When validating time series models, the distribution of the observations represents a potentially important assumption. In this Master's Thesis, the advocated approach uses smooth goodness-of-fit test statistics. This research provides theoretical and empirical developments of the smooth goodness of fit tests for vector autoregressive moving average models (VARMA). In previous work, Ducharme and Lafaye de Micheaux (2004) developed smooth goodness-of-fit tests designed for the residuals of univariate ARMA models. Later, Tagne Tatsinkou (2016) generalized the work within the framework of vector ARMA (VARMA) models, which prove to be potentially useful in real applications. Structured parameterizations, which are considerations specific to the multivariate case, are discussed. The works of Tagne Tatsinkou (2016) are completed, according to theoretical angles, and additional simulation studies are also considered. The new smooth tests are based on families of orthogonal polynomials. In this study, special attention is given to Legendre's family and Hermite's family. The major theoretical contribution in this work is a complete proof that the test statistic is invariant to linear affine transformations when the Hermite family is adopted. The results of Tagne Tatsinkou (2016) represent an important first step, but they were incomplete with respect to the use of the model residuals. The proposed tests are based on a family of densities under alternative hypotheses of order k. A data driven method to choose the maximal order, based on the results of Ledwina (1994), is discussed. In our simulation studies, the automatic selection is also implemented. Our simulation studies include bivariate models and a trivariate model. In the level study, we can appreciate the good performance of the smooth tests. In the power study, several competitors were considered. We found that the smooth tests displayed interesting power properties when the data came from VARMA models with innovations in the class of contaminated normal distributions.

Wavelet-based test procedures for lack-of-fit in autoregressive moving average (ARMA) models with conditionally heteroskedastic martingale difference innovations are investigated. In the framework of so-called semistrong ARMA models, the innovations are not independent, but are presumed a martingale difference, see for example Francq et al. (2005). Consequently, the Chi-square distributions of the popular Box-Pierce-Ljung test statistics, established under the hypothesis of independence in the error term, are not necessarily adequate. Moreover, the seasonal irregularities in the spectral density of the innovation can affect the power of the classical tests. This provides motivations for studying wavelet methods in the current context, where the error term is presumed more general than independent random variables. To find the asymptotic distribution of the new wavelet-based tests, we begin by establishing the asymptotic distribution of the residual autocovariances based on the residuals of the fitted model. The wavelet-based spectral density relies on certain coefficients, the so-called wavelet coefficients. We derive the asymptotic distribution of the empirical versions of those wavelet coefficients when the ARMA model is correctly specified. We show that the asymptotic variances and covariances of the empirical wavelet coefficients depend on both the coefficients of the ARMA model, and the covariance structure of the second moment of the innovations. Those results are used to construct new tests. We demonstrate that those test statistics have Chi-square distributions asymptotically. The performances of the proposed test statistics are investigated via Monte Carlo simulation studies for several sample sizes. The results of our simulations suggest that the new tests converge reasonably quickly to the Chi-square distributions and offer favorable power properties over the classical statistical tests, at least in some models. An example with real data illustrates the methodology.

This master’s thesis focuses on the study of dependent data. Classical literature has been widely focused on models described as linear. These models find great use when applied to macroeconomics data that are measured with relatively long time span (months, years, etc). When the data are measured using a shorter time span, and when the data are plentiful, it is then possible to describe the underlying stochastic process using more sophisticated models, which allows for proper modelling of the data’s nonlinear characteristics. It is in that contemporary setting that this master’s thesis takes place. It focuses on smooth transition autoregressive models, introduced and popularized by many authors including Teräsvirta (1994). We focus our study on modelling and forecasting for these models. As they are characterized by multiple regimes and a specific transition between those regimes, the smooth transition autoregressive models may allow for a more adequate modelling of a stochastic process using time series models that show nonlinear components. Our goal is to compare these models with the linear autoregressive models, and to study if they allow for more accurate forecasts. Thus, a main focus of this thesis is the elaboration of a forecasting system for these models. Although the use of smooth transition autoregressive models may seem enticing, applying the models to data brings its own share of complications, especially regarding parameters estimation. The estimation of the model sometimes has trouble converging due to a nonlinear component in the model, therefore requiring non-linear optimisation which is more complex. Of course, this is also true for the class of autoregressive moving average models (ARMA), which rely on the choice of autoregressive order p and moving average q. When q > 0, it is well-known that optimisation is nonlinear as well. However, empirical evidence suggests that numerical problems are less difficult than for smooth transition autoregressive models. The estimation of standard deviations of the parameters may also be difficult to obtain since the calculation of the variance-covariance matrix may be affected by computing issues. Forecasting for time series also poses issues for nonlinear time series. Classic linear theory is not applicable due to the nonlinear component in the model, and therefore point forecasting for smooth transition autoregressive models requires the use of other methods that vary in complexity, some which cause a bias for forecasts of lead times strictly larger than one which increase drastically in difficulty for longer lead times. For prediction intervals, because the linear theory is not applicable, they must be obtained using resampling methods. In fact, it can be asserted that an original contribution from this thesis is an elaborate and detailed study of point forecasting and forecast intervals for the models. In the first chapter, we introduce time series and classic linear models that are used to model time series. We elaborate nonlinear models and the univariate smooth transition autoregressive models, as well as the main characteristics and stationarity conditions for these models. In the second chapter, we develop estimation methods for smooth transition autoregressive models. The modelling process includes specification, estimation and evaluation of the model. Specification includes linearity tests for the data, which is required and justifies the use of nonlinear time series models. Estimation is done by nonlinear optimisation, and by finding its starting values using a grid search. In the third chapter, we present forecasting methods for classic linear autoregressive models and smooth transition autoregressive models. We elaborate the problems encountered when applying conventional theory to nonlinear models, and the methods to overcome these issues. We also define prediction intervals and how to obtain them. In the fourth chapter, we apply the theory of prior chapters to empirical simulations, with the goal of comparing linear autoregressive models to smooth transition autoregressive ones, focusing on modelling and both point and interval forecasting. We discuss our results. In the fifth chapter, we apply the theory to a time series representing daily returns of the SPDR S&P 500 stock market index (SPY ). We compare our results with those available in the literature, for both estimation of the smooth transition autoregressive models and point forecasting and prediction intervals performance. We then conclude the thesis.

In statistical modeling, we assume that the phenomenon of interest is generated by a model that can be fitted to the observed data. The part of the phenomenon not explained by the model is called error or innovation. There are two parts in the model. The main part is supposed to explain the observed data, while the unexplained part which is supposed to be negligible is also called error or innovation. In order to simplify the structures, the model are often assumed to rely on a finite set of parameters. The quality of a model depends also on the parameter estimators and their properties. For example, are the estimators relatively close to the true parameters ? Some questions also address the goodness-of-fit of the model to the observed data. This question is answered by studying the statistical and probabilistic properties of the innovations. On the other hand, it is also of interest to evaluate the presence or the absence of relationships between the observed data. Portmanteau or diagnostic type tests are useful to address such issue. The thesis is presented in the form of three projects. The first project is written in English as a scientific paper. It was recently submitted for publication. In that project, we study the class of vector multiplicative error models (vMEM). We use the properties of the Generalized Method of Moments to derive the asymptotic distribution of sample autocovariance function. This allows us to propose a new test statistic. Under the null hypothesis of adequacy, the asymptotic distributions of the popular Hosking-Ljung-Box (HLB) test statistics are found to converge in distribution to weighted sums of independent chi-squared random variables. A generalized HLB test statistic is motivated by comparing a vector spectral density estimator of the residuals with the spectral density calculated under the null hypothesis. In the second project, we derive the asymptotic distribution under weak dependence of cross covariances of covariance stationary processes. The weak dependence is defined in term of the limited effect of a given information on future observations. This recalls the notion of stability and geometric moment contraction. These conditions of weak dependence defined here are more general than the invariance of conditional moments used by many authors. A test statistic based on cross covariances is proposed and its asymptotic distribution is established. In the elaboration of the test statistics, the covariance matrix of the cross covariances is obtained from a vector autoregressive procedure robust to autocorrelation and heteroskedasticity. Simulations are also carried on to study the properties of the proposed test and also to compare it to existing tests. In the third project, we consider a cointegrated periodic model. Periodic models are present in the domain of meteorology, hydrology and economics. When modelling many processes, it can happen that the processes are just driven by a common trend. This situation leads to spurious regressions when the series are integrated but have some linear combinations that are stationary. This is called cointegration. The number of stationary linear combinations that are linearly independent is called cointegration rank. So, to model the real relationship between the processes, it is necessary to take into account the cointegration rank. In the presence of periodic time series, it is called periodic cointegration. It occurs when time series are periodically integrated but have some linear combinations that are periodically stationary. A two step estimation method is considered. The first step is the full rank estimation method that ignores the cointegration rank. It provides initial estimators to the second step estimation which is the reduced rank estimation. It is non linear and iterative. Asymptotic properties of the estimators are also established. In order to check for model adequacy, portmanteau type tests and their asymptotic distributions are also derived and their asymptotic distribution are studied. Simulation results are also presented to show the behaviour of the proposed test.

This thesis focuses on the treatment of representative outliers in two important aspects of surveys: small area estimation and imputation for item non-response. Concerning small area estimation, robust estimators in unit-level models have been studied. Sinha & Rao (2009) proposed estimation procedures designed for small area means, based on robustified maximum likelihood parameters estimates of linear mixed model and robust empirical best linear unbiased predictors of the random effect of the underlying model. Their robust methods for estimating area means are of the plug-in type, and in view of the results of Chambers (1986), the resulting robust estimators may be biased in some situations. Biascorrected estimators have been proposed by Chambers et al. (2014). In addition, these robust small area estimators were associated with the estimation of the Mean Square Error (MSE). Sinha & Rao (2009) proposed a parametric bootstrap procedure based on the robust estimates of the parameters of the underlying linear mixed model to estimate the MSE. Analytical procedures for the estimation of the MSE have been proposed in Chambers et al. (2014). However, their theoretical validity has not been formally established and their empirical performances are not fully satisfactorily. Here, we investigate two new approaches for the robust version the best empirical unbiased estimator: the first one relies on the work of Chambers (1986), while the second proposal uses the concept of conditional bias as an influence measure to assess the impact of units in the population. These two classes of robust small area estimators also include a correction term for the bias. However, they are both fully bias-corrected, in the sense that the correction term takes into account the potential impact of the other domains on the small area of interest unlike the one of Chambers et al. (2014) which focuses only on the domain of interest. Under certain conditions, non-negligible bias is expected for the Sinha-Rao method, while the proposed methods exhibit significant bias reduction, controlled by appropriate choices of the influence function and tuning constants. Monte Carlo simulations are conducted, and comparisons are made between: the new robust estimators, the Sinha-Rao estimator, and the bias-corrected estimator. Empirical results suggest that the Sinha-Rao method and the bias-adjusted estimator of Chambers et al (2014) may exhibit a large bias, while the new procedures offer often better performances in terms of bias and mean squared error. In addition, we propose a new bootstrap procedure for MSE estimation of robust small area predictors. Unlike existing approaches, we formally prove the asymptotic validity of the proposed bootstrap method. Moreover, the proposed method is semi-parametric, i.e., it does not rely on specific distributional assumptions about the errors and random effects of the unit-level model underlying the small-area estimation, thus it is particularly attractive and more widely applicable. We assess the finite sample performance of our bootstrap estimator through Monte Carlo simulations. The results show that our procedure performs satisfactorily well and outperforms existing ones. Application of the proposed method is illustrated by analyzing a well-known outlier-contaminated small county crops area data from North-Central Iowa farms and Landsat satellite images. Concerning imputation in the presence of item non-response, some single imputation methods have been studied. The deterministic regression imputation, which includes the ratio imputation and mean imputation are often used in surveys. These imputation methods may lead to biased imputed estimators if the imputation model or the non-response model is not properly specified. Recently, doubly robust imputed estimators have been developed. However, in the presence of outliers, the doubly robust imputed estimators can be very unstable. Using the concept of conditional bias as a measure of influence (Beaumont, Haziza and Ruiz-Gazen, 2013), we propose an outlier robust version of the doubly robust imputed estimator. Thus this estimator is denoted as a triple robust imputed estimator. The results of simulation studies show that the proposed estimator performs satisfactorily well for an appropriate choice of the tuning constant.

Several phenomena from natural and social sciences rely on distribution’s assumption among which the normal distribution is the most popular. The validity of that assumption is useful to setting up forecast intervals or for checking model adequacy of the underlying model. The goodness-of-fit procedures are tools to assess the adequacy of the data’s underlying assumptions. Autoregressive and moving average time series models are often used to find the mathematical behavior of these phenomena from natural and social sciences, and especially in the finance area. These models are based on some assumptions including normality distribution for the innovations. Normality assumption may be helpful for some testing procedures. Furthermore, stronger conclusions can be drawn from the adjusted model if the white noise can be assumed Gaussian. In this work, goodness-of-fit tests for checking normality for the innovations from autoregressive moving average time series models are proposed for both univariate and multivariate cases (ARMA and VARMA models). In our first project, a smooth test of normality for ARMA time series models with unknown mean based on a least square type estimator is proposed. We derive the asymptotic null distribution of the test statistic. The result here is an extension of the paper of Ducharme et Lafaye de Micheaux (2004), where they supposed the mean known and equal to zero. We use the least square type estimator proposed by Brockwell et Davis (1991, section 10.8) and we provide a rigorous proof that it is almost surely convergent. We show that the covariance matrix of the test is nonsingular regardless if the mean is known. We have also studied a data driven approach for the choice of the dimension of the family and we gave a finite sample approximation of the null distribution. Finally, the finite and asymptotic sample properties of the proposed test statistic are studied via a small simulation study. In the second project, goodness-of-fit tests for checking multivariate normality for the innovations from vector autoregressive moving average time series models are proposed. Since these time series models may rely on a large number of parameters, structured parameterization of the functional form is allowed. The methodology also relies on the smooth test paradigm and on families of orthonormal functions with respect to the multivariate normal density. It is shown that the smooth tests converge to convenient chi-square distributions asymptotically. An important special case makes use of Hermite polynomials, and in that situation we demonstrate that the tests are invariant under linear transformations. We observed that the test is not invariant under linear transformations with Legendre polynomials. A consistent data driven method is discussed to choose the family order from the data. In a simulation study, exact levels are studied and the empirical powers of the smooth tests are compared to those of other methods. Finally, an application to real data is provided, specifically on Canadian labour market data and annual global temperature. These works were exposed at several meeting (see for example Tagne, Duchesne and Lafaye de Micheaux (2013a, 2013b, 2014) for more details). A paper based on the first project is submitted in a refereed journal (see Duchesne, Lafaye de Micheaux et Tagne (2016)).

The objective of this master thesis is to present models for multivariate time series involving random vectors where each component is non-negative. We consider vMEM models (vector multiplicative error models) presented by Cipollini, Engle and Gallo (2006) and Cipollini and Gallo (2010). These models represent a generalization to the multivariate case of MEM models introduced by Engle (2002). These models are applied especially with financial time series. The vMEM models can be used to model time series involving asset volumes, durations, conditional variances, among these applications. It is also possible to model the variables jointly and to study the dynamics between these time series forming the system under study. To model multivariate time series with non-negative components, several specifications for the term vector error have been proposed in the literature. One approach is to consider the use of random vectors where the distribution of the error term is such that each component is non-negative. However, finding a sufficiently flexible multivariate distribution defined on the positive support is rather difficult, at least with the applications mentioned above. As indicated by Cipollini, Engle and Gallo (2006), a possible candidate is the multivariate gamma distribution, which however imposes severe restrictions on the contemporaneous correlations between variables. Since the possibilities are limited, one possible approach is to use the theory of copulas. Thus, according to this approach, margins can be specified, such that the distributions in question have non-negative supports, and a copula function takes into account the dependency between components. A possible estimation technique is the method of maximum likelihood. An alternative approach is the generalized method of moments (GMM). This latter method has the advantage of being semi-parametric in the sense that unlike the approach imposing a multivariate distribution, it is not necessary to specify a multivariate distribution for the error term. Generally, estimating vMEM models is complicated. Existing algorithms must take into account the large number of parameters and the elaborate nature of the likelihood function. In the case of the GMM estimation method, solving the system requires the use of solvers for non linear systems. In this project, considerable energy has been devoted to the development of computer code (in the R language) to estimate the different parameters of the model. In the first chapter, we define the stationary processes, autoregressive processes, the autoregressive conditional heteroskedasticity processes (ARCH) and the generalized ARCH processes (GARCH). We also present duration models ACD and MEM models. In the second chapter, we present the theory of copulas needed for our work, under the vector multiplicative error models vMEM. We also discuss the possible estimation methods. In the third chapter, we discuss the simulation results for several estimation methods. In the last chapter, applications to financial time series are presented. The R code is provided in an Appendix. A conclusion completes this thesis.

In this master thesis, time series models are studied, which have also a spatial component, in addition to the usual time index. More particularly, we study a certain class of models, the Generalized Space-Time AutoRegressive (GSTAR) time series models. First, links are considered between Vector AutoRegressive models(VAR) and GSTAR models. We obtain explicitly the asymptotic distribution of the residual autocovariances for the GSTAR models, assuming that the error term is a Gaussian white noise, which is a first original contribution. From that result, test statistics of the portmanteau type are proposed, and their asymptotic distributions are studied. In order to illustrate the behaviour of the test statistics, a simulation study is conducted where GSTAR models are simulated and correctly fitted. The methodology is illustrated with monthly real data concerning the production of tea in west Java for 24 cities from the period January 1992 to December 1999.

Time series models with conditionnaly heteroskedastic variances have become almost inevitable to model financial time series. In many applications, to confirm the existence of a relationship between two time series is very important. In this Master thesis, we generalize in several directions and in a multivariate framework, the method developed by Cheung and Ng (1996) designed to examine causality in variance in the case of two univariate series. Based on the work of El Himdi and Roy (1997) and Duchesne (2004), we propose a test based on residual cross-correlation matrices of squared residuals and cross-products of these residuals. Under the null hypothesis of no causality in variance, we establish that the test statistics converge in distribution to chi-square random variables. In a second approach, we define as in Ling and Li (1997) a transformation of the residuals for each residual time series. The test statistics are built from the cross-correlations of these transformed residuals. In both approaches, test statistics at individual lags are presented and also portmanteau-type test statistics. That methodology is also used to determine the direction of causality in variance. The simulation results show that the proposed tests provide satisfactory empirical properties. An application with real data is also presented to illustrate the methods

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