Assignment (MComm)--Stellenbosch University, 2005. / ENGLISH ABSTRACT: Volatilty was originally seen to be constant and deterministic, but it was later realised that return series are non-stationary. Owing to this non-stationarity nature of returns, there were no reliable ex-post volatility measurements. Subsequently, researchers focussed on ex-ante volatility models. It was only then realised that before good volatility models can be created, reliable ex-post volatility measuremetns need to be defined. In this study we examine non-parametric ex-post volatility measurements in order to obtain approximations of the variances of non-stationary return series. A detailed mathematical derivation and discussion of the already developed volatility measurements, in particular the realised volatility- and DST measurements, are given In theory, the higher the sample frequency of returns is, the more accurate the measurements are. These volatility measurements referred to above, however, all have short-comings in that the realised volatility fails if the sample frequency becomes to high owing to microstructure effects. On the other hand, the DST measurement cannot handle changing instantaneous volatility. In this study we introduce a new volatility measurement, termed microstructure realised volatility, that overcomes these shortcomings. This measurement, as with realised volatility, is based on quadratic variation theory, but the underlying return model is more realistic. / AFRIKAANSE OPSOMMING: Volatiliteit is oorspronklik as konstant en deterministies beskou, dit was eers later dat besef is dat opbrengste nie-stasionêr is. Betroubare volatiliteits metings was nie beskikbaar nie weens die nie-stasionêre aard van opbrengste. Daarom het navorsers gefokus op vooruitskattingvolatiliteits modelle. Dit was eers op hierdie stadium dat navorsers besef het dat die definieering van betroubare volatiliteit metings 'n voorvereiste is vir die skepping van goeie vooruitskattings modelle. Nie-parametriese volatiliteit metings word in hierdie studie ondersoek om sodoende benaderings van die variansies van die nie-stasionêre opbrengste reeks te beraam. 'n Gedetaileerde wiskundige afleiding en bespreking van bestaande volatiliteits metings, spesifiek gerealiseerde volatiliteit en DST- metings, word gegee. In teorie salopbrengste wat meer dikwels waargeneem word tot beter akkuraatheid lei. Bogenoemde volatilitieits metings het egter tekortkominge aangesien gerealiseerde volatiliteit faal wanneer dit te hoog raak, weens mikrostruktuur effekte. Aan die ander kant kan die DST meting nie veranderlike oombliklike volatilitiet hanteer nie. Ons stel in hierdie studie 'n nuwe volatilitieits meting bekend, naamlik mikro-struktuur gerealiseerde volatiliteit, wat nie hierdie tekortkominge het nie. Net soos met gerealiseerde volatiliteit sal hierdie meting gebaseer wees op kwadratiese variasie teorie, maar die onderliggende opbrengste model is meer realisties.

Thesis (MBA)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: Forecasting the future has always been one of the man's strongest desires. The aim to determine the future has resulted in scientifically based forecasting models of human health, behaviour, economics, weather, etc. The main purpose of forecasting is to reduce the range of uncertainty within which management decisions must be made. Forecasts are only effective if they are utilized by those who have decisionmaking authority. Forecasts need to be understood and appreciated by decision makers so that they find their way into management of the firm. Companies still predominantly rely on judgemental forecasting methods, most often on an informal basis. There is a large literature base that point to the numerous biases inherent in judgemental forecasting. Most companies know that their forecasts are incorrect but don't know what to do about it and choose to ignore the issue, hoping that the problem will solve itself. The collaborative forecasting process attempts to use history as a baseline, but supplement current knowledge about specific trends, events and other items. This approach integrates the knowledge and information that exists internally and externally into a single, more accurate forecast that supports the entire supply chain. Demand forecasting is not just a matter of duplicating or predicting history into the future. It is important that one person should lead and manage the process. Accountability needs to be established. An audit on the writer's own organization indicated that no formal forecasting process was present. The company's forecasting process was very political, since values were entered just to add up to the required targets. The real gap was never fully understood. Little knowledge existed regarding statistical analysis and forecasting within the marketing department who is accountable for the forecast. The forecasting method was therefore a top-down approach and never really checked with a bottom up approach. It was decided to learn more about the new demand planning process prescribed by the head office, and to start implementing the approach. The approach is a form of a collaborative approach which aims to involve all stakeholders when generating the forecast, therefore applying a bottom up approach. Statistical forecasting was applied to see how accurate the output was versus that of the old way of forecasting. The statistical forecast approach performed better with product groups where little changed from previous years existed, while the old way performed better where new activities were planned or known by the marketing team. This indicates that statistical forecasting is very important for creating the starting point or baseline forecast, but requires qualitative input from all stakeholders. Statistical forecasting is therefore not the solution to improved forecasting, but rather part of the solution to create robust forecasts. / AFRIKAANSE OPSOMMING: Vooruitskatting van die toekoms was nog altyd een van die mens se grootste begeertes. Die doel om die toekoms te bepaal het gelei tot wiskundige gebaseerde modelle van die mens se gesondheid, gedrag, ekonomie, weer, ens. The hoofdoel van vooruitskatting is om die reeks van risikos te verminder waarbinne bestuur besluite moet neem. Vooruitskattings is slegs effektief as dit gebruik word deur hulle wat besluitnemingsmag het. Vooruitskattings moet verstaan en gewaardeer word deur die besluitnemers sodat dit die weg kan vind na die bestuur van die firma. Maatskappye vertrou nog steeds hoofsaaklik op eie oordeel vooruitskatting metodes, en meestal op 'n informele basis. Daar is 'n uitgebreide literatuurbasis wat daarop dui dat heelwat sydigheid betrokke is by vooruitskattings wat gebaseer is op eie oordeel. Baie organisasies weet dat hulle vooruitskattings verkeerd is, maar weet nie wat daaromtrent te doen nie en kies om die probleem te ignoreer, met die hoop dat die probleem vanself sal oplos. Die geïntegreerde vooruitskattingsproses probeer om die verlede te gebruik as 'n basis, maar voeg huidige kennis rakende spesifieke neigings, gebeurtenisse, en ander items saam. Hierdie benadering integreer die kennis en informasie wat intern en ekstern bestaan in 'n enkele, meer akkurate vooruitskatting wat die hele verskaffingsketting ondersteun. Vraagvooruitskatting is nie alleen 'n duplisering of vooruitskatting van die verlede in die toekoms in nie. Dit is belangrik dat een persoon die proses moet lei en bestuur. Verantwoordelikhede moet vasgestel word. 'n Oudit op die skrywer se organisasie het getoon dat geen formele vooruitskattingsprosesse bestaan het nie. Die maatskappy se vooruitskattingsproses was hoogs gepolitiseerd, want getalle was vasgestel wat in lyn was met die nodige teikens. Die ware gaping was nooit werklik begryp nie. Min kennis was aanwesig rakende statistiese analises en vooruitskatting binne die bemarkingsdepartement wat verantwoordelik is vir die vooruitskatting. Die vooruitskatting is dus eerder gedoen op 'n globale vlak en nie noodwendig getoets deur die vooruitskatting op te bou uit detail nie. Daar is besluit om meer te leer rakende die nuwe vraagbeplanningsproses, wat voorgeskryf is deur hoofkantoor, en om die metode te begin implementeer. Die metode is 'n vorm van 'n geïntegreerde model wat beoog om alle aandeelhouers te betrek wanneer die vooruitskatting gedoen word, dus die vooruitskatting opbou met detail. Statistiese vooruitskatting was toegepas om te sien hoe akkuraat die uitset was teenoor die ou manier van vooruitskatting. Die statistiese proses het beter gevaar waar die produkgroepe min verandering van vorige jare ervaar het, terwyl die ou manier beter gevaar het waar bemarking self die nuwe aktiwiteite beplan het of bewus was daarvan. Dit bewys dat statistiese vooruitskatting baie belangrik is om die basis vooruitskatting te skep, maar dit benodig kwalitatiewe insette van all aandeelhouers. Statistiese vooruitskattings is dus nie die oplossing vir beter vooruitskattings nie, maar deel van die oplossing om kragtige vooruitskattings te skep.

South Africa is rich in platinum group metals (PGMs) and these metals are important in providing jobs as well as investments some of which have been seen in the Johannesburg Securities Exchange (JSE). In this country this sector has experienced some setbacks in recent times. The most notable ones are the 2008/2009 global nancial crisis and the 2012 major nationwide labour unrest. Worrisomely, these setbacks keep simmering. These events usually introduce jumps and breaks in data which changes the structure of the underlying information thereby inducing spurious long memory (long range dependence). Thus it is recommended that these two phenomena must be addressed together. Further, it is well-known that nancial returns are dominated by stylized facts. In this thesis we carried out an investigation on distributional properties of platinum returns, structural changes, long memory and stylized facts in platinum returns and volatility series. To understand the distributional properties of the returns, we used two classes of heavy tailed distributions namely the alpha-Stable distributions and generalized hyperbolic distributions. We then investigated structural changes in the platinum return series and changes in long range dependence and volatility. Using Akaike information criterion, the ARFIMA-FIAPARCH under the Student distribution was selected as the best model for platinum although the ARCH e ects were slightly signi cant, while using the Schwarz information criteria the ARFIMA-FIAPARCH under the Normal distribution. Further, ARFIMA-FIEGARCH under the skewed Student distribution and ARFIMA-HYGARCH under the Normal distribution models were able to capture the ARCH effects. The best models with respect to prediction excluded the ARFIMA-FIGARCH model and were dominated by ARFIMA-FIAPARCH model with non-Normal error distributions which indicates the importance of asymmetry and heavy tailed error distributions. / Statistics / M. Sc. (Statistics)

A diameter distribution prediction procedure for single species stands was developed based on the bivariate S_B distribution model. The approach not only accounted for and described the relationships between initial and future diameters and their distributions, but also assumed future diameter given initial diameter to be a random variable. While this method was the most theoretically correct, comparable procedures based on the definition of growth equations which assumed future diameter given initial diameter to be a constant, sometimes provided somewhat better results. Both approaches performed as well, and in some cases, better than the established methods of diameter distribution prediction such as parameter recovery, percentile prediction, and parameter prediction. The approaches based on the growth equations are intuitively and biologically appealing in that the future distribution is determined from an initial distribution and a specified initial-future diameter relationship. ln most appropriate. While this result simplified some procedures, it also implied that the initial and future diameter distributions differed only in location and scale, not in shape. This is a somewhat unrealistic assumption, however, due to the relatively short growth periods and the alterations in stand structure and growth due to the repeated thinnings, the data did not provide evidence against the linear growth equation assumption. The growth equation procedures not only required the initial and future diameter distributions to be of a particular form, but they also restricted the initial-future diameter relationship to be of a particular form. The individual tree model, which required no distributional assumptions or restrictions on the growth equation, proved to be the better approach to use in terms of predicting future stand tables as it performed better than all of the distribution-based approaches. For the bivariate distribution, the direct fit, parameter recovery, parameter prediction and percentile prediction diameter distribution prediction techniques, implied diameter relationships were defined. Evaluations revealed that these equations were both accurate and precise, indicating that the accurate specification of the initial distribution and the diameter diameter distribution. / Ph. D.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2017. / With an increase in opportunistically-collected data, statistical methods that can accommodate unstructured designs are increasingly useful. Spatial capturerecapture (SCR) has such potential, but its applicability for species that are strongly gregarious is uncertain. It assumes that average animal locations are spatially random and independent, which is violated for gregarious species. I used a data set for African elephants (Loxodonta africana) and data simulation to assess bias and precision of SCR population density estimates given violations in location independence. I found that estimates were negatively biased and likely too precise if non-independence was ignored. Encounter heterogeneity models produced more realistic precision but density estimates were positively biased. Lowest bias was achieved by estimating density of groups, group size, and then multiplying to estimate overall population density. Such findings have important implications for the reliability of population density estimates where data are collected by unstructured means. / LG2017

In this study a two-part mixed probability density function was derived which described the relative changes in the Standard and Poor's 100 Stock Index over various intervals of time. The density function is a mixture of two different halves of normal distributions. Optimal values for the standard deviations for the two halves and the mean are given. Also, a general form of the function is given which uses linear regression models to estimate the standard deviations and the means. The density functions allow stock market participants trading index options and futures contracts on the S & P 100 Stock Index to determine probabilities of success or failure of trades involving price movements of certain magnitudes in given lengths of time.

This study uses aspects of time series methodology to model and forecast major taxes such as Personal Income Tax (PIT), Corporate Income Tax (CIT), Value Added Tax (VAT) and Total Tax Revenue(TTAXR) in the South African Revenue Service (SARS). The monthly data used for modeling tax revenues of the major taxes was drawn from January 1995 to March 2010 (in sample data) for PIT, VAT and TTAXR. Due to higher volatility and emerging negative values, the CIT monthly data was converted to quarterly data from the rst quarter of 1995 to the rst quarter of 2010. The competing ARIMA/SARIMA and Holt-Winters models were derived, and the resulting model of this study was used to forecast PIT, CIT, VAT and TTAXR for SARS fiscal years 2010/11, 2011/12 and 2012/13. The results show that both the SARIMA and Holt-Winters models perform well in modeling and forecasting PIT and VAT, however the Holt-Winters model outperformed the SARIMA model in modeling and forecasting the more volatile CIT and TTAXR. It is recommended that these methods are used in forecasting future payments, as they are precise about forecasting tax revenues, with minimal errors and fewer model revisions being necessary. / Statistics / M.Sc. (Statistics)

This study uses aspects of time series methodology to model and forecast major taxes such as Personal Income Tax (PIT), Corporate Income Tax (CIT), Value Added Tax (VAT) and Total Tax Revenue(TTAXR) in the South African Revenue Service (SARS). The monthly data used for modeling tax revenues of the major taxes was drawn from January 1995 to March 2010 (in sample data) for PIT, VAT and TTAXR. Due to higher volatility and emerging negative values, the CIT monthly data was converted to quarterly data from the rst quarter of 1995 to the rst quarter of 2010. The competing ARIMA/SARIMA and Holt-Winters models were derived, and the resulting model of this study was used to forecast PIT, CIT, VAT and TTAXR for SARS fiscal years 2010/11, 2011/12 and 2012/13. The results show that both the SARIMA and Holt-Winters models perform well in modeling and forecasting PIT and VAT, however the Holt-Winters model outperformed the SARIMA model in modeling and forecasting the more volatile CIT and TTAXR. It is recommended that these methods are used in forecasting future payments, as they are precise about forecasting tax revenues, with minimal errors and fewer model revisions being necessary. / Statistics / M.Sc. (Statistics)

The main objective of the thesis was to investigate the feasibility of accurately and precisely fore- casting the number of both national and provincial bene ciaries of social security grants in South Africa, using simple autoregressive integrated moving average (ARIMA) models. The series of the monthly number of bene ciaries of the old age, child support, foster care and disability grants from April 2004 to March 2010 were used to achieve the objectives of the thesis. The conclusions from analysing the series were that: (1) ARIMA models for forecasting are province and grant-type spe- ci c; (2) for some grants, national forecasts obtained by aggregating provincial ARIMA forecasts are more accurate and precise than those obtained by ARIMA modelling national series; and (3) for some grants, forecasts obtained by modelling the latest half of the series were more accurate and precise than those obtained from modelling the full series. / Mathematical Sciences / M.Sc. (Statistics)

The main objective of the thesis was to investigate the feasibility of accurately and precisely fore- casting the number of both national and provincial bene ciaries of social security grants in South Africa, using simple autoregressive integrated moving average (ARIMA) models. The series of the monthly number of bene ciaries of the old age, child support, foster care and disability grants from April 2004 to March 2010 were used to achieve the objectives of the thesis. The conclusions from analysing the series were that: (1) ARIMA models for forecasting are province and grant-type spe- ci c; (2) for some grants, national forecasts obtained by aggregating provincial ARIMA forecasts are more accurate and precise than those obtained by ARIMA modelling national series; and (3) for some grants, forecasts obtained by modelling the latest half of the series were more accurate and precise than those obtained from modelling the full series. / Mathematical Sciences / M.Sc. (Statistics)