O aumento dos fluxos globais comerciais e financeiros, a partir da década de 90, e as diversas crises ocorridas até o atual período fizeram da avaliação de contágio um tema extremamente relevante, tanto para investidores quanto para formuladores de política. Nesse sentido, a presente dissertação tem como objetivo testar a hipótese de contágio financeiro para os mercados de Brasil, Inglaterra e Espanha em face à última crise americana de 2008. Para tanto, desenvolveu-se o artigo que integra o Capítulo 2 - a espinha dorsal deste trabalho - com dados diários dos retornos dos índices de Jan/2004 a Jun/2011. No âmbito da metodologia de cópulas, adotou-se uma estratégia empírica com base em duas etapas: i) a estimativa não paramétrica de cópulas, via kernel, utilizando o método desenvolvido em Fermanian et al. (2002) e a avaliação através de uma abordagem de bootstrap, sobre a ocorrência de um aumento significativo nas medidas de dependência delas extraídas; ii) testes sobre a igualdade entre cópulas empíricas, conforme proposto por Remillard e Scaillet (2009), a fim de verificar se houve mudança na estrutura de dependência a partir da crise. Os resultados obtidos nas duas etapas da estratégia empírica são semelhantes e sugerem a existência de contágio financeiro para os países analisados no período estudado. / The increase in global trade and financial flows since the 90’s, and the various crises in the current period until these days made contagion an extremely important issue for both investors and policy makers. Accordingly, this dissertation aims to test the hypothesis of financial contagion between USA and markets in Brazil, England and Spain in the face of the last USA crisis of 2008. To this end, we produce the article in Chapter 2 - the backbone of this work - with daily data of index-returns from Jan/2004 to Jun/2011. Under the scope of copula methodology, we addopt an empirical strategy based on two steps: i) estimating nonparametric copulas via kernel, using the method developed in Fermanian et al. (2002) and assessing through a bootstrap approach whether a significant change in dependence measures extracts thereof, ii) testing whether two empirical estimated copulas are the same, as proposed by Remillard e Scaillet (2009), to check again whether dependence structures change with crisis. The results obtained in these two steps of the empirical strategy are similar and suggest the existence of financial contagion between the countries analysed in the studied period.

A mensuração do risco de um investimento é uma das mais importantes etapas para a tomada de decisão de um investidor. Em virtude disto, este trabalho comparou três métodos de estimação (tradicional, através da analise univariada dos retornos do portfólio; cópulas estáticas e cópulas dinâmicas) de duas medidas de risco: Value at Risk (VaR) e Expected Shortfall (ES). Tais medidas foram estimadas para o portfólio composto pelos índices BOVESPA e S&P500 no período de janeiro de 1998 a maio de 2012. Para as modelagens univariadas, incluindo as marginais das cópulas, foram comparados os modelos GARCH e EGARCH. Para cada modelo univariado, utilizamos as cópulas Normal, t-Student, Gumbel rotacionada e Joe-Clayton simetrizada, com isso totalizando 36 modelos comparados. Nas comparações do VaR e ES foram utilizados, respectivamente, o teste de Chritoffersen e o teste de Mcneil e Frey. Os principais resultados encontrados foram a superioridade de modelos que supõem erros com distribuição t-Student, assim como a identificação de mudança no comportamento dos parâmetros dinâmicos nos períodos de crise. / Measuring the risk of an investment is one of the most important steps in an investor's decision-making. With this in light, this study compared three estimation methods (traditional; by univariate analysis of portfolio returns; dynamic copulas and static copulas), of two risk measurements: Value at Risk (VaR) and Expected Shortfall (ES). Such estimated measures are performed for a portfolio composed by the BOVESPA and S&P500 indexes, ranging from January 1998 to May 2012. For univariate modelling (including copulas marginals), the GARCH and EGARCH models were compared,. Regarding copulas, we use Normal, t-Student, rotated Gumbel and symmetric Joe-Clayton, leading to a total of 36 models being compared. For the comparison of VaR and ES were used, respectively, the Christoffersen test, and the Mcneil and Frey test. The main results found were the superiority of models assuming the t-Student distributed errors, as well as the identification of a change in the behaviour of dynamic parameters in periods of crisis.

Cópulas tem se tornado uma importante ferramenta para descrever e analisar a estrutura de dependência entre variáveis aleatórias e processos estocásticos. Recentemente, surgiram alguns métodos de estimação não paramétricos, utilizando kernels e ondaletas. Neste contexto, sabendo que cópulas podem ser escritas como expansão em ondaletas, foi proposto um estimador não paramétrico via ondaletas para a função cópula para dados independentes e de séries temporais, considerando processos alfa-mixing. Este estimador tem como característica principal estimar diretamente a função cópula, sem fazer suposição alguma sobre a distribuição dos dados e sem ajustes prévios de modelos ARMA - GARCH, como é feito em ajuste paramétrico para cópulas. Foram calculadas taxas de convergência para o estimador proposto em ambos os casos, mostrando sua consistência. Foram feitos também alguns estudos de simulação, além de aplicações a dados reais. / Copulas are important tools for describing the dependence structure between random variables and stochastic processes. Recently some nonparametric estimation procedures have appeared, using kernels and wavelets. In this context, knowing that a copula function can be expanded in a wavelet basis, we have proposed a nonparametric copula estimation procedure through wavelets for independent data and times series under alpha-mixing condition. The main feature of this estimator is the copula function estimation without assumptions about the data distribution and without ARMA - GARCH modeling, like in parametric copula estimation. Convergence rates for the estimator were computed, showing the estimator consistency. Some simulation studies were made, as well as analysis of real data sets.

In this thesis we will deal with the term of operational risk, as it is presented in the directives Basel 2 that are mandatory for financial institutions in the European Union. The main problem is operational risk modeling, therefore, how to measure and manage it. In the first part we will look at the possibility of calculating the capital requirements for operational risk under Basel 2, mainly the calculation with the internal model. We will describe the specific procedures for the development of the internal model and we will focus on Loss Distribution Approach. The internal model will be based on modeling of loss in each risk cell separately. In the second part we will show, how to include modeling of dependence structure between risk cells to the internal model with using copulas. Finally, we will show the illustrative example, where we will see, whether the modeling of dependence leads to a reduction of the total capital requirement. Powered by TCPDF (www.tcpdf.org)

Diploma thesis abstract Thesis title: Statistical inference in multivariate distributions based on copula models Author: Vojtěch Kika This diploma thesis aims for statistical inference in copula based models. Ba- sics of copula theory are described, followed by methods for statistical inference. These are divided into three main groups. First of them are parametric methods for copula parameter estimation which assume fully parametric structure, thus for both joint and marginal distributions. The second group consists of semi- parametric methods for copula parameter estimation which, unlike parametric methods, do not require parametric structure for marginal distributions. The last group describes goodness-of-fit tests used for testing the hypothesis that consi- dered copula belongs to some specific copula family. The thesis is accompanied by a simulation study that investigates the dependence of the observed coverage of the asymptotic confidence intervals for copula parameter on the sample size. Pseudolikelihood method was chosen for the simulation study since it is one of the most popular semiparametric methods. It is shown that sample size of 50 seems to be sufficient for the observed coverage to be close to the theoretical one. For Frank and Gumbel-Hougaard copula families even sample size of 30 gives us...

Une classe générale de statistiques de rangs basées sur la fonction caractéristique est introduite afin de tester l'hypothèse composite d'appartenance à une famille de copules multidimensionnelles. Ces statistiques d'adéquation sont définies comme des distances fonctionnelles de type L_2 pondérées entre une version non paramétrique et une version semi-paramétrique de la fonction caractéristique que l'on peut associer à une copule. Il est démontré que ces statistiques de test se comportent asymptotiquement comme des V-statistiques dégénérées d'ordre quatre et que leurs lois limites s'expriment en termes de sommes pondérées de variables khi-deux indépendantes. La convergence des tests sous des alternatives générales est établie, de même que la validité du bootstrap paramétrique pour le calcul de valeurs critiques. Le comportement des nouveaux tests sous des tailles d'échantillons faibles et modérées est étudié à l'aide de simulations et est comparé à celui d'un test concurrent fondé sur la copule empirique. La méthodologie est finalement illustrée sur un jeu de données à plusieurs dimensions.

Dans cette thèse, on considère la modélisation de la loi jointe des statistiques d'ordre, c.à.d. des vecteurs aléatoires avec des composantes ordonnées presque sûrement. La première partie est dédiée à la modélisation probabiliste des statistiques d'ordre d'entropie maximale à marginales fixées. Les marginales étant fixées, la caractérisation de la loi jointe revient à considérer la copule associée. Dans le Chapitre 2, on présente un résultat auxiliaire sur les copules d'entropie maximale à diagonale fixée. Une condition nécessaire et suffisante est donnée pour l'existence d'une telle copule, ainsi qu'une formule explicite de sa densité et de son entropie. La solution du problème de maximisation d'entropie pour les statistiques d'ordre à marginales fixées est présentée dans le Chapitre 3. On donne des formules explicites pour sa copule et sa densité jointe. On applique le modèle obtenu pour modéliser des paramètres physiques dans le Chapitre 4.Dans la deuxième partie de la thèse, on étudie le problème d'estimation non-paramétrique des densités d'entropie maximale des statistiques d'ordre en distance de Kullback-Leibler. Le chapitre 5 décrit une méthode d'agrégation pour des densités de probabilité et des densités spectrales, basée sur une combinaison convexe de ses logarithmes, et montre des bornes optimales non-asymptotiques en déviation. Dans le Chapitre 6, on propose une méthode adaptative issue d'un modèle exponentiel log-additif pour estimer les densités considérées, et on démontre qu'elle atteint les vitesses connues minimax. L'application de cette méthode pour estimer des dimensions des défauts est présentée dans le Chapitre 7 / In this thesis we consider the modelling of the joint distribution of order statistics, i.e. random vectors with almost surely ordered components. The first part is dedicated to the probabilistic modelling of order statistics of maximal entropy with marginal constraints. Given the marginal constraints, the characterization of the joint distribution can be given by the associated copula. Chapter 2 presents an auxiliary result giving the maximum entropy copula with a fixed diagonal section. We give a necessary and sufficient condition for its existence, and derive an explicit formula for its density and entropy. Chapter 3 provides the solution for the maximum entropy problem for order statistics with marginal constraints by identifying the copula of the maximum entropy distribution. We give explicit formulas for the copula and the joint density. An application for modelling physical parameters is given in Chapter 4.In the second part of the thesis, we consider the problem of nonparametric estimation of maximum entropy densities of order statistics in Kullback-Leibler distance. Chapter 5 presents an aggregation method for probability density and spectral density estimation, based on the convex combination of the logarithms of these functions, and gives non-asymptotic bounds on the aggregation rate. In Chapter 6, we propose an adaptive estimation method based on a log-additive exponential model to estimate maximum entropy densities of order statistics which achieves the known minimax convergence rates. The method is applied to estimating flaw dimensions in Chapter 7

Depuis le début de la décennie 2000, le marché du maïs connaît un changement profond. D'une part, le prix enregistre une volatilité extrême sans précédent. D'autre part, ce marché bénéficie d'un déferlement massif des investisseurs financiers. Il offre des opportunités d'investissements financiers rentables en raison des crises récurrentes sur le marché boursier. Il est intéressant pour des investisseurs (spéculateurs, fondamentalistes, arbitragistes) d'avoir connaissance des résultats d'analyse des variations extrêmes du prix du marché du maïs. La maîtrise des variations extrêmes du prix permet une meilleure gestion du risque. Des études ont déjà été menées dans cette direction en utilisant des techniques du type « VaR ». Cependant, les différents modèles de gestion du risque par la VaR souffrent de certaines limites. Ils supposent l'hypothèse de la normalité des distributions. Or, la distribution des rendements du maïs montre des valeurs extrêmes. Cela ne permet pas une bonne appréciation du risque. Afin de contribuer à l'analyse des variations extrêmes de prix sur le marché du maïs, nous faisons appel aux modèles GARCH et à la théorie des valeurs extrêmes. Puis, dans un cadre multi-varié, le lien entre rendements spots et futures exprime le degré de la dépendance. Il permet ainsi d'analyser l'effet de la spéculation. Pour cela, nous utilisons la théorie des valeurs extrêmes couplée à la mesure de la dépendance qu'on appelle « copule » pour cerner les mouvements extrêmes des variations du prix au delà d'un seuil. En effet, la théorie des copules propose toute une gamme de fonctions capable de mesurer la dépendance asymétrique aux queues de la distribution des rendements spots et futures du maïs. / Since the early 2000s, the corn market is undergoing a profound change. On the one hand, the price has experienced unprecedented extreme volatility. Moreover, this market has a massive outpouring of financial investors. The corn market offers profitable financial investments due to recurrent crises in the stock market opportunities. It is interesting for investors (speculators, fundamentalists, arbitrageurs) to be aware of the analysis of extreme price changes in corn results. The mastery of extreme price changes provides better risk management. Studies have already been conducted in this direction by using techniques such as "VaR". However, the different models of risk management VaR suffer from certain limitations. They assume the assumption of normality of distributions. However, the distribution of return corn shows extreme values. This does not allow a proper assessment of risk. To contribute to the analysis of extreme price changes in the corn market, we use the GARCH models and the theory of extreme values. Then, in a multi-varied context, the link between returns and future spots expresses the degree of dependence. It allows analyzing the effect of speculation. We use extreme value theory coupled to the measure of dependence called "copula" to identify extreme movements of price changes beyond a threshold. Indeed, copula theory offers a range of features that can measure the asymmetric dependence tails of the distribution of spot return and futures of corn.

Available statistical methodologies focus more on accommodating continuous variables, however recently dealing with count data has received high interest in the statistical literature. In this dissertation, we propose some statistical approaches to investigate linear and nonlinear dependencies between two discrete random variables, or between a discrete and continuous random variables. Copula functions are powerful tools for modeling dependencies between random variables. We derive copula-based population version of Spearman’s rho when at least one of the marginal distribution is discrete. In each case, the functional relationship between Kendall’s tau and Spearman’s rho is obtained. The asymptotic distributions of the proposed estimators of these association measures are derived and their corresponding confidence intervals are constructed, and tests of independence are derived. Then, we propose a Bayesian copula factor autoregressive model for time series mixed data. This model assumes conditional independence and shares latent factors in both mixed-type response and multivariate predictor variables of the time series through a quadratic timeseries regression model. This model is able to reduce the dimensionality by accommodating latent factors in both response and predictor variables of the high-dimensional time series data. A semiparametric time series extended rank likelihood technique is applied to the marginal distributions to handle mixed-type predictors of the high-dimensional time series, which decreases the number of estimated parameters and provides an efficient computational algorithm. In order to update and compute the posterior distributions of the latent factors and other parameters of the models, we propose a naive Bayesian algorithm with Metropolis-Hasting and Forward Filtering Backward Sampling methods. We evaluate the performance of the proposed models and methods through simulation studies. Finally, each proposed model is applied to a real dataset.

The thesis presents fundamental characteristics of survival analysis in the case of competing risks and their relationships. In the case without regression, basic nonparametric estimates and a logarithmic likelihood function for parameter estimates is given. The main focus is on Cox's proportional hazards model (PH), a model with accelerated time (AFT) and a flexible regression model (FG) are also mentioned. The identifiability of the associated survival function is solved using copulas. Basics of copula theory and the measurement of dependence by correlation coefficients (Pearson, Spearman and Kendal) are described in a separate chapter. A substantial part of the theory is practically used in a generated case without regression.