an appropriate way to each class. Based on the definition, a vector nonlinear time series model with common features is set up for testing for common features. The proposed models are available for forecasting as well after being well specified. The first paper addresses a testing procedure on nonstationary time series. A class of nonlinear cointegration, smooth-transition (ST) cointegration, is examined. The ST cointegration nests the previously developed linear and threshold cointegration. An Ftypetest for examining the ST cointegration is derived when stationary transition variables are imposed rather than nonstationary variables. Later ones drive the test standard, while the former ones make the test nonstandard. This has important implications for empirical work. It is crucial to distinguish between the cases with stationary and nonstationary transition variables so that the correct test can be used. The second and the fourth papers develop testing approaches for stationary time series. In particular, the vector ST autoregressive (VSTAR) model is extended to allow for common nonlinear features (CNFs). These two papers propose a modeling procedure and derive tests for the presence of CNFs. Including model specification using the testing contributions above, the third paper considers forecasting with vector nonlinear time series models and extends the procedures available for univariate nonlinear models. The VSTAR model with CNFs and the ST cointegration model in the previous papers are exemplified in detail,and thereafter illustrated within two corresponding macroeconomic data sets.

This thesis consists of four manuscripts in the area of nonlinear time series econometrics on topics of testing, modeling and forecasting nonlinear common features. The aim of this thesis is to develop new econometric contributions for hypothesis testing and forecasting in thesearea. Both stationary and nonstationary time series are concerned. A definition of common features is proposed in an appropriate way to each class. Based on the definition, a vector nonlinear time series model with common features is set up for testing for common features. The proposed models are available for forecasting as well after being well specified. The first paper addresses a testing procedure on nonstationary time series. A class of nonlinear cointegration, smooth-transition (ST) cointegration, is examined. The ST cointegration nests the previously developed linear and threshold cointegration. An F-type test for examining the ST cointegration is derived when stationary transition variables are imposed rather than nonstationary variables. Later ones drive the test standard, while the former ones make the test nonstandard. This has important implications for empirical work. It is crucial to distinguish between the cases with stationary and nonstationary transition variables so that the correct test can be used. The second and the fourth papers develop testing approaches for stationary time series. In particular, the vector ST autoregressive (VSTAR) model is extended to allow for common nonlinear features (CNFs). These two papers propose a modeling procedure and derive tests for the presence of CNFs. Including model specification using the testing contributions above, the third paper considers forecasting with vector nonlinear time series models and extends the procedures available for univariate nonlinear models. The VSTAR model with CNFs and the ST cointegration model in the previous papers are exemplified in detail, and thereafter illustrated within two corresponding macroeconomic data sets.

Nested by linear cointegration first provided in Granger (1981), the definition of nonlinear cointegration is presented in this paper. Sequentially, a nonlinear cointegrated economic system is introduced. What we mainly study is testing no nonlinear cointegration against nonlinear cointegration by residual-based test, which is ready for detecting stochastic trend in nonlinear autoregression models. We construct cointegrating regression along with smooth transition components from smooth transition autoregression model. Some properties are analyzed and discussed during the estimation procedure for cointegrating regression, including description of transition variable. Autoregression of order one is considered as the model of estimated residuals for residual-based test, from which the teststatistic is obtained. Critical values and asymptotic distribution of the test statistic that we request for different cointegrating regressions with different sample sizes are derived based on Monte Carlo simulation. The proposed theoretical methods and models are illustrated by an empirical example, comparing the results with linear cointegration application in Hamilton (1994). It is concluded that there exists nonlinear cointegration in our system in the final results.

This paper describes the "strucplot" framework for the visualization of multi-way contingency tables. Strucplot displays include hierarchical conditional plots such as mosaic, association, and sieve plots, and can be combined into more complex, specialized plots for visualizing conditional independence, GLMs, and the results of independence tests. The framework's modular design allows flexible customization of the plots' graphical appearance, including shading, labeling, spacing, and legend, by means of "graphical appearance control" functions. The framework is provided by the R package vcd.

This paper describes the `strucplot' framework for the visualization of multi-way contingency tables. Strucplot displays include hierarchical conditional plots such as mosaic, association, and sieve plots, and can be combined into more complex, specialized plots for visualizing conditional independence, GLMs, and the results of independence tests. The framework's modular design allows flexible customization of the plots' graphical appearance, including shading, labeling, spacing, and legend, by means of graphical appearance control (`grapcon') functions. The framework is provided by the R package vcd. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Les solveurs de calcul en mécanique des fluides numérique (solveurs CFD) ont atteint leur maturité en termes de précision et d'efficacité de calcul. Toutefois, des progrès restent à faire pour les écoulements instationnaires surtout lorsqu'ils sont régis par de grandes structures cohérentes. Pour ces écoulements, les solveurs CFD actuels n'apportent pas de solutions assez précises à moins d'utiliser des maillages très fins. De plus, la haute précision est une caractéristique cruciale pour l'application des stratégies avancées de simulation de turbulence, comme la Simulation des Grandes Echelles (LES). Afin d'appliquer les méthodes d'ordre élevé pour les écoulements instationnaires complexes plusieurs points doivent être abordés dont la robustesse numérique et la capacité à gérer des géométries complexes.Dans cette thèse, nous étudions une famille d'approximations compactes qui offrent une grande précision non pour chaque dérivée spatiale traitée séparement mais pour le résidu r complet, c'est à dire la somme de tous les termes des équations considérées. Pour des problèmes stationnaires résolus par avancement temporelle, r est le résidu à l'état stationnaire ne comprenant que des dérivées spatiales; pour des problèmes instationnaires r comprend également la dérivée temporelle. Ce type de schémas sont appelés schémas Compacts Basés sur le Résidu (RBC). Plus précisément, nous développons des schémas RBC d'ordre élevé pour des écoulements instationnaires compressibles, et menons une étude approfondie de leurs propriétés de dissipation. Nous analysons ensuite les erreurs de dissipation et la dispersion introduites par les schémas RBC afin de quantifier leur capacité à résoudre une longueur d'onde donnée en utilisant un nombre minimal de points de maillage. Les capacités de la dissipation de RBC à drainer seulement l'énergie aux petites échelles sous-résolues sont également examinées en vue de l'application des schémas RBC pour des simulations LES implicites (ILES). Enfin, les schémas RBC sont étendus à la formulation de type volumes finis (FV) afin de gérer des géométries complexes. Une formulation FV des schémas RBC d'ordre trois préservant une précision d'ordre élevé sur des maillages irréguliers est présentée et analysée. Des applications numériques, dont la simulation d'écoulements instationnaires complexes de turbomachines régis par les équations de Navier-Stokes moyennées et des simulations ILES d'écoulements turbulents dominés par des structures cohérentes dynamiques ou en décroissance, confirment les résultats théoriques. / Computational Fluid Dynamics (CFD) solvers have reached maturity in terms of solution accuracy as well as computational efficiency. However, progress remains to be done for unsteady flows especially when governed by large, coherent structures. For these flows, current CFD solvers do not provide accurate solutions unless very fine mesh are used. Moreover, high-accuracy is a crucial feature for the application of advanced turbulence simulation strategies, like Large Eddy Simulation (LES). In order to apply high-order methods to complex unsteady flows several issues needs to be addressed among which numerical robustness and the capability of handling complex geometries.In the present work, we study a family of compact approximations that provide high accuracy not for each space derivative treated apart but for the complete residual r, i.e. the sum of all of the terms in the governing equations. For steady problems solved by time marching, r is the residual at steady state and it involves space derivatives only; for unsteady problems, r also includes the time derivative. Schemes of this type are referred-to as Residual-Based Compact (RBC). Precisely, we design high-order finite difference RBC schemes for unsteady compressible flows, and provide a comprehensive study of their dissipation properties. The dissipation and dispersion errors introduced by RBC schemes are investigated to quantify their capability of resolving a given wave length using a minimal number of grid-points. The capabilities of RBC dissipation to drain energy only at small, ill-resolved scales are also discussed in view of the application of RBC schemes to implicit LES (ILES) simulations. Finally, RBC schemes are extended to the Finite Volume (FV) framework in order to handle complex geometries. A high-order accuracy preserving FV formulation of the third-order RBC scheme for general irregular grids is presented and analysed. Numerical applications, including complex Reynolds-Averaged Navier-Stokes unsteady simulation of turbomachinery flows and ILES simulations of turbulent flows dominated by coherent structure dynamics or decay, support the theoretical results.

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.

Thesis title: Flexibility, Robustness and Discontinuity in Nonparametric Regression Approaches Author: Mgr. Matúš Maciak, M.Sc. Department: Department of Probability and Mathematical Statistics, Charles University in Prague Supervisor: Prof. RNDr. Marie Hušková, DrSc. huskova@karlin.mff.cuni.cz Abstract: In this thesis we focus on local polynomial estimation approaches of an unknown regression function while taking into account also some robust issues like a presence of outlying observa- tions or heavy-tailed distributions of random errors as well. We will discuss the most common method used for such settings, so called local polynomial M-smoothers and we will present the main statistical properties and asymptotic inference for this method. The M-smoothers method is especially suitable for such cases because of its natural robust flavour, which can nicely deal with outliers as well as heavy-tailed distributed random errors. Another important quality we will focus in this thesis on is a discontinuity issue where we allow for sudden changes (discontinuity points) in the unknown regression function or its derivatives respectively. We will propose a discontinuity model with different variability structures for both independent and dependent random errors while the discontinuity points will be treated in a...

Chorin’s method, Incremental Pressure Correction Scheme (IPCS) and Crank-Nicolson’s method (CN) are three numerical methods that were investigated in this study. These methods were here used for solving the incompressible Navier-Stokes equations, which describe the motion of an incompressible fluid, in three different benchmark problems. The methods were stabilized using residual based artificial viscosity, which was introduced to avoid instability. The methods were compared in terms of accuracy and computational time. Furthermore, a theoretical study of adaptivity was made, based on an a posteriori error estimate and an adjoint problem. The implementation of the adaptivity is left for future studies. In this study we consider the following three well-known benchmark problems: laminar 2D flow around a cylinder, Taylor-Green vortex and lid-driven cavity problem. The difference of the computational time for the three methods were in general relatively small and differed depending on which problem that was investigated. Furthermore the accuracy of the methods also differed in the benchmark problems, but in general Crank-Nicolson’s method gave less accurate results. Moreover the stabilization technique worked well when the kinematic viscosity of the fluid was relatively low, since it managed to stabilize the numerical methods. In general the solution was affected in a negative way when the problem could be solved without stabilization for higher viscosities.