Stochastic Weather Generators (SWG) try to reproduce the stochastic patterns of climatological variables characterized by high dimensionality, non-normal probability density functions and non-linear dependence relationships. However, conventional SWGs usually typify weather variables with unjustified probability distributions assuming linear dependence between variables. This research proposes an alternative SWG that introduces the advantages of the Copula modeling into the reproduction of stochastic weather patterns. The Copula based SWG introduces more flexibility allowing researcher to model non-linear dependence structures independently of the marginals involved, also it is able to model tail dependence, which results in a more accurate reproduction of extreme weather events. Statistical tests on weather series simulated by the Copula based SWG show its capacity to replicate the statistical properties of the observed weather variables, along with a good performance in the reproduction of the extreme weather events. In terms of its use in crop growth models for the ratemaking process of new insurance schemes with no available historical yield data, the Copula based SWG allows one to more accurately evaluate the risk. The use of the Copula based SWG for the simulation of yields results in higher crop insurance premiums from more frequent extreme weather events, while the use of the conventional SWG for the yield estimation could lead to an underestimation of risks.

Ein Ausfallkorb (Default Basket, Basket Default Swap, BDS) ist die Bündelung einer relativ geringen Anzahl einzelner Kreditpositionen. Der Sicherungsgeber (Investor) verpflichtet sich, den i-ten Forderungsausfall zu übernehmen und wird als ith-to-default-Käufer bezeichnet. Da es sich um die Bündelung einer relativ geringen Anzahl von, möglicherweise sehr heterogenen Kreditpositionen handelt, lassen sich herkömmliche Modellierungsansätze aus dem Kreditrisiko nicht direkt zur Risikobewertung anwenden. Der vorliegende Beitrag stellt deshalb Möglichkeiten vor, eine Risikobewertung für Ausfallkörbe vorzunehmen. Der Modellierungsansatz über das Ein-Faktormodell ist dabei sehr stark an die typische Kreditrisikomodellierung angelehnt, weicht aber in einigen Punkten auch erheblich davon ab. Zentrales Anliegen dieses Artikels ist es daher, die wesentlichen Mechanismen zu verdeutlichen, welche die Risikobewertung in diesem Modell beeinflussen. Hierbei wird insbesondere das Zusammenspiel von Abhängigkeitsstruktur (in Form der Korrelation), Ausfallwahrscheinlichkeiten der Einzelpositionen und den Ausfallwahrscheinlichkeiten für die Risikogeber betrachtet.

Thesis advisor: Zhijie Xiao / The first two chapters study the copula Markov model combined with nonstationarity. The last chapter proposes a new structural break test with good size and power. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.

One of the fundamental tasks in science is to find explainable relationships between observed phenomena. Recent work has addressed this problem by attempting to learn the structure of graphical models - especially Gaussian models - by the imposition of sparsity constraints. The graphical lasso is a popular method for learning the structure of a Gaussian model. It uses regularisation to impose sparsity. In real-world problems, there may be latent variables that confound the relationships between the observed variables. Ignoring these latents, and imposing sparsity in the space of the visibles, may lead to the pruning of important structural relationships. We address this problem by introducing an expectation maximisation (EM) method for learning a Gaussian model that is sparse in the joint space of visible and latent variables. By extending this to a conditional mixture, we introduce multiple structures, and allow side information to be used to predict which structure is most appropriate for each data point. Finally, we handle non-Gaussian data by extending each sparse latent Gaussian to a Gaussian copula. We train these models on a financial data set; we find the structures to be interpretable, and the new models to perform better than their existing competitors. A potential problem with the mixture model is that it does not require the structure to persist in time, whereas this may be expected in practice. So we construct an input-output HMM with sparse Gaussian emissions. But the main result is that, provided the side information is rich enough, the temporal component of the model provides little benefit, and reduces efficiency considerably. The GWishart distribution may be used as the basis for a Bayesian approach to learning a sparse Gaussian. However, sampling from this distribution often limits the efficiency of inference in these models. We make a small change to the state-of-the-art block Gibbs sampler to improve its efficiency. We then introduce a Hamiltonian Monte Carlo sampler that is much more efficient than block Gibbs, especially in high dimensions. We use these samplers to compare a Bayesian approach to learning a sparse Gaussian with the (non-Bayesian) graphical lasso. We find that, even when limited to the same time budget, the Bayesian method can perform better. In summary, this thesis introduces practically useful advances in structure learning for Gaussian graphical models and their extensions. The contributions include the addition of latent variables, a non-Gaussian extension, (temporal) conditional mixtures, and methods for efficient inference in a Bayesian formulation.

The field of time-series analysis has made important contributions to a wide spectrum of applications such as tide-level studies in hydrology, natural resource prospecting in geo-statistics, speech recognition, weather forecasting, financial trading, and economic forecasts and analysis. Nevertheless, the analysis of the non-Gaussian and non-stationary features of time-series remains challenging for the current state-of-art models. This thesis proposes an innovative framework that leverages the theory of copula, combined with a probabilistic framework from the machine learning community, to produce a versatile tool for multiple time-series analysis. I coined this new model Kernel-based Copula Processes (KCPs). Under the new proposed framework, various idiosyncracies can be modeled compactly via a kernel function for each individual time-series, and long-range dependency can be captured by a copula function. The copula function separates the marginal behavior and serial dependency structures, thus allowing them to be modeled separately and with much greater flexibility. Moreover, the codependent structure of a large number of time-series with potentially vastly different characteristics can be captured in a compact and elegant fashion through the notion of a binding copula. This feature allows a highly heterogeneous model to be built, breaking free from the homogeneous limitation of most conventional models. The KCPs have demonstrated superior predictive power when used to forecast a multitude of data sets from meteorological and financial areas. Finally, the versatility of the KCP model is exemplified when it was successfully applied to non-trivial classification problems unaltered.

The field of time-series analysis has made important contributions to a wide spectrum of applications such as tide-level studies in hydrology, natural resource prospecting in geo-statistics, speech recognition, weather forecasting, financial trading, and economic forecasts and analysis. Nevertheless, the analysis of the non-Gaussian and non-stationary features of time-series remains challenging for the current state-of-art models. This thesis proposes an innovative framework that leverages the theory of copula, combined with a probabilistic framework from the machine learning community, to produce a versatile tool for multiple time-series analysis. I coined this new model Kernel-based Copula Processes (KCPs). Under the new proposed framework, various idiosyncracies can be modeled compactly via a kernel function for each individual time-series, and long-range dependency can be captured by a copula function. The copula function separates the marginal behavior and serial dependency structures, thus allowing them to be modeled separately and with much greater flexibility. Moreover, the codependent structure of a large number of time-series with potentially vastly different characteristics can be captured in a compact and elegant fashion through the notion of a binding copula. This feature allows a highly heterogeneous model to be built, breaking free from the homogeneous limitation of most conventional models. The KCPs have demonstrated superior predictive power when used to forecast a multitude of data sets from meteorological and financial areas. Finally, the versatility of the KCP model is exemplified when it was successfully applied to non-trivial classification problems unaltered.

In this thesis, an attempt is made to build on the quantitative research in the field of Islamic Finance. Firstly, univariate modelling using special GARCH-type models is performed on both the FTSE All World and FTSE Shari'ah All World indices. The AR(1) + APARCH(1,1) model with standardized skewed student-t innovations provided the best overall fit and was the most successful at VaR modelling for long and short trading positions. A risk assessment is done using the Conditional Tail Expectation (CTE) risk measure which concluded that in short trading positions the FTSE Shari'ah All World index was riskier than the FTSE All World index but, in long trading positions the results were not conclusive as to which is riskier. Secondly, under the Markowitz model of risk and return the performance of Islamic equity is compared to conventional equity using various Dow Jones indices. The results indicated that even though the Islamic portfolio is relatively less diversified than the conventional portfolio, due to several investment restrictions, the Shari'ah screening process excluded various industries whose absence resulted in risk reduction. As a result, the Islamic portfolio provided a basket of stocks with special and favourable risk characteristics. Lastly, copulas are used to model the dependency structure between the filtered returns of the FTSE All World and FTSE Shari'ah All World indices after fitting the AR(1) + APARCH(1,1) model with standardized skewed student-t innovations. The t copula outperformed the others and a demonstration of forecasting using the copula-extended model is done.

In this thesis, an attempt is made to build on the quantitative research in the field of Islamic Finance. Firstly, univariate modelling using special GARCH-type models is performed on both the FTSE All World and FTSE Shari'ah All World indices. The AR(1) + APARCH(1,1) model with standardized skewed student-t innovations provided the best overall fit and was the most successful at VaR modelling for long and short trading positions. A risk assessment is done using the Conditional Tail Expectation (CTE) risk measure which concluded that in short trading positions the FTSE Shari'ah All World index was riskier than the FTSE All World index but, in long trading positions the results were not conclusive as to which is riskier. Secondly, under the Markowitz model of risk and return the performance of Islamic equity is compared to conventional equity using various Dow Jones indices. The results indicated that even though the Islamic portfolio is relatively less diversified than the conventional portfolio, due to several investment restrictions, the Shari'ah screening process excluded various industries whose absence resulted in risk reduction. As a result, the Islamic portfolio provided a basket of stocks with special and favourable risk characteristics. Lastly, copulas are used to model the dependency structure between the filtered returns of the FTSE All World and FTSE Shari'ah All World indices after fitting the AR(1) + APARCH(1,1) model with standardized skewed student-t innovations. The t copula outperformed the others and a demonstration of forecasting using the copula-extended model is done.

In this paper, we study on the stock indexes return¡¦s dependence structure of the U.S. versus other G5 members during the 2008 subprime mortgage financial crisis. The sample series are weekly returns of the MSCI stock price indexes from 2003 to 2011. The model structure is combined with marginal model and copula model. We model the marginal distributions of our returns using the univariate skewed Student t AR¡]1¡^-GARCH model of Hansen¡]1994¡^, and we model the time-varying copula of Patton¡]2006¡^to measure the dependence structure between stock indexes returns. By analyzing the time series behavior of the dynamic copula parameters, we find that,¡]1¡^the dependence of stock indexes returns increased significantly between U.S. and other G5 members in early subprime mortgage financial crisis, which means the dependence structure has contagion effect.¡]2¡^Except the dependence structure between U.S. and Japan, the other dependence structure between U.S. and other G5 members in later subprime mortgage financial crisis have the phenomenon of interdependence, and their average tail dependence increased significantly.¡]3¡^By the above, international portfolio constructed by correlation coefficient will failed to diversify the downside risk and the systematic risk will be increased in financial crisis period, which is similar with the 2008 subprime mortgage financial crisis. Therefore, the construction of an international portfolio must consider the asymmetric dependence structure between the stock indexes returns.

Financial market and its various components are currently in turmoil. Many large corporations are devising new ways to overcome the current market instability. Consequently, any study fostering the understanding of financial markets and the dependencies of various market components would greatly benefit both the practitioners and academicians. To understand different parts of the financial market, this dissertation employs time series methods to model causality and structure and degree of dependence. The relationship of housing market prices for nine U.S. census divisions is studied in the first essay. The results show that housing market is very interrelated. The New England and West North Central census divisions strongly lead house prices of the rest of the country. Further evidence suggests that house prices of most census divisions are mainly influenced by house price changes of other regions. The interdependence of oil prices and stock market indices across countries is examined in the second essay. The general dependence structure and degree is estimated using copula functions. The findings show weak dependence between stock market indices and oil prices for most countries except for the large oil producing nations which show high dependence. The dependence structure for most oil consuming (producing) countries is asymmetric implying that stock market index and oil price returns tend to move together more during the market downturn (upturn) than a market boom (downturn). In the third essay, the relationship among stock returns of ten U.S. sectors is studied. Copula models are used to explore the non-linear, general association among the series. The evidence shows that sectors are strongly related to each other. Energy sector is relatively weakly connected with the other sectors. The strongest dependence is between the Industrials and Consumer Discretionary sectors. The high dependence suggests small (if any) gains from industry diversification in U.S. In conclusion, the correct formulation of relationships among variables of interest is crucial. This is one of the fundamental issues in portfolio analysis. Hence, a thorough examination of time series models that are used to understand interactions of financial markets can be helpful for devising more accurate investment strategies.