The estimation of univariate and multiple regression models with stochastic trend components has been considered in the time domain and in the frequency domain. Such models assume as regressors weakly exogenous variables. However if the regression equations are part of a simultaneous equation system some of the regressors will no longer be weakly exogenous and estimators obtained by ignoring this fact will be inconsistent. One way of proceeding in such situations is to estimate the whole system, that is, to construct full information maximum (FIML) estimators. Alternatively, single equation estimators such as limited information maximum likelihood (LIML) can be constructed, as well as estimators based on the instrumental variable (IV) principle which possess the merit of consistency. As in the analogous situation in classical simultaneous equation systems, within this class of limited information estimators, LIML is asymptotically efficient. Hence it is appropriate to study the asymptotic properties of LIML and review the possibility of alternative consistent estimators, using LIML as a benchmark. The purpose of the thesis is thus: to examine the issues of identifiability when stochastic trends are present in simultaneous equation systems; -to examine the computational issues associated with FIML, LIML and various IV estimators in simultaneous equation systems with stochastic trends and derive the asymptotic properties in the frequency domain of these estimators; to compare the performance of IV and LIML via Monte Carlo experiments; to apply the methods to real data.

The computation of polynomial greatest common divisors (GCDs) is a fundamental problem in algebraic computing and has important widespread applications in areas such as computing theory, control, image processing, signal processing and computer-aided design (CAD).

This thesis studies two types of polynomials with random coeffcients, random algebraic polynomials and random trigonometric polynomials are the most studied cases in here. Six different problems have been discussed by obtaining the expected number of level crossings with the employment of the Kac-Rice for- mula, which is based on the knowledge of stochastic processes. The study is concentrated on the normal distribution with different assumptions set for the coefficients of the polynomials.

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Random Matrix Theory (RMT) is the study of matrices with random variables determining the entries, and various additional symmetry conditions imposed on the matrices.

Critical phenomena relate to the behaviour of systems as a phase transition is approached. They are believed to be universal, in that the behaviour of a large class of different systems fits the same description. The aim of this thesis is to establish universality of critical phenomena for both one-dimensional quantum and two-dimensional classical systems with the use of random matrix theory. The central focus of this work is a general class of quantum spin chains which are quadratic in Fermi operators and have been exactly solved under certain symmetry constraints. We compute the critical properties of this general class of systems, and obtain expressions for various correlators as well as the dynamic and correlation length critical exponent, which we calculate explicitly for specific parameter restrictions. This provides us with a demonstration of how symmetries of the system dictate the critical behaviour. We then exploit mappings between quantum and classical systems enabling us to transport these results into the classical regime by obtaining a class of two-dimensional classical systems whose critical properties are related to those of the quantum system. In particular we make use of two types of equivalence; one type is established by commuting the quantum Hamiltonian with the transfer matrix of a classical system, and the other using the Trotter-Suzuki mapping.

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Contributions to the design and analysis of statistical experiments are grouped into five chapters. The minimisation of mean squared error over a design region has been given as a criterion for designing an experiment by Box and Draper. Their fundamental theorem is extended to the case of the general linear hypothesis. Some numerical results are given for polynomial regression designs. The bordering method of inverting matrices is used to solve the usual least squares linear algebra equations. It is found that this method will in theory provide accurate orthogonal functions and tests for linear dependence and the rank of the matrix. Rounding errors are measured in a numerical example solved on a computer. Chapter three is an investigation of the usefulness of order statistics obtained from the analysis of variance, in the case where we test "treatment" sums of squares all having the same number of degrees of freedom. Considerable use is made of the idea that the ordered data should be plotted on the relevant probability paper as a pictorial aid. This assists the investigator to avoid the pitfalls of carrying out standard tests such as F tests under conditions where it is disadvantageous to do so. Even in unreplicated experiments one may see immediately that the usual hypothesis of normality is wrong, or that the assumption of additivity may not hold. In chapter four a solution is given to the problem of fitting a curve subject to the constraint that the slope should be non-negative throughout the region of interest, This is treated as a problem in quadratic programming with a single linear parametric constraint. Chapter five contains a brief investigation into the problems of fitting a model which is non-linear overall but which consists of separate linear models joined at location parameters which have to be estimated.