The tail dependence coefficient (TDC) is a natural tool to describe extremal dependence. Estimation of the tail dependence coefficient can be performed via empirical process theory. In case of extremal independence, the limit degenerates and hence one cannot construct a test for extremal independence. In order to deal with this issue, we consider an analog of the covariance matrix, namely the extremogram matrix, whose entries depend only on extremal observations. We show that under the null hypothesis of extremal independence and for finite dimension d ≥ 2, the largest eigenvalue of the sample extremogram matrix converges to the maximum of d independent normal random variables. This allows us to conduct an hypothesis testing for extremal independence by means of the asymptotic distribution of the largest eigenvalue. Simulation studies are performed to further illustrate this approach.

This paper studies the dependence structure between the real estate and financial markets in the United States from roughly 1975 to 2010, including the stock, bond and foreign exchange markets. This analysis uses dynamic copulas, including the Gaussian, Gumbel and Clayton copula. The Gumbel and Clayton copulas are used to separately capture the tail dependence of data. The dependence between the property indices (HPI and NCREIF) and the three financial markets is analyzed using the parameters of the copula. The property indices are divided in two different ways: by different regions and by different types of real estate. Although we study the dependence between the real estate and the financial markets in the U.S., the main objective of this paper is to analyze the change in the dependence structure when financial disasters occur. This study indicates that the real estate and the stock markets were positively related during this time period, and this dependence drove extreme movement when financial crises occurred. This dependence differed depending on the type of financial crisis, such as the Internet bubble crisis or the financial crisis in 2008. The dependence between the real estate and bond markets was also positively related, and extreme movement also occurred during financial crises. As for the dependence between the real estate and foreign exchange markets, although the results shows that dependence decreased when financial crises occurred, this is because the value of U.S. dollars are opposite to those of the index, and the left tail dependence exists as previous result. When looking at different regions or types of property, the differences in dependence structure were not obvious, although they were positively related. Both right and left tail dependences existed for most regions and property types, although some regions or types showed either right or left tail dependences alone. Therefore, investors should focus on the relationship between different markets, not on the region or type of real estate.

<p>Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here. </p><p>Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.</p><p>One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.</p><p>Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.</p><p>In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models. </p><p>Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data. </p><p>The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.</p><p>Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.</p> / Dissertation

The dependence structures in financial markets count among the most frequently discussed topics in the recent literature. However, no general consensus on modeling of the cross-market linkages has been reached. This thesis analyses the dependence structure and contagion in the financial markets in Central and Eastern Europe. Tail dependence, symmetry and dynamics of the dependence structure are examined. A conditional copula framework extended by recently developed dynamic generalized autoregressive score (GAS) model is used to capture the conditional time-varying joint distribution of stock market returns. Considering the Czech, Croatian, Hungarian, Austrian and Polish stock market indices over the 2005-2012 period, we find that time-varying Student's t GAS copula provides the best fit. The results show, that the degree of dependence increases substantially during the global financial crisis, having a direct impact on portfolio optimization.

In this study, we analyze the quantile dependence for green bond returns and renewable energy stock returns with three major asset classes: corporate bonds, stocks and oil. Furthermore, we control the dependence structure for technology, uncertainties as well as lag structures and time-varying effects. We apply the cross-quantilogram developed by Han et al. (2016) that allows us to study the dependence structures between two time series in arbitrary quantiles. The results led us to three key findings: 1) The returns of thegreen bond market are tail-dependent on the returns of both long and short-term maturities for the corporate bond market but are not dependent on the stock market nor the oil market. The tail-dependence indicates that while investors may hold green bonds due to moral incentives, it is not enough during times of turbulence. Further, the dependence structures are short-lived. 2)The renewable energy market is dependent on oil returns of similar quantiles, suggesting that renewable energy substitutes oil when oil prices increase. However, renewable energy does not influence the oil market, indicating that oil is not a substitutional energy source for renewable energy driven firms. Renewable energy stocks are further highly dependent on the returns of the general stock market but are not influenced by the returns on the corporate bond market. 3) The dependence of both renewable energy and green bonds with the asset markets are time-varying. Our overall results obtained by this paper provides information that could help facilitate new investment allocations towards green investments. Further, the results may have immediate and important implications for investors. For those in the corporate bond market, adding green bonds does not add diversification benefits during turbulence. Similarly, renewable energy stock does not add diversification benefits to investors in the oil or stock market.

DeMiguel, Garlappi e Uppal (2009) fizeram a comparação da regra 1/N ou de Talmud com 14 modelos de otimização que vieram depois do trabalho de Markowitz (1952). As conclusões mostraram que todos os modelos de alocação ótima analisados tiveram um desempenho inferior ao da regra de Talmud. Tu e Zhou (2011) propuseram uma combinação entre Markowitz e Talmud para que tal modelo superasse Talmud. Os resultados obtidos foram satisfatórios. A desconsideração dos eventos extremos (dependência assimétrica ou caudal) durante o processo de construção de carteiras poderá diminuir as habilidades dos gestores de ativos em reduzir o risco através da diversificação. A modelagem de cópulas sobre os retornos dos ativos nos permite calcular uma alternativa para medir a dependência dos ativos em eventos extremos através do índice de dependência caudal inferior. Hatherley e Alcock (2007) relataram que o modelo de Markowitz tende a subestimar as perdas potenciais que venham a ocorrer na presença de eventos extremos de mercado (crashes) para um determinado nível de retorno esperado. Verificamos se as estratégias com dependência caudal superaram Talmud, o modelo de Markowitz e o modelo de Tu e Zhou (2011) através da simulação de 1.000 carteiras com 3, 5, 10 e 20 ativos escolhidos ao acaso do índice DJIA no período de 03/1990 até 12/2012. Concluímos que os modelos de dependência caudal e o de Markowitz tiveram uma desempenho fora da amostra superior ao Talmud e ao modelo de Tu e Zhou (2011) para as carteiras com 3, 5, 10 e 20 ativos. A estratégia com dependência caudal superou Markowitz, em termos de retorno acumulado, em mais de 60% dos meses considerados em todas as análises. Os resultados apontam que a regra de Talmud deve ser descartada num contexto de construção de carteiras com ações frente à estratégia com dependência caudal. / DeMiguel, Garlappi and Uppal (2009) made a comparison of rule 1 / N or Talmud with most optimization techniques that followed the work of Markowitz (1952). The conclusions were devastating for all asset allocation models in the context of portfolios combined with other portfolios. Tu and Zhou (2011) proposed a combination between Markowitz and Talmud to overcome such a rule Talmud. The results were satisfactory. In the presence of extreme events, the Pearson correlation coefficient tends to increase in magnitude, making spurious results diversification based solely on this factor. The elimination of extreme events (asymmetric or tail dependence) during the portfolio construction process can reduce the skills of asset managers to reduce risk through diversification. The copula theory allows us to calculate an alternative to measure the dependence of extreme events in assets through the index lower tail dependence. Hatherley and Alcock (2007) reported that the Markowitz model tends to underestimate the potential losses that may occur in the presence of extreme market events (crashes) for a given level of expected return. We check that the strategies with tail dependence overcame Talmud rule, the Markowitz model and the model of Tu and Zhou (2011) by simulating 1,000 portfolios with 3, 5, 10 and 20 randomly selected assets from DJIA for the period 03/1990 until 12/2012. We conclude that models of tail dependence and Markowitz had more performance ex-ante than Talmud and the Tu and Zhou (2011) model for portfolios with 3, 5, 10 and 20 assets. Tail dependence models overcome Markowitz, in terms of cumulative return, in over 60% of months considered in the analysis. The results indicate that the Talmud rule should be discarded in a context of constructing portfolios with individual stocks ahead strategies with tail dependence.

DeMiguel, Garlappi e Uppal (2009) fizeram a comparação da regra 1/N ou de Talmud com 14 modelos de otimização que vieram depois do trabalho de Markowitz (1952). As conclusões mostraram que todos os modelos de alocação ótima analisados tiveram um desempenho inferior ao da regra de Talmud. Tu e Zhou (2011) propuseram uma combinação entre Markowitz e Talmud para que tal modelo superasse Talmud. Os resultados obtidos foram satisfatórios. A desconsideração dos eventos extremos (dependência assimétrica ou caudal) durante o processo de construção de carteiras poderá diminuir as habilidades dos gestores de ativos em reduzir o risco através da diversificação. A modelagem de cópulas sobre os retornos dos ativos nos permite calcular uma alternativa para medir a dependência dos ativos em eventos extremos através do índice de dependência caudal inferior. Hatherley e Alcock (2007) relataram que o modelo de Markowitz tende a subestimar as perdas potenciais que venham a ocorrer na presença de eventos extremos de mercado (crashes) para um determinado nível de retorno esperado. Verificamos se as estratégias com dependência caudal superaram Talmud, o modelo de Markowitz e o modelo de Tu e Zhou (2011) através da simulação de 1.000 carteiras com 3, 5, 10 e 20 ativos escolhidos ao acaso do índice DJIA no período de 03/1990 até 12/2012. Concluímos que os modelos de dependência caudal e o de Markowitz tiveram uma desempenho fora da amostra superior ao Talmud e ao modelo de Tu e Zhou (2011) para as carteiras com 3, 5, 10 e 20 ativos. A estratégia com dependência caudal superou Markowitz, em termos de retorno acumulado, em mais de 60% dos meses considerados em todas as análises. Os resultados apontam que a regra de Talmud deve ser descartada num contexto de construção de carteiras com ações frente à estratégia com dependência caudal. / DeMiguel, Garlappi and Uppal (2009) made a comparison of rule 1 / N or Talmud with most optimization techniques that followed the work of Markowitz (1952). The conclusions were devastating for all asset allocation models in the context of portfolios combined with other portfolios. Tu and Zhou (2011) proposed a combination between Markowitz and Talmud to overcome such a rule Talmud. The results were satisfactory. In the presence of extreme events, the Pearson correlation coefficient tends to increase in magnitude, making spurious results diversification based solely on this factor. The elimination of extreme events (asymmetric or tail dependence) during the portfolio construction process can reduce the skills of asset managers to reduce risk through diversification. The copula theory allows us to calculate an alternative to measure the dependence of extreme events in assets through the index lower tail dependence. Hatherley and Alcock (2007) reported that the Markowitz model tends to underestimate the potential losses that may occur in the presence of extreme market events (crashes) for a given level of expected return. We check that the strategies with tail dependence overcame Talmud rule, the Markowitz model and the model of Tu and Zhou (2011) by simulating 1,000 portfolios with 3, 5, 10 and 20 randomly selected assets from DJIA for the period 03/1990 until 12/2012. We conclude that models of tail dependence and Markowitz had more performance ex-ante than Talmud and the Tu and Zhou (2011) model for portfolios with 3, 5, 10 and 20 assets. Tail dependence models overcome Markowitz, in terms of cumulative return, in over 60% of months considered in the analysis. The results indicate that the Talmud rule should be discarded in a context of constructing portfolios with individual stocks ahead strategies with tail dependence.

This thesis examines the dependence structures between commodity futures and corresponding commodity producer equity indices in bearish, bullish and normal market conditions. We study commodity futures and producer indices in the energy, precious metals, gold and agriculture commodity markets using daily return data that ranges from 16 December 2005 to 28 June 2019. We employ the cross-quantilogram approach developed by Han et al. (2016) to examine dependence structures in the full quantile range, which represents different market states. Furthermore, we control for different lag structures, uncertainties and time-varying dependence structures. From our results we conclude the following: 1) There are time-varying asymmetric and symmetric dependencies in different commodity markets. There is asymmetric dependence between commodity futures and producer indices in the precious metals, gold and agricultural markets. In the oil market, the relationship is symmetrical. No relationship is found in the natural gas market. 2) Heterogenous dependence structures are identified in the gold, precious metals and agricultural commodity markets. The oil market uncovers homogenous dependence structures. 3) The observed spillover in all markets occur in the very short run, at one day, and dissipates after a week and additionally after a month. Our results provide new information regarding commodity diversification attributes which can be useful to investors. Our results also provide important policy implications: Since volatility spillovers between commodity futures and producer indices may deter investors from including commodities in their portfolios, as they might lose their diversifier qualities, it is important to enforce policies that will prevent the spillovers between the assets. Further, regulations of the commodity futures markets could be an alternative to reduce the spillovers.

[pt] A representação e interpretação claras da estrutura de dependência presente em vetores aleatórios, em particular em vetores bivariados, podem ser feitas com o uso do conceito de cópulas. Na análise bivariada, os coeficientes de dependência homogênea e heterogênea de cauda têm por objetivo estudar uma medida de dependência quando as variáveis assumem valores extre- mos. Obtemos as expressões dos coeficientes de dependência heterogênea de cauda a partir da função de distribuição acumulada condicional e apresen- tamos a demonstração de que os coeficientes de dependência homogênea de cauda de uma distribuição normal assimétrica são iguais a zero. Com o uso do conceito de cópulas e de dependência de cauda total, estudamos a estru- tura de dependência entre as seguintes variáveis: (i) log-retornos das taxas, interpoladas, para a estrutura a termo pré-fixada de 1 ano e de 2 anos; (ii) log-retorno das taxas para a estrutura a termo pré-fixada de 1 (um) ano e log-retorno do índice do Ibovespa; e (iii) log-retorno das taxas para a estru- tura a termo pré-fixada de 1 (um) ano e log-retorno da expectativa da taxa PTAX, 6 meses a frente. / [en] Using the concepts of copula we can represent and interpret the dependence structure presented in random vectors with clarity, particularly in bivariate vectors. In bivariate analysis, the role of both heterogeneous tail-dependence coefficient and homogenous tail- dependence coefficient are to study a measure of dependence when variables reach extreme values. We find expressions for the heterogeneous tail-dependence coefficients from the conditional cumulative distribution function and prove that the homoge- neous tail-dependence coefficients of a skewed normal distribution are equal to zero. Using the concepts of copula and the total tail dependence, we study the dependence structure between the following variables: (i) log- return of interpolated rates for the 1-year and 2-year fixed term structure; (ii) log-return of interpolated rate for the 1-year and log- return for the Bo- vespa index; e (iii) log-return of interpolated rate for the 1-year fixed term structure and log-return of expected PTAX, 6 months ahead.

In the present thesis we deal with dependence among extremal values within time series. Concerning this type of relations the commonly used autocorrelation function does not provide sufficient information. Moreover, autocorrelation function is suitable for Gaussian processes while nowadays we often work with heavy-tailed time series. In this thesis we cover two measures of extremal dependence that are used for this type of data. We introduce the coefficient of tail dependence, measure of extremal dependence based on tail characteristics of joint survival function. The second measure is called extremogram, which depends only on the extreme values in the sequence. In addition to the theoretical part, simulation study and application to real data of both described measures including their comparison are performed. Results are stated together with tables and graphical output.