The log-linear models have been a significant area of research in the field of categorical data analysis since the 1950s. However, until the mid-1970s, log-linear models only considered the modelling of nominal variables and did not make any assumption about the ordering of categories of an ordinal variable. Therefore, the log-linear models have been modified to incorporate the structure of any ordinal variable. This issue is especially relevant in most fields of social science. The ordinal log-linear models are amid the most widely used and powerful techniques to model association among the ordinal variables in categorical data analysis. Traditionally, the parameters from such models are estimated using iterative algorithms (such as the Newton-Raphson method, and iterative proportional fitting), but issues such as choice of poor initial values and contingency tables of larger dimensions can reduce the convergence rate as well as highly increase the number of iterations required for the algorithms to converge.

More recent advances have suggested a method of non-iterative estimation that gives numerically similar estimates as that of the iterative methods for the estimation of linear- by-linear association parameter in an ordinal log-linear model for a two-way table. This presentation will highlight the iterative and non-iterative techniques commonly used to estimate the linear-by-linear association parameter from two-dimensional ordinal log- linear models. It will provide an overview of how the growing number of non-iterative estimation techniques fit into the problem. Several possibilities to extend the research on the non-iterative estimates in order to validate their further use are discussed. The presentation will also highlight the research undertaken so far to achieve this objective. This includes considering the two fundamental estimates for the analysis of the association between two categorical variables forming a contingency table and to determine their asymptotic characteristics. A computational study is carried out for contingency tables of varying sizes to show that these two estimates are asymptotically unbiased. It is also shown that both estimates are asymptotically normally distributed. On the basis of the standard errors, their relative efficiency has been established for 13 commonly analysed contingency tables that appear throughout the literature.

Keywords: Ordinal log-linear models, non-iterative estimation, linear-by-linear association parameter, orthogonal polynomials, Newton Raphson, Iterative proportional fitting.

The main aim of this thesis is to develop an automatic MCMC estimation procedure for STAR-GARCH models that can be applied to any data set that could normally be modeled by any of the sub-classes of models that the STAR-GARCH model generalises. This will remove the need for linearity testing and model specification. The project will achieve three specific objectives:
(i) to formulate a fully Bayesian analysis of univariate STAR models, followed by univariate and multivariate STAR-GARCH models respectively;
(ii) to perform posterior analysis obtained in (i) by creating an efficient MCMC algorithm and using simulation studies, implemented by novel computer code in R;
(iii) to apply the MCMC estimation algorithm to real world data to examine regime-switching and varying volatility in the data.