Modelling and forecasting stock market volatility is a critical issue in various fields of finance and economics. Forecasting volatility in stock markets find extensive use in portfolio management, risk management and option pricing. The primary objective of this study was to describe the volatility in the Johannesburg Stock Exchange (JSE) index using univariate and multivariate GARCH models. We used daily log-returns of the JSE index over the period 6 June 1995 to 30 June 2012. In the univariate GARCH modelling, both asymmetric and symmetric GARCH models were employed. We investigated volatility in the market using the simple GARCH, GJR-GARCH, EGARCH and APARCH models assuming di erent distributional assumptions in the error terms. The study indicated that the volatility in the residuals and the leverage effect was present in the JSE index returns. Secondly, we explored the dynamics of the correlation between the JSE index, FTSE-100 and NASDAQ-100 index on the basis of weekly returns over the period 6 June 1995 to 30 June 2012. The DCC-GARCH (1,1) model was employed to study the correlation dynamics. These results suggested that the correlation between the JSE index and the other two indices varied over time. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

The purpose of this study is to investigate and understand data which are grouped into categories. Various statistical methods was studied for categorical binary responses to investigate the causes of death from diarrhoea in South Africa. Data collected included death type, sex, marital status, province of birth, province of death, place of death, province of residence, education status, smoking status and pregnancy status. The objective of this thesis is to investigate which of the above explanatory variables was most affected by diarrhoea in South Africa. To achieve this objective, different sample survey data analysis techniques are investigated. This includes sketching bar graphs and using several statistical methods namely, logistic regression, surveylogistic, generalised linear model, generalised linear mixed model, and generalised additive model. In the selection of the fixed effects, a bar graph is applied to the response variable individual profile graphs. A logistic regression model is used to identify which of the explanatory variables are more affected by diarrhoea. Statistical applications are conducted in SAS (Statistical Analysis Software). Hosmer and Lemeshow (2000) propose a statistic that they show, through simulation, is distributed as chi‐square when there is no replication in any of the subpopulations. Due to the similarity of the Hosmer and Lemeshow test for logistic regression, Parzen and Lipsitz (1999) suggest using 10 risk score groups. Nevertheless, based on simulation results, May and Hosmer (2004) show that, for all samples or samples with a large percentage of censored observations, the test rejects the null hypothesis too often. They suggest that the number of groups be chosen such that G=integer of {maximum of 12 and minimum of 10}. Lemeshow et al. (2004) state that the observations are firstly sorted in increasing order of their estimated event probability. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

This research is based on a detailed description of model building for multivariate time series models. Under the assumption of stationarity, identification, estimation of the parameters and diagnostic checking for the Vector Auto regressive (p) (VAR(p)), Vector Moving Average (q) (VMA(q)) and Vector Auto regressive Moving Average (VARMA(p, q) ) models are described in detail. With reference to the non-stationary case, the concept of cointegration is explained. Procedures for testing for cointegration, determining the cointegrating rank and estimation of the cointegrated model in the VAR(p) and VARMA(p, q) cases are discussed. The utility of multivariate time series models in the field of economics is discussed and its use is demonstrated by analysing quarterly South African inflation and wage data from April 1996 to December 2008. A review of the literature shows that multivariate time series analysis allows the researcher to: (i) understand phenomenon which occur regularly over a period of time (ii) determine interdependencies between series (iii) establish causal relationships between series and (iv) forecast future variables in a time series based on current and past values of that variable. South African wage and inflation data was analysed using SAS version 9.2. Stationary VAR and VARMA models were run. The model with the best fit was the VAR model as the forecasts were reliable, and the small values of the Portmanteau statistic indicated that the model had a good fit. The VARMA models by contrast, had large values of the Portmanteau statistic as well as unreliable forecasts and thus were found not to fit the data well. There is therefore good evidence to suggest that wage increases occur independently of inflation, and while inflation can be predicted from its past values, it is dependent on wages. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

Lesotho, like any other country of the world, is faced with the task of improving the / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2007.

The agriculture sector in Amhara National Regional State is characterised by producing cereal crops which occupy the largest percentage (84.3%) of the total crop area cultivated in the region. As a result, it is imperative to investigate which factors influence the yields of cereal crops particularly in relation to the five major types of cereals in the study region namely barley, maize, sorghum, teff and wheat. Therefore, in this thesis, using data collected by the Central Statistical Agency of Ethiopia, various statistical methods such as multiple regression analysis were applied to investigate the factors which influence the mean yields of the major cereal crops. Moreover, a mixed model analysis was implemented to assess the effects associated with the sampling units (enumeration areas), and a cluster analysis to classify the region into similar groups of zones. The multiple regression results indicate that all the studied cereals mean yields are affected by zone, fertilizer type and crop damage effects. In addition to this, barley is affected by extension programme; maize crop by seed type, irrigation, and protection of soil erosion; sorghum and teff crops are additionally affected by crop prevention method, extension programme, protection of soil erosion, and gender of the household head; and wheat crop by crop prevention methods, extension programme and gender of the household head. The results from the mixed model analysis were entirely different from the regression results due to the observed dependencies of the cereals mean yields on the sampling unit. Based on the hierarchical cluster analysis, five groups of classes (clusters) were identified which seem to be in agreement with the geographical neighbouring positions of the locations and the similarity of the type of crops produced. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

The current work deals with modelling longitudinal or repeated non-Gaussian measurements for a respiratory disease. The analysis of longitudinal data for non-Gaussian binary disease outcome data can broadly be modeled using three different approaches; the marginal, random effects and transition models. The marginal type model is used if one is interested in estimating population averaged effects such as whether a treatment works or not on an average individual. On the other hand random effects models are important if apart from measuring population averaged effects a researcher is also interested in subject specific effects. In this case to get marginal effects from the subject-specific model we integrate out the random effects. Transition models are also called conditional models as a general term. Thus all the three types of models are important in understanding the effects of covariates and disease progression and distribution of outcomes in a population. In the current work the three models have been researched on and fitted to data. The random effects or subject-specific model is further modified to relax the assumption that the random effects should be strictly normal. This leads to the so called hierarchical generalized linear model (HGLM) based on the h-likelihood formulation suggested by Lee and Nelder (1996). The marginal model was fitted using generalized estimating equations (GEE) using PROC GENMOD in SAS. The random effects model was fitted using PROC GLIMMIX and PROC NLMIXED in SAS (generalized linear mixed model). The latter approach was found to be more flexible except for the need of specifying initial parameter values. The transition model was used to capture the dependence between outcomes in particular the dependence of the current response or outcome on the previous response and fitted using PROC GENMOD. The HGLM was fitted using the GENSTAT software. Longitudinal disease outcome data can provide real and reliable data to model disease progression in the sense that it can be used to estimate important disease i parameters such as prevalence, incidence and others such as the force of infection. Problem associated with longitudinal data include loss of information due to loss to follow up such as dropout and missing data in general. In some cases cross-sectional data can be used to find the required estimates but longitudinal data is more efficient but may require more time, effort and cost to collect. However the successful estimation of a given parameter or function depends on the availability of the relevant data for it. It is sometimes impossible to estimate a parameter of interest if the data cannot its estimation. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

The objective of this dissertation is to model the volatility of financial time series data using ARCH, GARCH and stochastic volatility models. It is found that the ARCH and GARCH models are easy to fit compared to the stochastic volatility models which present problems with respect to the distributional assumptions that need to be made. For this reason the ARCH and GARCH models remain more widely used than the stochastic volatility models. The ARCH, GARCH and stochastic volatility models are fitted to four data sets consisting of daily closing prices of gold mining companies listed on the Johannesburg stock exchange. The companies are Anglo Gold Ashanti Ltd, DRD Gold Ltd, Gold Fields Ltd and Harmony Gold Mining Company Ltd. The best fitting ARCH and GARCH models are identified along with the best error distribution and then diagnostics are performed to ensure adequacy of the models. It was found throughout that the student-t distribution was the best error distribution to use for each data set. The results from the stochastic volatility models were in agreement with those obtained from the ARCH and GARCH models. The stochastic volatility models are, however, restricted to the form of an AR(1) process due to the complexities involved in fitting higher order models. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

Malawi’s economy relies heavily on agriculture which is threatened by declines in soil fertility. Measures to ensure increased crop productivity at household level include the increased use of inorganic fertilizers. To supplement the Government’s effort in ensuring food security, Rural Livelihood Diversification Project (RLDP) was implemented in Kasungu and Lilongwe Districts in Malawi. The RLDP Project was aimed at increasing accessibility and utilisation of inorganic fertilizers. We used the data collected by the International Center for Tropical Agriculture (CIAT), to investigate if there could be any significant impacts of the interventions carried out by the project. A general linear model was initially used to model the data. Terms in the model were selected using the automatic stepwise procedure in GLMSELECT procedure of SAS. Other models that were used included a transformed response general linear model, gamma model based on log link and its alternative inverse link, and quantile regression procedures were used in modelling the amount of fertilizer use per acre response given a set of fixed effect predictors where households were only sampled at baseline or impact assessment study. The general linear model failed to comply with the model assumption of normality and constant variance. The gamma model was affected by influential observations. Quantile regression model is robust to outliers and influential observations. Quantile regression provided that number of plots cultivated, timeline, household saving and irrigation interaction, and the interaction between plots and timeline significantly affected the amounts of fertilizers applied per acre amongst the 25% of the households who apply lower levels of fertilizer per acre. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Just over half (50:6%) of the population of South Sudan lives on less than one US Dollar a day. Three quarters of the population live below the poverty line (NBS, Poverty Report, 2010). Generally, effective government policy to reduce unemployment and eradicate poverty focuses on stimulating new businesses. Micro and small enterprises (MSEs) are the major source of employment and income for many in under-developed countries. The objective of this study is to identify factors that determine business success and failure in South Sudan. To achieve this objective, generalized linear models, survey logistic models, the generalized linear mixed models and multiple correspondence analysis are used. The data used in this study is generated from the business survey conducted in 2010. The response variable, which is defined as business success or failure was measured by profit and loss in businesses. Fourteen explanatory variables were identified as factors contributing to business success and failure. A main effect model consisting of the fourteen explanatory variables and three interaction effects were fitted to the data. In order to account for the complexity of the survey design, survey logistic and generalized linear mixed models are refitted to the same variables in the main effect model. To confirm the results from the model we used multiple correspondence analysis. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

Longitudinal data tend to be correlated and hence posing a challenge in the analysis since the correlation has to be accounted for to obtain valid inference. We study various statistical methods for such correlated longitudinal binary responses. These models can be grouped into five model families, namely, marginal, subject-specific, transition, joint and semi-parametric models. Each one of the models has its own strengths and weaknesses. Application of these models is carried out by analyzing data on patient’s adherence status to highly active antiretroviral therapy (HAART). One other complicating issue with the HAART adherence data is missingness. Although some of the models are flexible in handling missing data, they make certain assumptions about missing data mechanisms, the most restrictive being missing completely at random (MCAR). The test for MCAR revealed that dropout did not depend on the previous outcome. A logistic regression model was used to identify predictors for the patients’ first month’s adherence status. A marginal model was then fitted using generalized estimating equations (GEE) to identify predictors of long-term HAART adherence. This provided marginal population-based estimates, which are important for public health perspective. We further explored the subject’s specific effects that are unique to a particular individual by fitting a generalized linear mixed model (GLMM). The GLMM was also used to assess the association structure of the data. To assess whether the current optimal adherence status of a patient depended on the previous adherence measurements (history) in addition to the explanatory variables, a transition model was fitted. Moreover, a joint modeling approach was used to investigate the joint effect of the predictor variables on both HAART adherence status of patients and duration between successive visits. Assessing the association between the two outcomes was also of interest. Furthermore, longitudinal trajectories of observed data may be very complex especially when dealing with practical applications and as such, parametric statistical models may not be flexible enough to capture the main features of the longitudinal profiles, and so a semiparametric approach was adopted. Specifically, generalized additive mixed models were used to model the effect of time as well as interactions associated with time non-parametrically. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.