This thesis considers two important problems in finance, namely, correlation stress testing and multivariate volatility modeling. Correlation stress testing refers to the correlation matrix adjustment to evaluate potential impact of the changes in correlations under financial crises. Very often, some correlations are explicitly adjusted (core correlations), with the remainder left unspecified (peripheral correlations), although it would be more natural for both core correlations and peripheral correlations to vary. However, most existing methods ignored the potential change in peripheral correlations. Inspiring from this idea, two methods are proposed in which the stress impact on the core correlations is transmitted to the peripheral correlations through the dependence structure of the empirical correlations. The first method is based on a Bayesian framework in which a prior for a population correlation matrix is proposed that gives flexibility in specifying the dependence structure of correlations. In order to increase the rate of convergence, the algorithm of posterior simulation is extended so that two correlations can be updated in one Gibbs sampler step. To achieve this, an algorithm is developed to find the region of two correlations keeping the correlation matrix positive definite given that all other correlations are held fixed. The second method is a Black-Litterman approach applied to correlation matrices. A new correlation matrix is constructed by maximizing the posterior density. The proposed method can be viewed as a two-step procedure: first constructing a target matrix in a data-driven manner, and then regularizing the target matrix by minimizing a matrix norm that reasonably reflects the dependence structure of the empirical correlations. Multivariate volatility modeling is important in finance since variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high frequency returns as the ex-post realization of the covariance matrix of low frequency returns. For the analysis of dynamics of RCOV matrices, the generalized conditional autoregressive Wishart model is proposed. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The proposed model is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments and stationarity are discussed. The forecasting performance of the proposed model is compared with the existing models. Outliers exist in the series of realized volatility which is often decomposed into continuous and jump components. The vector multiplicative error model is a natural choice to jointly model these two non-negative components of the realized volatility, which is also a popular multivariate time series model for other non-negative volatility measures. Diagnostic checking of such models is considered by deriving the asymptotic distribution of residual autocorrelations. A multivariate portmanteau test is then devised. Simulation experiments are carried out to investigate the performance of the asymptotic result in finite samples. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / Applied Statistics / Master / Master of Social Sciences

published_or_final_version / Applied Statistics / Master / Master of Social Sciences

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This work involves numerical integration as well as mathematical integration techniques in computing the bivariate and trivariate normal probabilities. The probabilities have been computed and tabulated for desired ranges of the non-negative values of the random variables, and for some specific values of correlation coefficients, taking the size of the tables into consideration. Formulas, however, also been given to derive the probabilities for negative values of the random variables.Computer programs have also been developed and included to compute the probabilities so as to produce the above mentioned probability tables. These computer programs were run on DECSYSTEM-10.

Bibliography: leaves 220-224. / 224 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Statistics, 1997?

This thesis considers location and scale parameter modelling of the heteroscedastic t-distribution. This new distribution is an extension of the heteroscedastic Gaussian and provides robust analysis in the presence of outliers as well accommodates possible heteroscedasticity by flexibly modelling the scale parameter using covariates existing in the data. To motivate components of work in this thesis the Gaussian linear mixed model is reviewed. The mixed model equations are derived for the location fixed and random effects and this model is then used to introduce Restricted Maximum Likelihood ( REML ). From this an algorithmic scheme to estimate the scale parameters is developed. A review of location and scale parameter modelling of the heteroscedastic Gaussian distribution is presented. In this thesis, the scale parameters are a restricted to be a function of covariates existing in the data. Maximum Likelihood ( ML ) and REML estimation of the location and scale parameters is derived as well as an efficient computational algorithm and software are presented. The Gaussian model is then extended by considering the heteroscedastic t distribution. Initially, the heteroscedastic t is restricted to known degrees of freedom. Scoring equations for the location and scale parameters are derived and their intimate connection to the prediction of the random scale effects is discussed. Tools for detecting and testing heteroscedasticity are also derived and a computational algorithm is presented. A mini software package " hett " using this algorithm is also discussed. To derive a REML equivalent for the heteroscedastic t asymptotic likelihood theory is discussed. In this thesis an integral approximation, the Laplace approximation, is presented and two examples, with the inclusion of ML for the heteroscedastic t, are discussed. A new approximate integral technique called Partial Laplace is also discussed and is exemplified with linear mixed models. Approximate marginal likelihood techniques using Modified Profile Likelihood ( MPL ), Conditional Profile Likelihood ( CPL ) and Stably Adjusted Profile Likelihood ( SAPL ) are also presented and offer an alternative to the approximate integration techniques. The asymptotic techniques are then applied to the heteroscedastic t when the degrees of freedom is known to form two distinct REMLs for the scale parameters. The first approximation uses the Partial Laplace approximation to form a REML for the scale parameters, whereas, the second uses the approximate marginal likelihood technique MPL. For each, the estimation of the location and scale parameters is discussed and computational algorithms are presented. For comparison, the heteroscedastic t for known degrees of freedom using ML and the two new REML equivalents are illustrated with an example and a comparative simulation study. The model is then extended to incorporate the estimation of the degrees of freedom parameter. The estimating equations for the location and scale parameters under ML are preserved and the estimation of the degrees of freedom parameter is integrated into the algorithm. The approximate REML techniques are also extended. For the Partial Laplace approximation the estimation of the degrees of freedom parameter is simultaneously estimated with the scale parameters and therefore the algorithm differs only slightly. The second approximation uses SAPL to estimate the parameters and produces approximate marginal likelihoods for the location, scale and degrees of freedom parameters. Computational algorithms for each of the techniques are also presented. Several extensive examples, as well as a comparative simulation study, are used to illustrate ML and the two REML equivalents for the heteroscedastic t with unknown degrees of freedom. The thesis is concluded with a discussion of the new techniques derived for the heteroscedastic t distribution along with their advantages and disadvantages. Topics of further research are also discussed. / Thesis (Ph.D.)--School of Agriculture and Wine, 2005.

Thesis (Ph.D. ) -- McMaster University, 1998. / Includes bibliographical references (leaves 75-81). Also available via World Wide Web.