This dissertation examines a number of issues related to Static and Dynamic Student t Regression Models. The Static Student t Regression Model is derived and transformed to an operational form. The operational form is then examined in a series of Monte Carlo experiments. The model is judged based on its usefulness for estimation and testing and its ability to model the heteroskedastic conditional variance. It is also compared with the traditional Normal Linear Regression Model. Subsequently the analysis is broadened to a dynamic setup. The Student t Autoregressive Model is derived and a number of its operational forms are considered. Three forms are selected for a detailed examination in a series of Monte Carlo experiments. The models’ usefulness for estimation and testing is evaluated, as well as their ability to model the conditional variance. The models are also compared with the traditional Dynamic Linear Regression Model. / Ph. D.

[Abstract]: This thesis investigates the prediction distributions of future response(s), conditional on a set of realized responses for some linear models havingstudent-t error distributions by the Bayesian approach under the uniform priors. The models considered in the thesis are the multiple regression modelwith multivariate-t errors and the multivariate simple as well as multiple re-gression models with matrix-T errors. For the multiple regression model, results reveal that the prediction distribution of a single future response anda set of future responses are a univariate and multivariate Student-t distributions respectively with appropriate location, scale and shape parameters.The shape parameter of these prediction distributions depend on the size of the realized responses vector and the dimension of the regression parameters' vector, but do not depend on the degrees of freedom of the error distribu-tion. In the multivariate case, the distribution of a future responses matrix from the future model, conditional on observed responses matrix from the realized model for both the multivariate simple and multiple regression mod-els is matrix-T distribution with appropriate location matrix, scale factors and shape parameter. The results for both of these models indicate that prediction distributions depend on the realized responses only through the sample regression matrix and the sample residual sum of squares and products matrix. The prediction distribution also depends on the design matricesof the realized as well as future models. The shape parameter of the prediction distribution of the future responses matrix depends on size of the realized sample and the number of regression parameters of the multivariatemodel. Furthermore, the prediction distributions are derived by the Bayesian method as multivariate-t and matrix-T are identical to those obtained under normal errors' distribution by the di®erent statistical methods such as the classical, structural distribution and structural relations of the model approaches. This indicates not only the inference robustness with respect todepartures from normal error to Student-t error distributions, but also indicates that the Bayesian approach with a uniform prior is competitive withother statistical methods in the derivation of prediction distribution.

Banking is a risk and return business; in order to obtain the desired returns, banks are required to take on risks. Following the demise of Lehman Brothers in September 2008, the Basel III Accord proposed considerable increases in capital charges for banks. Whilst this ensures greater economic stability, banks now face an increasing risk of becoming capital inefficient. Furthermore, capital analysts are not only required to estimate capital requirements for individual business lines, but also for the organization as a whole. Copulas are a popular technique to model joint multi-dimensional problems, as they can be applied as a mechanism that models relationships among multivariate distributions. Firstly, a review of the Basel Capital Accord will be provided. Secondly, well known risk measures as proposed under the Basel Accord will be investigated. The penultimate chapter is dedicated to the theory of copulas as well as other measures of dependence. The final chapter presents a practical illustration of how business line losses can be simulated by using the Gaussian, Cauchy, Student t and Clayton copulas in order to determine capital requirements using 95% VaR, 99% VaR, 95% ETL, 99% ETL and StressVaR. The resultant capital estimates will always be a function of the choice of copula, the choice of risk measure and the correlation inputs into the copula calibration algorithm. The choice of copula, the choice of risk measure and the conservativeness of correlation inputs will be determined by the organization’s risk appetite. / Sc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

Banking is a risk and return business; in order to obtain the desired returns, banks are required to take on risks. Following the demise of Lehman Brothers in September 2008, the Basel III Accord proposed considerable increases in capital charges for banks. Whilst this ensures greater economic stability, banks now face an increasing risk of becoming capital inefficient. Furthermore, capital analysts are not only required to estimate capital requirements for individual business lines, but also for the organization as a whole. Copulas are a popular technique to model joint multi-dimensional problems, as they can be applied as a mechanism that models relationships among multivariate distributions. Firstly, a review of the Basel Capital Accord will be provided. Secondly, well known risk measures as proposed under the Basel Accord will be investigated. The penultimate chapter is dedicated to the theory of copulas as well as other measures of dependence. The final chapter presents a practical illustration of how business line losses can be simulated by using the Gaussian, Cauchy, Student t and Clayton copulas in order to determine capital requirements using 95% VaR, 99% VaR, 95% ETL, 99% ETL and StressVaR. The resultant capital estimates will always be a function of the choice of copula, the choice of risk measure and the correlation inputs into the copula calibration algorithm. The choice of copula, the choice of risk measure and the conservativeness of correlation inputs will be determined by the organization’s risk appetite. / Sc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

The purpose with this study is to compare four different models to VaR in terms of accuracy, namely Historical Simulation (HS), Simple Moving Average (SMA), Exponentially Weighted Moving Average (EWMA) and Exponentially Weighted Historical Simulation (EWHS). These VaR models will be applied to one underlying asset which is the Brent Blend Oil using these confidence levels 95 %, 99 % and 99, 9 %. Concerning the return of the asset the models under two different assumptions namely student t-distribution and normal distribution will be studied

In this study, different from previous studies, the explanatory power of Student-t distribution is compared to normal distribution by employing both standard GARCH and EGARCH models to dollar/ lira (USD/TRY) exchange rate. Then the impact of Central Bank of Republic of the Turkey&rsquo / s (CBRT) decisions and actions on both the level of exchange rate and the volatility is investigated. Moreover the relationship between volatility and market liquidity is examined using spot foreign exchange (FX) market volume as a proxy. The results reveal that, in contrast to preceding findings, Student-t could not capture the leptokurtic property better than normal distribution does. Furthermore, an increase in Turkish government benchmark bond rates, CBRT FX purchase interventions and announcement of suspending/ decreasing-the-amount-of FX auctions lead Turkish lira to depreciate. Because of the significant positive leverage effect, the results of GARCH and EGARCH variance equations differ so much. Thereby the results should be evaluated cautiously. In addition it is observed that, only EGARCH model gives significant results when the spot market trading volume is included in the models

在市場上最常被用來評價抵押債務債券(Collateralized Debt Obligation, CDO)的分析方法即為應用大樣本同質性資產組合(Large Homogeneous Portfolio, LHP)假設之單因子關聯結構模型(One Factor Copula Model)。由過去文獻指出，自2008年起，抵押債務債券的商品結構已漸漸出現改變，而目前所延伸之各種單因子關聯結構模型在新型商品的評價結果中皆仍有改善空間。 在本文中使用標準化偏斜Student-t分配(Standardized Skew Student-t distribution, SSTD)取代傳統的高斯分配進行抵押債務債券之分券的評價，此分配擁有控制分配偏態與峰態的參數。但是與Student-t分配相同，SSTD同樣不具備穩定的摺積(convolution)性質，因此在評價過程中會額外消耗部分時間。而在實證分析中，以單因子SSTD關聯結構模型評價擔保債務債券新型商品之分券時得到了較佳的結果，並且比單因子高斯關聯結構模型擁有更多參數以符合實際需求。 / The most widely used method for pricing collateralized debt obligation(CDO) is the one factor copula model with Large Homogeneous Portfolio assumption. Based on the literature of discussing, the structure of CDO had been changed gradually since 2008. The effects for pricing new type CDO tranches in the current extended one factor copula models are still improvable. In this article, we substitute the Gaussian distribution with the Standardized Skew Student-t distribution(SSTD) for pricing CDO tranches, and it has the features of heavy-tail and skewness. However, similar to the Student-t distribution, the SSTD is not stable under convolution as well. For this reason, it takes extra time in the pricing process. The empirical analysis shows that the one factor SSTD copula model has a good effect for pricing new type CDO tranches, and furthermore it brings more flexibility to the one factor Gaussian copula model.

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].

Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are widely-used mathematical models for decomposing multivariate data. They capture spatial relationships between variables, but ignore any temporal relationships that might exist between observations. Probabilistic PCA (PPCA) and Probabilistic CCA (ProbCCA) are versions of these two models that explain the statistical properties of the observed variables as linear mixtures of an alternative, hypothetical set of hidden, or latent, variables and explicitly model noise. Both the noise and the latent variables are assumed to be Gaussian distributed. This thesis introduces two new models, named PPCA-AR and ProbCCA-AR, that augment PPCA and ProbCCA respectively with autoregressive processes over the latent variables to additionally capture temporal relationships between the observations. To make PPCA-AR and ProbCCA-AR robust to outliers and able to model leptokurtic data, the Gaussian assumptions are replaced with infinite scale mixtures of Gaussians, using the Student-t distribution. Bayesian inference calculates posterior probability distributions for each of the parameter variables, from which we obtain a measure of confidence in the inference. It avoids the pitfalls associated with the maximum likelihood method: integrating over all possible values of the parameter variables guards against overfitting. For these new models the integrals required for exact Bayesian inference are intractable; instead a method of approximation, the variational Bayesian approach, is used. This enables the use of automatic relevance determination to estimate the model orders. PPCA-AR and ProbCCA-AR can be viewed as linear dynamical systems, so the forward-backward algorithm, also known as the Baum-Welch algorithm, is used as an efficient method for inferring the posterior distributions of the latent variables. The exact algorithm is tractable because Gaussian assumptions are made regarding the distribution of the latent variables. This thesis introduces a variational Bayesian forward-backward algorithm based on Student-t assumptions. The new models are demonstrated on synthetic datasets and on real remote sensing and EEG data.

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].