MODELLING GHANA STOCK EXCHANGE INDICES AND EXCHANGE RATES WITH STABLE DISTRIBUTIONS BY GABRIEL KALLAH-DAGADU (10220578) THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA, LEGON IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF THE MPHIL STATISTICS DEGREE JUNE, 2013 University of Ghana http://ugspace.ug.edu.gh ii DECLARATION Candidate’s Declaration This is to certify that, this thesis is the result of my own research work and that no part of it has been presented for another degree in this University or elsewhere. SIGNATURE: …………………………. DATE………………………..…. GABRIEL KALLAH-DAGADU (10220578) Supervisors’ Declaration We hereby certify that this thesis was prepared from the candidate’s own work and supervised in accordance with guidelines on supervision of thesis laid down by the University of Ghana. SIGNATURE: …………………………. DATE………………………..…. DR EZEKIEL N.N. NORTEY (Principal Supervisor) SIGNATURE: …………………………. DATE………………………..…. DR KWABENA DOKU-AMPONSAH (Co-Supervisor) University of Ghana http://ugspace.ug.edu.gh iii ABSTRACT Most of the concepts in theoretical and empirical finance that have been developed over the last 50 years rest upon the assumption that the return or price distribution for financial data follows a normal distribution. But this assumption is not justified by empirical data. Rather, the empirical observations (financial returns) exhibit excess kurtosis, more colloquially known as fat tails or heavy tails. This research first described the stable distribution family - stable,  Levy stable, Cauchy and Gaussian or Normal distributions. The study presented three methods of estimating parameters of stable  distributions, namely Maximum Likelihood estimation, Empirical Characteristic function and Sample Quantile methods, and goodness of fit tests- K-S and Chi-square, were used to quantitatively assess the quality performance of their respective estimates. A sample of weekly financial data (GSE All-Shares index, USD/GHC, GBP/GHC and EUR/GHC exchange rates) covering the period of 02/01/2000 − 31/12/2011 was analysed, and fitted to stable,  Cauchy and Normal distributions. Diagnostic tests such as P-P and Q-Q plots and goodness of fit tests (K-S, Chi-square, Anderson-Darling and Shapiro-Wilk) were graphically and quantitatively used to assess fitness to the returns of the data respectively. The study concludes that the weekly return distributions of Ghana financial data are heavy tailed and asymmetry and the maximum likelihood estimation method produce the most accurate and efficient estimates for the stable  fit to the data. The weekly financial data considered were modelled with stable  distribution and recommends that for efficient risk and assets returns management, analysts should explore and discover actual return distributions of financial data and not desist from speculative assumptions. University of Ghana http://ugspace.ug.edu.gh iv DEDICATION I dedicate this thesis to my guardian Madam Jessica Reath, Philipine Mordzinu and my beloved late parents; Mr. Samuel Maama K. Dagadu and Mrs. Margret A. Adjei Dagadu. University of Ghana http://ugspace.ug.edu.gh v ACKNOWLEDGEMENT I thank the Almighty God who has given me the care, knowledge and the opportunity to pursue education up to this level. There are many people without whom this work could not have been undertaken. I render my heart-felt thanks to my Supervisors; Dr. E.N.N. Nortey and Dr. K. Doku-Amponsah for their countless guidance, advice and constructive criticisms throughout this work. I would also thank all the lectures of Statistics Department, especially Mr. R. Minkah for their services and pieces of advice throughout my two years study in this University. To Madam Jessica, Mr Sammy Appiah and my family, I say thank you all for your support, encouragement, advice and patience throughout my studies and may the good Lord continue to bless you all. Finally, to the management of Ghana Stock Exchange, Bank of Ghana and all my friends especially 2013 batch of MPhil students of Statistics Department, and to all of you including those not mentioned here, I ask for Gods guidance and mercies. Thank you and God bless you. University of Ghana http://ugspace.ug.edu.gh vi TABLE OF CONTENTS Contents Declaration .......................................................................................................................... ii Candidate’s Declaration .................................................................................................. ii Supervisors’ Declaration ................................................................................................. ii Abstract ............................................................................................................................. iii Dedication .......................................................................................................................... iv Acknowledgement ............................................................................................................... v Table of Contents ............................................................................................................... vi List of Figures .................................................................................................................... ix List of Tables ....................................................................................................................... x List of Abbreviations .......................................................................................................... xi Chapter One ........................................................................................................................ 1 Introduction ......................................................................................................................... 1 1.0 Background of The Study ..................................................................................... 1 1.1 Problem Statement ................................................................................................ 3 1.2 Objectives of The Study ....................................................................................... 5 1.3 Scope of The Study .............................................................................................. 6 1.4 Significance of The Study .................................................................................... 6 1.5 Limitations of The Study ...................................................................................... 7 1.6 Organization of The Study ................................................................................... 7 Chapter Two ........................................................................................................................ 8 Review of Related Literature .............................................................................................. 8 Chapter Three .................................................................................................................... 17 Methodology ..................................................................................................................... 17 3.0 Introduction ........................................................................................................ 17 3.1.0 The Stable Distribution Family ....................................................................... 17 University of Ghana http://ugspace.ug.edu.gh vii 3.1.1   Stable Distribution ................................................................................ 18 3.1.2 Stable Density and Distribution Functions ................................................. 20 3.2 Special Cases of the Stable Distribution ............................................................ 24 3.2.1 Levy Stable Distribution ............................................................................. 25 3.2.2 The Cauchy Distribution ............................................................................. 26 3.2.3 The Gaussian (Normal) Distribution ........................................................... 27 3.3 Probability-Probability (P-P) Plot ...................................................................... 29 3.4 Quantile-Quantile (Q-Q) Plot ............................................................................. 29 3.5 Goodness-of-Fit Tests ........................................................................................ 30 3.5.1 Anderson-Darling Test ................................................................................ 30 3.5.2 Kolmogorov-Smirnov Goodness-of-Fit Test .............................................. 31 3.5.3 Chi-Square Goodness of Fit Test ................................................................ 33 3.5.4 Shapiro-Wilk Test ....................................................................................... 34 3.6 Estimation The Parameters of  - Stable Distribution ....................................... 35 3.6.1 Maximum Likelihood Estimation Method .................................................. 36 3.6.2 Sample Quantile Method ............................................................................. 38 3.6.3 Empirical Characteristic Function Method ................................................. 39 3.7.1 Central Limit Theorem ................................................................................... 42 3.7.2 Generalized Central Limit Theorem ............................................................... 43 3.7.3 Infinite Variance ............................................................................................. 44 3.7.4 Infinite Variance Central Limit Theorem ....................................................... 44 Chapter Four ...................................................................................................................... 45 Analysis and Discussions .................................................................................................. 45 4.0 Introduction ........................................................................................................ 45 4.1 Ghana Stock Exchange All-Shares Index ........................................................... 45 4.2 Currency Exchange Rates ................................................................................... 55 Chapter Five ...................................................................................................................... 81 University of Ghana http://ugspace.ug.edu.gh viii Summary, Conclusion And Recommendations ................................................................ 81 5.0 Introduction ........................................................................................................ 81 5.1 Summary ............................................................................................................. 81 5.2 Conclusion .......................................................................................................... 82 5.3 Recommendations .............................................................................................. 83 5.4 Further Studies .................................................................................................... 84 References ......................................................................................................................... 85 Appendix A ....................................................................................................................... 91 Appendix B ..................................................................................................................... 104 University of Ghana http://ugspace.ug.edu.gh LIST OF FIGURES Figure 1: The GSE All-Shares index evolution from 02/01/00 to 31/12/10 ..................... 46 Figure 2: Logarithm returns of GSE All-Shares index ..................................................... 47 Figure 3: GSE All-Shares index from 02/01/00- 31/12/10 ............................................... 48 Figure 4: The density plots of log returns of GSE All-Shares index ................................ 49 Figure 5: The Normal Q-Q plot GSE All-Shares index ................................................... 50 Figure 6: Stable Q-Q plot of GSE All-Shares index ......................................................... 51 Figure 7: The Stable P-P plot of GSE all-Shares index .................................................... 51 Figure 8: USD/GHC progression for the 11 years period ................................................. 55 Figure 9: USD/GHC of log returns of volatility over the 11 years period ........................ 56 Figure 10: GBP/GHC progression for the 11 years period ............................................... 57 Figure 11: GBP/GHC of log returns of volatility over the 11 years period ...................... 58 Figure 12: EUR/GHC progression for the 10 years period ............................................... 59 Figure 13: EUR/GHC log returns of volatility over the 10 years period .......................... 60 Figure 14: Histogram plot of log returns of USD/GHC .................................................... 61 Figure 15: Histogram plot of log returns of GBP/GHC .................................................... 62 Figure 16: Histogram plot logarithm returns of EUR/GHC .............................................. 63 Figure 17: The density plots of log returns of the data ..................................................... 64 Figure 18: The density plots of log returns of the data ..................................................... 65 Figure 19: The density plots of log returns of the data ..................................................... 66 Figure 20 : Normal Q-Q plot of USD/GHC exchange rate ............................................... 67 Figure 21: The diagnostics tests of USD/GHC data ........................................................ 68 Figure 22: Normal Q-Q plot of GBP/GHC exchange rate ................................................ 69 Figure 23: The diagnostics tests of GBP/GHC data .......................................................... 70 Figure 24: Normal Q-Q plot of EUR/GHC exchange rate ................................................ 71 University of Ghana http://ugspace.ug.edu.gh x Figure 25: The diagnostics tests of EUR/GHC data ........................................................ 72 LIST OF TABLES Table 1: Estimated parameters of the Stable distribution ................................................. 52 Table 2: Goodness of fit tests ............................................................................................ 53 Table 3: Goodness of fit tests for Normal, Cauchy and Stable distributions .................... 54 Table 4: Estimated parameters of the Stable distribution ................................................. 73 Table 5: Estimated parameters of the Stable distribution ................................................. 73 Table 6: Estimated parameters of the Stable distribution ................................................. 74 Table 7: Goodness of fit tests for USD/GHC exchange rates ........................................... 75 Table 8: Goodness of fits test for GBP/GHC exchange rates ........................................... 76 Table 9: Goodness of fit tests for EUR/GHC exchange rates ........................................... 76 Table 10: Goodness of fit tests for the distributions to USD/GHC data ........................... 78 Table 11: Goodness of fit tests for the distributions to GBP/GHC data ........................... 78 Table 12: Goodness of fit tests for distributions to the EUR/GHC Exchange data .......... 78 Table 13: Estimated parameters of the Normal distribution ............................................. 79 University of Ghana http://ugspace.ug.edu.gh xi LIST OF ABBREVIATIONS APT-------------------Arbitrage Pricing Theory CAPM ---------------Capital Asset Pricing Model CF--------------------Characteristic Function ECF-------------------Empirical Characteristic Function EUR-------------------European Euro EUR/GHC-----------Euro to Ghana Cedi exchange rate FFT -------------------Fast Fourier Transform GBP-------------------Great Britain Pound GBP/GHC-----------British pound to Ghana Cedi exchange rate GCLT-----------------General Central Limit Theorem GSE-------------------Ghana Stock Exchange GSEI------------------Ghana Stock Exchange all-shares index i.i.d. -------------------independent identically distributed K-S--------------------Kolmogorov-Smirnov test MCMC---------------Markov Chain Monte Carlo MLE------------------Maximum Likelihood Estimation P-P--------------------Probability-Probability plot Q-Q-------------------Quantile-Quantile plot S. Quantile-----------Sample Quantile USD-------------------United State of America Dollar USD/GHC-----------US dollar to Ghana Cedi exchange rate University of Ghana http://ugspace.ug.edu.gh 1 Chapter One INTRODUCTION 1.0 Background of the Study Most of the concepts in theoretical and empirical finance that have been developed over the last 50 years rest upon the assumption that the return or price distribution for financial assets follows a normal distribution. Yet, with rare exception, studies that have investigated the validity of this assumption since the 1960s fail to find support for the normal distribution or Gaussian distribution as it is also called. Moreover, there is ample empirical evidence that many, if not most, financial return series are heavy-tailed and possibly skewed (Rachev, Menn & Fabozzi, 2005). The “tails” of the distribution are where the extreme values occur. Empirical distributions for stock prices and returns have found that the extreme values are more likely than would be predicted by the normal distribution. This means that, between periods where the market exhibits relatively modest changes in prices and returns, there will be periods where there are changes that are much higher (i.e., crashes and booms) than predicted by the normal distribution. This is not only of concern to financial theorists, but also to practitioners who are, in view of the frequency of sharp market down turns in the equity markets, troubled by the compelling evidence that something is wrong in the foundation of the statistical edifice used, for example, to produce probability estimates for financial risk assessment by Hope’s study (Rachev et al., 2005). Stable distributions have been widely used for fitting data in which extreme values are frequent due to the fact that it accommodates heavy-tailed financial series and therefore University of Ghana http://ugspace.ug.edu.gh 2 produces more reliable measures of tail risk such as value at risk (Garcia, Renault & Veredas, 2010). Today it is widely acknowledged that the proper management of assets and prices or other related investment risks, requires the proper modelling of the return distribution of financial assets. For instance, the answer to whether it is possible to beat the market except by chance depends on whether stock market prices display long memory and how probable are very large price fluctuations (Alfonso, Mansilla & Terrero-Escalante, 2011). The crucial difficulty, however, is that the financial market is a very complex system; it has a large number of non-linearly interacting internal elements, and is highly sensible to the action of external forces. Even more, the real challenge here is that the number of the system constituents and the details of their interactions and of the external factors acting upon it is actually barely known (Alfonso et al., 2011). It is often argued that financial asset returns are the cumulative outcome of a vast number of pieces of information and individual decisions arriving almost continuously in time. Hence, in the presence of heavy-tails it is natural to assume that they are approximately governed by a stable non-Gaussian distribution. Stable distributions have been proposed as a model for many types of physical and economic systems. There are several reasons for using a stable distribution to describe a system. The first is where there are solid theoretical reasons for expecting a non-Gaussian stable model, for example, reflection of rotating mirror yielding Cauchy distribution, hitting times from a Brownian motion yielding Levy distribution, the gravitational field of stars yielding the Holtsmark distribution (Feller, 1971; Uchaikin & Zolotarev, 1999). University of Ghana http://ugspace.ug.edu.gh 3 The second reason is the Generalized Central Limit Theorem, which states that the only possible non-trivial limit of normalized sums of independent identically distributed terms is stable. It is argued that some observed quantities are the sum of many small terms, for example, the price of a stock, and hence a stable model should be used to describe such systems. The third argument for modelling with stable distributions is empirical: many large data sets exhibit heavy tails and skewness. The strong empirical evidence for these features combined with the Generalized Central Limit Theorem are used to justify the use of stable models. Stable distributions have been successfully fit to stock returns, excess bond returns, foreign exchange rates, commodity price returns and real estate returns (McCulloch, 1996; Rachev & Mittnik, 2000). For a developing country like Ghana, there is a need for the proper management of assets and prices (and the related investment risks) and that requires the proper modelling of the return distribution of financial assets. 1.1 Problem Statement The behaviour of extreme variations of economic indices, stock prices or even currencies, has been a topic of interest in finance and economics, and its study has become relevant in the context of risk management and financial risk theory. However, the analyses of stock prices, asset returns and exchange rates are usually difficult to perform due to the small number of extreme values in the tails of the distributions of financial time series variations. Many techniques in modern finance rely heavily on the assumption that the random variables under investigation follow a Gaussian distribution. However, time series data observed in finance, and in other applications, often deviate from the normal distribution, University of Ghana http://ugspace.ug.edu.gh 4 in that their marginal distributions are heavy-tailed and, possibly asymmetric. In such situations, the appropriateness of the commonly adopted normal assumption is highly questionable (Borak, Härdle, Wolfgang & Weron, 2005). Many of the concepts in theoretical and empirical finance developed over the past decades rest upon the assumption that asset returns follow a normal distribution. However, it has been long known that asset returns practically are not normally distributed. Rather, the empirical observations of financial asset returns and stock price indices exhibit fat tails. In response to the empirical evidence, Mandelbrot (1963) and Fama (1965) proposed the stable distribution as an alternative model. Frain (2009) states that, the use of α-stable distribution in Finance was originally proposed by Mandelbrot to model various goods and asset prices and it became popular in the sixties and seventies but interest waned thereafter. This decline in interest was due not only to its mathematical complexity and the considerable computation resources required but to the considerable success of the Merton-Black-Scholes Gaussian approach to finance theory which was developed at the same time (Frain, 2009). It is widely recognized that the key to develop successful strategies for risk management and asset pricing is to parsimoniously describe the stochastic process governing asset dynamics (Xu, Xiao, Wu & Dong, 2011). The financial market of Ghana remains under developed as compared to emerging markets of developed countries, due to lack of ability to manage risk and pricing of assets. Managing risk of an asset depends on knowing riskiness of the asset and the distribution of the returns of asset. Tsay (2005) states emphatically that, proper managing of risky assets depends on both the positive and negative returns of the asset or stock. An investor University of Ghana http://ugspace.ug.edu.gh 5 may be able to hold a short or long position of an asset, or put short or put long of an asset if he/she knows the return distribution of the asset/stock. Investors take positions in currency swaps, options, futures or forward because they know the return distributions of that currency exchange rate. Studying the return distribution of assets and stocks make the exchange market efficient and vibrant, and prevent arbitrageurs from taking advantage of the market. The study anticipates finding out the distribution of the financial data of Ghana stock exchange. It is observed that due to mathematical complexity of computing parameters of stable distributions, financial, statistical analysts and economists resort to the assumptions that financial data follows the Gaussian (normal) distribution. 1.2 Objectives of the Study The main objective of this study is to investigate the empirical performance of α-stable distribution in fitting the behaviour of asset returns and exchange rates and also to explore which of the existing methods for estimating the parameters of -stable distribution is much better in terms of fitting stable models. This study specifically seeks to; 1. Determine the return distribution of financial assets (currency exchange rates and GSE all-shares index). 2. Investigate which of the following methods (Maximum Likelihood Estimation, Sample Quantile and Empirical Characteristic Function) of estimating the four parameters of stable distribution produces the best fit. 3. Fit an -stable model to the Ghana Stock Exchange All-Shares Index and Currency Exchange rates. University of Ghana http://ugspace.ug.edu.gh 6 1.3 Scope of the Study The analysts have rarely studied the modelling of financial data of Ghana using stable laws. The study will consider a sample of data covering a period of eleven years (2000- 2011) and it will consist of daily stock indices from Ghana Stock Exchange and daily or monthly exchange rates from Bank of Ghana. The study will consider the Ghana Stock Exchange (GSE) All-Shares index and the three major exchange rates; US Dollar to Ghana Cedi, Euro to Ghana Cedi and British Pound to Ghana Cedi. 1.4 Significance of the Study Although there are several studies that have examined the performance of stock prices and assets returns, there are a few or no studies that have investigated the performance of returns in the Ghana stock market using stable distributions. Findings from this study will go a long way in assisting economists, financial analysts and policy makers to make decent government economic or financial policies that will yield impressive gains in all the sectors of economic. The results from this study will be very important in assisting investors to develop successful strategies for risk management and asset pricing, especially for investors who adopt chasing index returns and exchange rates investment strategy in the Ghanaian stock market. It will also help the economists and financial analysts to stabilise the Ghanaian currency (Cedi) and also enable them to make accurate future forecast and predictions, since empirical results have shown that, assets returns with heavy tails give a much better stable model fit than Gaussian model fit (Nolan, 2010). On the whole, it will add knowledge to the academic field, since there is little or no work that has been carried out in Ghana. University of Ghana http://ugspace.ug.edu.gh 7 1.5 Limitations of the Study The research intended to model the daily logarithm returns of Ghanaian financial data, but due to the constant nature and low volatility of the data, the study couldn’t modelled the daily logarithm returns of the data. The returns of the financial data could not satisfied some of the assumptions of stable laws, therefore the fractional moment method of estimating stable parameters was not applicable because the returns of the financial data contains zero. The Ghana Stock Exchange change the calculation of the indices from All- Shares index to composite index and financial index and that took effect from second January, 2011 and so that year data was not considered which reduces the sample size of GSE all-shares index. 1.6 Organization of the Study The rest of the thesis is organized into four chapters, chapter two comprised of reviews of related literature on the topic and chapter three looks at the family of stable distributions, methods of estimating the parameters of stable distribution and goodness of fit tests of stable model. Chapter four analyzes and discusses the findings of the study and finally chapter five presents the summary, conclusion and recommendations of the study. University of Ghana http://ugspace.ug.edu.gh 8 Chapter Two REVIEW OF RELATED LITERATURE It is often argued that financial asset returns are the cumulative outcome of a vast number of pieces of information and individual decisions arriving almost continuously in time. Hence, in the presence of heavy-tails it is natural to assume that they are approximately governed by a stable non-Gaussian distribution. McCulloch (1997) states that other leptokurtic distributions, including Student's t , Weibull and Hyperbolic, do not obey the central limit theorem. Tsay (2005) disputes the traditional assumption that simple financial returns are independently and identically distributed as normal with mean and variance. He explained that the simple returns have a lower bound of -1 but the normal distribution has a lower bound of -∞. Also, he said if the simple returns are normally distributed, then the multi- period simple return is not normally distributed because it is a product of one-period returns. Tsay concluded that, the normality assumption is not supported by many empirical asset returns (Tsay, 2005). The most common assumption in some literature is that the logarithm returns of an asset are independent and identically distributed as normal with mean  and variance 2 but Tsay (2005) dispute that the lognormal assumption is not consistent with all the properties of stock returns, which lean towards positive excess kurtosis (Tsay, 2005). Stable distributions have been successfully fitted to stock returns, excess bond returns, foreign exchange rates, commodity price returns and real estate returns (McCulloch, 1996; Rachev & Mittnik, 2000). In recent years, however, several studies have found what University of Ghana http://ugspace.ug.edu.gh 9 appears to be strong evidence against the stable model (Gopikrishnan et al., 1999; McCulloch, 1997). The implication that returns of financial assets have a heavy-tailed distribution may be profound to a risk manager in a financial institution. Bradley and Taqqu (2003) argue that, a 3σ events may occur with a much larger probability when the return distribution is heavy- tailed than when it is normal. Quantile based measures of risk, such as value at risk, may also be drastically different if calculated for a heavy-tailed distribution. This is especially true for the highest quantiles of the distribution associated with very rare but very damaging adverse market movements (Bradley & Taqqu, 2003). DuMouchel (1973) estimated the Paretian tail index directly from the tail observations and using either the Pareto distribution or a generalization of the Pareto distribution proposed by DuMouchel, found a tail index that appears to be significantly greater than 2; the maximum permissible value for a stable distribution (McCulloch, 1997). An issue here is whether the underlying distributions are actually stable. Stability only holds for 0 2  and some authors have found that the tails of some financial time series have to be modelled with > 2 (Gopikrishnan, et al., 1999; Pagan, 1996; Coronel-Brizio & Hernandez-Montoya, 2005). Alfonso et al. (2011) and Cont (2001), argue that, in order for a parametric distributional model to reproduce the properties of the empirical distribution it must have at least four parameters: a location parameter, a scale parameter, a parameter describing the decay of the tails and an asymmetry parameter. There are other heavy-tailed distributions such as University of Ghana http://ugspace.ug.edu.gh 10 Student’s t , hyperbolic or normal inverse Gaussian which fulfilled this condition. Therefore, in order to grasp the universal laws behind markets dynamics, it is important to keep accumulating empirical facts about the statistics of different financial indices around the world (Alfonso et al., 2011). A rigorous statistical analysis of logarithm returns of daily fluctuations of the IPC; the leading Mexican Stock Market Index, shows that the stable Levy distribution best fits the data after considering the Gaussian, normal inverse Gaussian and Levy distributions and the corresponding  was indeed in the range for a stable Levy distribution (Alfonso et al., 2011). Bradley and Taqqu (2001) argue that despite Markowitz’s mean–variance portfolio theory, as well as the CAPM (Capital Asset Pricing Model) and APT (Arbitrage Pricing Theory) models, relies either explicitly or implicitly on the assumption of normally distributed asset returns. Today, with long histories of price or return data available for many financial assets, it is easy to see that this assumption is inadequate. Empirical evidence using NASDAQ composite index, suggests that asset returns have distributions which are heavier-tailed than the normal distribution (Bradley & Taqqu, 2003). Xu et al. (2011) fitted the Shanghai Composite Index and Shenzhen Component Index returns with α - stable distribution, and the empirical results show that the asymmetric leptokurtic features present in the Shanghai Composite index and Shenzhen Component index returns can be captured by α-stable law. Their findings of empirical result shows that α - stable distribution is better fitted to Chinese stock returns than the Black–Scholes model. University of Ghana http://ugspace.ug.edu.gh 11 Khindanova, Rachev and Schwartz (2001), argue that one of the most important tasks of financial institutions is evaluating the exposure to market risks, which arise from variations in prices of equities, commodities, exchange rates, and interest rates. The dependence on market risks can be measured by changes in the portfolio value, or profits and losses. The empirical observations of financial data exhibit “fat” tails and excess kurtosis. The historical methods (delta method, historical simulation, Monte Carlo simulation, and stress-testing) do not impose distributional assumptions but it is not reliable in estimating low quantiles of changes in prices with a small number of observations in the tails. The value-at-risk (VAR) measurements are widely applied to estimate the exposure to market risks. Khindanova et al. (2001) shows that the traditional approaches to VAR computations-the Delta method, Historical simulation, Monte Carlo simulation, and Stress-testing, do not provide satisfactory evaluation of possible losses. The Delta-normal methods do not describe well financial data with heavy tails. Hence, they underestimate VAR measurements in the tails. The Historical simulation does not produce robust VAR estimates since it is not reliable in approximating low quantiles with a small number of observations in the tails. The stable Paretian model, while sharing the main properties of the normal distribution leading to the CLT (central limit theorem), outperforms the normal modelling for high values of the VAR confidence level 99% and provides superior fit in modelling VAR (Khindanova et al., 2001). Many of the concepts in theoretical and empirical finance developed over the past decades such as the classical portfolio theory, the Black-Scholes-Merton option pricing model or the Risk Metrics variance-covariance approach to VaR, rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical University of Ghana http://ugspace.ug.edu.gh 12 data, but rather, the empirical observations exhibit excess kurtosis and was justified by Guillaume et al. (1997) and Rachev and Mittnik (2000) and Borak, Misiorek & Weron (2010). The Basle Committee on Banking Supervision’s study (as cited in Borak et al., 2010) suggested that for the purpose of determining minimum capital reserves, financial institutions should use a 10-day VaR at the 99% confidence level multiplied by a safety factor s ∈ [3, 4]. Stahl (1997) and Danielsson, Hartmann and De Vries’ study (as cited in Borak et al., 2010) also argue convincingly that the range of s is as a result of the heavy- tailed nature of asset returns (Borak, Misiorek & Weron , 2010). Borak et al. (2010) fitted two samples of financial data (Dow Jones Industrial Average (DJIA) index and the Polish WIG20 index) to Gaussian, Hyperbolic, Normal-Inverse Gaussian ( NIG) and Stable distributions and found that for both datasets, Kolmogorov and Anderson-Darling goodness-of-fit statistics suggest the Hyperbolic and NIG distributions as the best models for filtered and standardized returns, respectively. Peters, Sisson and Fan (2009) states that models constructed with α-stable distributions possess several useful properties, including infinite variance, skewness and heavy tails and also justified by Zolotarev (1986), Alder, Feldman and Taqqu (1998), Samorodnitsky and Taqqu (1994) and Nolan (2010). Peters et al. (2009) argue convincingly that, statistical inference for α-stable models is challenging due to the computational intractability of the density function. In practice this limits the range of models fitted, to univariate and bivariate cases. By adopting likelihood- University of Ghana http://ugspace.ug.edu.gh 13 free Bayesian methods they were able to circumvent this difficulty, and provide approximate, but credible posterior inference in the general multivariate case, at a moderate computational cost. They show that multivariate projections of data onto the unit hypersphere, in combination with sample quantile estimators, are adequate for this task and their method shows greater sampler consistency than alternative samplers, such as the auxiliary Gibbs or Markov Chain Monte Carlo (MCMC) inversion plus series expansion samplers (Peters, Sisson & Fan, 2009). Alfonso et al. (2011) found out that, despite the simplifications of the normal distribution provides in analytical calculation are very valuable, empirical studies by Mandelbrot (1963), Mantegna and Stanley (1995), Gopikrishnan et al. (1999) and Cont (2001) show that the distribution of returns has a tail heavier than that of a Gaussian. They illustrated this fact, with the histogram plot of daily logarithm differences of the Mexican IPC index from April 9th, 2000 to April 9th, 2010. The chi-square goodness of fit test, the Anderson-Darling test and the Kolmogorov- Smirnov (K-S) test rejected the Mexican financial index, IPC, being distributed normally or as a normal inverse Gaussian. On the other hand, the three tests show that IPC data comes from α-stable levy distribution (Alfonso et al., 2011). Alfonso et al. (2011) illustrate that the sample size have impact on estimating the tail index and that high frequency data is needed in order to determine whether or not a given distribution is stable. Cartea and Howison (2009) claim that based on the Generalised Central Limit Theorem (GCLT); there are two ways of modelling stock prices or stock returns, in general terms. If it is believed that stock returns are at least approximately governed by a Levy-Stable University of Ghana http://ugspace.ug.edu.gh 14 distribution then the accumulation of the random events is additive. On the other hand, if it is believed that the logarithms of stock prices are approximately governed by a Levy- stable distribution then the accumulation is multiplicative. McCulloch (1996) assumes that assets returns are log Levy-Stable and prices options using a utility maximisation argument also follow Levy-Stable process. Also, Carr and Wu (2003) states that priced European options of the log-stock price returns follow a maximally skewed Levy-Stable process (Cartea & Howison, 2009). Cartea and Howison, use the Black-Scholes model in finance for pricing assets as a benchmark to compare the option prices obtained when the returns follow Levy-Stable process. Their findings were consistent with the findings of Hull and White (1987) where the Black-Scholes model under prices in- and out of-the-money call option prices and overprices at-the-money options (Cartea & Howison, 2009). Until recently, it has been difficult to use stable laws in practical problems because of computational difficulties. Nolan (1997) developed a software program, known as STABLE which can compute stable densities, cumulative distribution functions, parameters and quantiles. Nolan (1997) described the basic method used in the program –Fourier transform and simulation. Later improvements to the program was made by incorporating the Chambers, Mallows and Stuck (1976) method of simulating stable random variables, which improved accuracy in the calculations, and estimation of stable parameters from data sets (Nolan, 2003). University of Ghana http://ugspace.ug.edu.gh 15 The basic estimation problem for stable laws is to estimate the four parameters  , , ,     from an i.i.d. random sample 1 2, ,..., .nX X X There are several methods available for this basic estimation problem: a Quantile method of McCulloch (1986), a Fractional Moment method of Nikias and Shao (1995), Empirical Characteristic Function (ECF) method of Kogon and Williams (1998) based on ideas of Koutrouvelis, and Maximum Likelihood (ML) estimation of DuMouchel (1971) and Nolan (2001). Ojeda (2001), compared these methods in a simulation study and found that the Maximum Likelihood estimates are almost always more accurate, with the Empirical Characteristic Function estimates next best, followed by the Sample Quantile method, and finally the Moment method. The ML method has the added advantage that one can give large sample confidence intervals for the parameters, based on numerical computations of the Fisher information matrix (Nolan, 2003). Standard exploratory data analysis of graphical techniques can be adapted to informally evaluate the closeness of a stable fit. Nolan observed that comparing smoothed data density plots to a proposed fit gives a good sense of how good the fit is near the centre of the data. Nolan argues that the P–P plots allow a comparison over the range of the data whiles Q– Q plots not as satisfactory for comparing heavy tailed data to the proposed fit and for technical reasons he recommend the “variance stabilized” P–P plot of Michael (1983). He claim that heavy tailed data set have many more extreme values than a typical sample from finite variance population and that it forces a Q–Q plot to be visually compressed, with a few extreme values dominating the plot. Also, the heavy tails imply that the extreme order statistics will have a lot of variability, and hence deviations from an ideal straight line Q– Q plot are hard to assess (Nolan, 2003). University of Ghana http://ugspace.ug.edu.gh 16 The above discussion has mainly highlighted the return distribution of financial data. It reviews a range of financial data that have been modelled with different heavy-tail distributions. The chapter also reviews different methods of estimating the parameters of stable  distribution and concluded with diagnostic tests for fitting the models. The next chapter elaborates further on this statistical technique and provides a detailed account of their procedures. University of Ghana http://ugspace.ug.edu.gh 17 Chapter Three METHODOLOGY 3.0 Introduction This chapter discusses the various stable distribution families which will best fit the GSE All-Shares index and the three currency exchange rates under study. These involve α- stable, Levy-stable, Cauchy and Gaussian (Normal) distributions. Specifically, this chapter examines the α-stable distribution and the normal distribution. Techniques for estimating the parameters of the distributions as well as Quantile-Quantile (Q-Q) and Probability-Probability (P-P) plots used in assessing goodness of fit to the data set are described. The Kolmogorov-Smirnov, Chi-square, Shapiro-Wilk and Anderson- Darling goodness of fit tests applied in this research work are also discussed. 3.1.0 The Stable Distribution Family This section gives detailed description of the four stable distributions that this study explores to model GSE all-shares index and exchange rates. The study considers the stable families of distributions due to their stability property and nature of the financial data. This research considers the α-stable distribution, Levy stable distribution, Cauchy distribution and Gaussian or normal distribution. This family of distributions has a very interesting pattern of shapes, allowing for asymmetry and thick tails, that makes them suitable for the modelling of several phenomena, ranging from the engineering (noise of degraded audio sources) to the financial; asset returns (Lombardi, 2007). University of Ghana http://ugspace.ug.edu.gh 18 3.1.1   Stable Distribution Stable distributions are a class of probability laws that have intriguing theoretical and practical properties. The   stable family of distributions stems from a more general version of the central limit theorem which replaces the assumption of the finiteness of the variance with a much less restrictive one concerning the regular behaviour of the tails (Gnedenko & Kolmogorov, 1954). Their applications to financial modelling comes from the fact that they generalize the normal (Gaussian) distribution and allow heavy tails and skewness, which are frequently seen in financial data. In general, there are no closed form formulas for α-stable density functions f and cumulative distribution functions F , but there are now reliable computer programs for working with these laws (Nolan, 2005) and α-stable distributions are represented by characteristic functions. Definition 1 The complex­valued function     3.01itxX t E e     is called the characteristic function (c.f.) of a real random variable. Here t is some real­valued variable and if the density  f x exists, then the Fourier transform of that density is given as;       3.02itxX t e f x dx   University of Ghana http://ugspace.ug.edu.gh 19 The density function  f x is derived using inverse Fourier transform which allows us to reconstruct the density of a distribution from a known c.f. by the uniqueness theorem and is given as;      12 3.03itx Xf x e t dx    Definition 2 A random variable X is α-stable distributed; denoted by 0( , , , ;0)S     , if it has the characteristic function          1 0 0 exp 1 tan 1 1, 2 exp 3.04 2 exp 1 ln 1, u i sign u u i u E iuX u i sign u u i u                                                      where sign(u) is 1 if u > 0, 0 if u = 0, and –1 if u < 0. Definition 3 A random variable X is α-stable distributed; denoted by 1( , , , ;1)S     , if it has characteristic function as given below;         1 1 exp 1 tan 1, 2 exp 3.05 2 exp 1 ln 1, u i sign u i u E iuX u i sign u u i u                                                  where sign(u) is 1 if u > 0, 0 if u = 0, and –1 if u < 0. University of Ghana http://ugspace.ug.edu.gh 20 The parameter  is called the index of the law or the index of stability or characteristic exponent and must be in the range  0 2 or 0,2 .  The parameter β is called the skewness of the law, and must be in the range 1 1   . If β = 0, the distribution is symmetric, if β > 0 it is skewed towards the right and if β < 0, it is skewed towards the left. The parameters α and β determines the shape of the distribution. The parameter γ is a scale parameter and it can be any positive number, i.e. 0.  The parameter δ is a location parameter and range < < ,  it shifts the distribution right if > 0 , and left if < 0. The two different definitions of α-stable distribution above, according to Zolotarev (1986) and Nolan (2003) are the two common different parameterizations, which this study will consider and are denoted by 0( , , , ;0)S     and 1( , , , ;1)S     . The first is use in applications, because it has better numerical behaviour and intuitive meaning but the formulation of the characteristic function is, in this case, quite more cumbersome, and the analytic properties have a less intuitive meaning. The second parameterization has a quite manageable expression of its characteristic function and can straightforwardly produce several interesting analytic results (Zolotarev, 1986), but has a major drawback for what concerns estimation and inferential purposes: it is not continuous with respect to the parameters, having a pole at α = 1, but is more commonly used in the literature. 3.1.2 Stable Density and Distribution Functions The lack of closed form formulas for most α-stable densities and distribution functions has negative consequences. For example, during maximum likelihood estimation computationally burdensome numerical approximations have to be used. There generally are two approaches to this problem. Either the Fast Fourier transform (FFT) has to be applied to the characteristic function (Mittnik, Rachev, Doganoglu & Chenyao, 1999) or University of Ghana http://ugspace.ug.edu.gh 21 direct numerical integration has to be utilized (Nolan, 1997). The FFT based approach is faster for large samples, whereas the direct integration method favours small data sets since it can be computed at any arbitrarily chosen point (Borak et al., 2005). The probability density function  ; ,f x   , of a standard α-stable random variable is given below as: Let 2tan ,    When 1, :x               11 2 1; , ; , exp ; , , 3.061xf x V x V d                        When 1, :x           12 1 2 1 cos; , , 3.07 1 f x           When 1 and :x        ; , ; , , 3.08f x f x       When 1, x   ℝ:           22 2 2 2 1 ;1, exp ;1, , 0 2 ;1, 3.09 1 , = 0 1 x x V V d f x x e e                                   University of Ghana http://ugspace.ug.edu.gh 22 where     1 arctan , 1 3.10 , = 1 2              and            11 1 1 cos 1cos cos , 1 sin cos ; , 3.11 2 2 exp tan , 1, 0 cos 2 V                                                                The cumulative distribution function (cdf)  ; ,F x   of a standard α-stable random variable can be expressed as:  When 1 and :x           121 (1 ); , ( , ) exp ; , , 3.12signF x c x V d             where    1 1 , 1 2 , 3.13 1, 1 c                       University of Ghana http://ugspace.ug.edu.gh 23  When 1, :x        1; , , 3.142F x      When 1, :x        ; , 1 ; , , 3.15F x F x        When 1, x   ℝ:          22 2 1 exp ;1, , 0, 1 1 ; , arctan , 0, 3.16 2 1 ;1, , 0. x V d F x x F x e                                 The density formula (eqn. 3.06) above requires numerical integration of the function     . exp .p p , where    1( ; , , ) ; , .p x x V        The integrand is 0 at , and increases monotonically to a maximum of 1e at point * for which *; , , ) 1p x   , and then decreases monotonically to 0 at 2  . However, in some cases the integrand becomes very peaked and numerical algorithms can miss the spike and underestimate the integral. To avoid this problem we need to find the argument * of the peak numerically University of Ghana http://ugspace.ug.edu.gh 24 and compute the integral as a sum of two integrals: one from  to * and the other from * to 2  (Nolan, 1997). The defining characteristic, and reason for the term stable, is that they retain their shape (up to scale and shift) under addition. If 1 2, ,..., nX X X are independent and identically distributed stable random variables, with distribution function F , then for every n ;  1 2 ... 3.17n n nX X X d c X d    for some constants 0 and .n nc d The symbol d means equality in distribution, i.e., the right and left hand sides have the same distribution. The law is called strictly stable if 0d  for all .n In terms of financial returns, one could say that the sum of daily returns is up to scale and location equally distributed as the monthly return or yearly return. 3.2 Special Cases of the Stable Distribution There are three special cases of the stable distributions; Levy stable, Cauchy and Gaussian distributions that have closed-form expression of density functions. The case where α = 2 (and β = 0, which plays no role in this case) and with the reparameterization in scale, where 2  , yields the normal distribution. The case where α = 1 and β = 0 yields the Cauchy distribution with much fatter tails than the normal distribution. The third and last special case is obtained for 12  and 1  , yields the Lévy distribution. University of Ghana http://ugspace.ug.edu.gh 25 3.2.1 Levy Stable Distribution A random variable X is Levy stable with parameters  , ; and        has the probability density function (pdf)  ; ,f x   given as:        3 2 1 2 1; , exp , 3.18 2 2 f x x xx                    The probability density of the Lévy distribution is concentrated on the interval (μ, +∞). The cumulative distribution function (cdf)  ; ,F x   of a Lévy distribution random variable can be expressed as:        3 2 1 2 1; , exp , 3.19 2 2 F x dx x xx                         ; , , 2F x erfc xx          where ( )erfc x is the complementary error function. Both the expected value (mean) and the variance of a Levy stable distribution does not exist, ( ). The characteristic function of a Levy stable distribution is given as      exp 2 3.20X t i t i t     University of Ghana http://ugspace.ug.edu.gh 26 3.2.2 The Cauchy Distribution A random variable X has the standard Cauchy distribution, if X has probability density function (pdf) given as:      2 1 , 3.211f x xx  The distribution function ( )F x of the standard Cauchy distribution is given by      1 1 arctan , 3.222F x x x   The Cauchy distribution is generalized by adding location and scale parameters. Suppose that Z has the standard Cauchy distribution and that   and 0,   . Then X Z   has the Cauchy distribution with location parameter  and scale parameter  and the probability density function of X is given as:      22 , 3.23h x xx          The distribution function ( )H x of the general Cauchy distribution is given by    1 1 arctan , 3.242 xH x x        The characteristic function of a standard Cauchy distribution is given as      exp , 3.25X t t t    University of Ghana http://ugspace.ug.edu.gh 27 The characteristic function of a general Cauchy distribution is given as      exp , 3.26X t it t t     The Cauchy distribution is a simple family of distributions for which the expected value (and other moments) do not exist. Secondly, the family is closed under sums of independent variables, and hence is an infinitely divisible family of distributions. It is a symmetrical ‘bell­shaped’ function like the Gaussian distribution, but it differs from that in the behaviour of their tails: the tails of the Cauchy density decrease as 2.x 3.2.3 The Gaussian (Normal) Distribution The most popular distribution in various physical and engineering applications is the Gaussian or Normal distribution. A random variable X has a Normal distribution with parameters    2 and mean Variance  if X has probability density function (pdf) given as:       2 2 1; , exp , - < , 0 and 3.2722 xf x x                  The distribution function ( )F x of the normal distribution is given as   2 2 1 ( ) exp 22 x t F x dt                     2 3.28 1 = 1 , - , > 0 and 2 x x erf x                   University of Ghana http://ugspace.ug.edu.gh 28 The random variable X having a standard normal distribution is given as:    21 exp , - < < 3.2922 xf x x        The distribution function ( )F x of the standard normal distribution is given as:       2 1 exp , - < < 22 3.30 = x t F x dt x x              The characteristic functions of both the standard normal distribution and the normal distribution are given respectively as:       2 21 2 exp , 2 exp ( ) t t t i t t              The expected value of X is denoted as µ and is called the location parameter and the variance of X denoted as 2 is known as the shape parameter. The Gaussian distribution is symmetrical ‘bell­shaped’ about the mean ( ) and it has the mean, mode and median been equal. University of Ghana http://ugspace.ug.edu.gh 29 3.3 Probability-Probability (P-P) Plot The probability-probability plot is a graphical technique for assessing whether or not a data set follows a specified theoretical distribution. The data are plotted against a theoretical distribution in such a way that the points should form approximately a straight line. Departures from this straight line indicate departures from the specified distribution. P-P plots tend to magnify deviations from the proposed distribution in the middle. The probability-probability plot is constructed using the theoretical cumulative distribution function, ( )iF x , where 1, 2, 3, ..., i n , of the specified model. The values in the sample of the data are ordered from the smallest to the largest and are denoted as; (1) (2) (3) ( ), , , ... , nX X X X. For 1, 2, 3, ..., i n , ( )( )iF x is plotted against 12i n     . The P-P plot was used in this study to illustrate how the returns of financial data fit the Gaussian distribution and the stable  distribution. 3.4 Quantile-Quantile (Q-Q) Plot The quantile-quantile (Q-Q) plot is a diagnostic graph of the input (observed) data values plotted against the theoretical (fitted) distribution quantiles. Both axes of this graph are in units of the input data set. The plotted points should be approximately linear if the specified theoretical distribution is the correct model. The quantile-quantile plot is more sensitive to the deviances from the theoretical distribution (normal) in the tails and tends to magnify deviation from the proposed distribution on the tails. The Q-Q plot is constructed using the theoretical cumulative distribution function, ( )F x , of the specified theoretical model or distribution. The values in the sample of the data are University of Ghana http://ugspace.ug.edu.gh 30 ordered from the smallest to the largest and are denoted as (1) (2) (3) ( ), , , ... , nX X X X. For 1, 2, 3, ..., i n , ( ) 'iX s are plotted against the inverse cumulative distribution function; 11 2iF n      . The Q-Q plot was used in this study to graphically assess goodness of fit of the returns of financial data to the Gaussian distribution and the stable  distribution. 3.5 Goodness-of-Fit Tests The goodness of fit (GOF) tests measures the compatibility of a random sample with a theoretical probability distribution function. In other words, a test for goodness of fit usually involves examining a random sample from some unknown distribution in order to test the null hypothesis that the unknown distribution function is in fact a known, specified distribution. The general procedure consists of defining a test statistic which is some function of the data measuring the distance between the hypothesised distribution and the data, and then calculating the probability of obtaining data which have a still larger value of this test statistic than the value observed. Assuming the hypothesis is true, this probability is called the confidence level. 3.5.1 Anderson-Darling Test The Anderson-Darling procedure is a general test to compare the fit of an observed cumulative distribution function to an expected cumulative distribution function. The Anderson-Darling test is used to test if a sample of data came from a population with a specific distribution. It is a modification of the Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than the K-S test does. The K-S test is distribution free in the sense that the critical values do not depend on the specific distribution being tested but the Anderson-Darling test makes use of the specific distribution in calculating the critical University of Ghana http://ugspace.ug.edu.gh 31 values. This has the advantage of allowing a more sensitive test and the disadvantage that critical values must be calculated for each distribution. The Anderson-Darling tests the hypothesis: 0 1 : The data follow the specified distribution. : The data do not follow the specified distribution. H H The Anderson-Darling test statistic ( 2A ) is defined as;    2 1 1 1 (2 1) ln ( ) ln(1 ( )) 3.31 n i n i i A n i F X F Xn        where F is the cumulative distribution function (cdf) of the specified distribution and iX is the ordered data. The hypothesis regarding the distributional form is rejected at the chosen significance level ( ) if the test statistic, 2 ,A is greater than the critical value obtained from a table. 3.5.2 Kolmogorov-Smirnov Goodness-of-Fit Test The Kolmogorov-Smirnov (K-S) goodness of fit test is a nonparametric test used to assess whether a sample comes from a population with a specified distribution. A hypothesis test involves calculation of a test statistic from the data and the probability of obtaining a value at least as large as a tail area if the correct distribution is chosen. The K-S test is based on the empirical cumulative distribution function (ECDF), it measures the Supremun distance between the cumulative distribution function of the theoretical distribution and the empirical distribution function, over all the sample points. The K-S test is distribution free since its critical values do not depend on the specific distribution being tested. The K-S test is relatively insensitive to differences in the tails but more sensitive to points near the median of the distribution. University of Ghana http://ugspace.ug.edu.gh 32 Definition 4 Let 1 2 3, , , ... , nX X X X be a random sample. The empirical distribution function ( )S x is a function of X , which equals the fraction of 'iX s that are less than or equal to X for each iX , ,iX       1 1( ) 3.32 i n x x i S x In   Where  ( )ix xI x is an indicator function and    0, if ( ) 3.331, if i i x x i x xI x x x    The empirical distribution function ( )S x is useful as an estimator of ( )F x , the unknown distribution function of the 'iX s . We compare the empirical distribution function ( )S x with the hypothesized distribution function *( )F x to investigate if there is good fit. Conover (1999) states that, Kolmogorov in 1933, suggested the test statistic as be the greatest (denoted by “sup” for Supremum) vertical distance between ( )S x and *( )F x , and is define as;  *( ) ( ) 3.34sup i i i D F x S x  For testing the hypothesis, * 0 * 1 : ( ) ( ), for all - : ( ) ( ), for at least one value of H F x F x x H F x F x x       University of Ghana http://ugspace.ug.edu.gh 33 The Kolmogorov-Smirnov goodness of fit test is used in this study to test the goodness of fit of the financial data to the Normal, Cauchy and stable  distributions. 3.5.3 Chi-square Goodness of Fit Test The Chi-square goodness of fit test is the test which makes a statement or claim concerning the nature of the distribution for the whole population and it is a parametric test. The data in the sample is examined in order to see whether this distribution is consistent with the hypothesized distribution of the population or not. One way in which the Chi-square goodness of fit test can be used is to examine how closely a sample matches a population of known distribution. The Chi-Squared test is used to determine if a sample comes from a population with a specific distribution. This test is applied to binned data, so the value of the test statistic depends on how the data is binned. Although there is no optimal choice for the number of bins ( ),k there are several formulas which can be used to calculate this number based on the sample size ( )N . The value of k can be determined by the empirical formula;  21 log 3.35k N  The data can be grouped into intervals of equal probability or equal width. The first approach is generally more acceptable since it handles peaked data much better. Each bin should contain at least 5 or more data points, so certain adjacent bins sometimes need to be joined together for this condition to be satisfied. University of Ghana http://ugspace.ug.edu.gh 34 Definition 5 The Chi-Square statistic is defined as;     2 2 1 , 3.36 k i i i i O E E   where iO is the observed frequency for bin i , and iE is the expected frequency for bin i calculated by 2 1[ ( ) ( )],i iiE N F x F x  where F is the cumulative distribution function (cdf) of the probability distribution being tested, N the total sample size and 1 2, x x are the limits for bin i . The test statistic is used for testing the hypothesis; 0 1 : The data follow the specified distribution. : The data do not follow the specified distribution. H H The hypothesis regarding the distributional form is rejected at the chosen significance level ( ) if the test statistic is greater than the critical value defined as 2,k c  , where ( )k c is the degrees of freedom and 1,c u  where u is the number of estimated parameters from the sample. 3.5.4 Shapiro-Wilk Test The Shapiro-Wilk test was proposed in 1965 by Shapiro and Wilk, for testing a random sample, 1 2, ,..., nX X X comes from (specifically) a normal distribution. The test statistic is denoted as W and small values of W are evidence of departure from normality and University of Ghana http://ugspace.ug.edu.gh 35 percentage points for the W statistic are obtained via Monte Carlo simulations and were reproduced by Pearson and Hartley (1972). The W statistic is defined as follows:     2 ( ) 1 2 1 , 3.37 n i i i n i i a x W x x            where ( )ix are the ordered sample values and 'ia s are constants generated from the means, variances and covariances of the order statistics of a sample of size n from a normal distribution. Let  1 2 3 ( ), , , ..., , where ( )n i im m m m m m E X   and V is x n n covariance matrix of ij where ( ) ( )Cov( , ).ij i jX X  The vector a of the weights 'ia s from eqn. (3.37) is defined as:     121 1 1 3.38a m V m V V m        The null hypothesis that the random sample comes the normal distribution is rejected, if the test statistic W is less than or equal tow , the critical value for the significance level ; the probability of type I error. 3.6 Estimation the Parameters of  - Stable Distribution This section discusses the methods of estimating the parameters of stable  distribution by the use of sample information. It is natural to expect that if a sample is at all representative of the population, one can use it to make an estimate that is better than a sheer guess (Lindgren, 1993). University of Ghana http://ugspace.ug.edu.gh 36 The lack of known closed-form density functions of   stable distribution complicates statistical inference estimation of stable distributions. For instance, maximum likelihood (ML) estimates have to be based on numerical approximations or direct numerical integration of the formulas presented in Section 3.1. Consequently, ML estimation is difficult to implement and time consuming for samples encountered in modern finance. However, there are also other numerical methods that have been found useful in practice are sample quantile and empirical characteristic function (Borak et al., 2010). The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. The estimator takes the measured data as input and produces an estimate of the parameters. It is also preferable to derive an estimator that exhibits optimality – achieving minimum average error over some class of estimators, for example, a minimum variance unbiased estimator. In this case, the class is the set of unbiased estimators, and the average error measure is variance (average squared error between the value of the estimate and the parameter). 3.6.1 Maximum Likelihood (ML) Estimation Method If a random sample is available from a population for which the probability distribution is known except for the value of one parameter 1 2, where = ( , , ..., ).k     Each specific value of  determines one particular distribution from among the large number of theoretically possible distribution. The estimation of  is therefore equivalent to selecting that particular distribution which applies to the population that produced the sample (Odoom, 2007). University of Ghana http://ugspace.ug.edu.gh 37 The Maximum Likelihood Principle suggests that the criterion for making the selection of an estimate should be the probability (or likelihood) which a particular distribution can produce the given sample. Thus, the distribution with the highest probability of producing the sample must be taken as the appropriate distribution that produced the sample. The value of  for that distribution is the Maximum Likelihood Estimate of the unknown  (Odoom, 2007). The Maximum Likelihood Estimation (MLE) technique is asymptotically consistent, that is, as the sample size gets larger, the estimates converge to the right values. It is asymptotically efficient, it produces the most precise estimates and asymptotically unbiased, that is, for large samples one expects to get the right value on average. Let  1 2, ,..., nX X X X be an i.i.d.   stable random sample of size n , then the maximum likelihood (ML) estimate of the parameter vector   , , ,      is obtained by maximizing the log-likelihood function:     1 ( ; ) log ; , 3.39 n i i L x f x    where  ;if x  is the   stable pdf. By maximizing ,L the maximum likelihood estimators (MLE) of ( , , , )     are the simultaneous solutions of four (4) equations such that:  0, where ( , , , ) 3.40L        University of Ghana http://ugspace.ug.edu.gh 38 In general, we do not know the explicit form of the   stable density and have to approximate it numerically. However, all of them have an appealing common feature – under certain regularity conditions that the ML estimator is asymptotically normal with the variance specified by the Fisher information matrix (DuMouchel, 1973). The latter can be approximated either by using the Hessian matrix arising in maximization or, as in Nolan (2001), by numerical integration. DuMouchel (1971) developed an approximate ML method, which was based on grouping the data set into bins and using a combination of means to compute the density (FFT for the central values of x and series expansions for the tails) to compute an approximate log-likelihood function. This function was then numerically maximized. 3.6.2 Sample Quantile Method Fama and Roll (1971), were the first to provide a very simple estimates for parameters of symmetric ( 0, = 0)  stable laws with 1.  A decade later McCulloch (1986) generalized their method and provided consistent estimators of all four stable parameters (with the restriction 0.6  ). McCulloch (1986) define the estimators as:     0.95 0.05 0.75 0.25 0.95 0.05 0.50 0.95 0.05 3.41 2 3.42 x x x x x x x x x             0.75 0.25 = 3.43x x   Where fx denotes the thf  population quantile and the ( )ix are ordered in ascending order and the fx matched, so that ( , , , )( ) ( ; ).fS x f x     The statistics and    University of Ghana http://ugspace.ug.edu.gh 39 are functions of and   only, i.e. they are independent of both and .  This relationships are inverted and the parameters and   is viewed as functions of and :    1 2( , ) and = ( , ) 3.44           Substituting and    by their sample values and applying linear interpolation between values found in tables given by McCulloch (1986) yields estimators ˆˆ and   . The scale parameters, ( ) , is also estimated in a similar way and finally, the location parameter, ( ), is estimated using the sample mean   when > 1 .x  However, due to the discontinuity of the characteristic function (CF) for = 1 and 0  , in the representation of stable distributions, this procedure is much more complicated (Borak et al., 2010). 3.6.3 Empirical Characteristic Function Method Press (1972) was the first to use the empirical characteristic function in the context of statistical inference for stable laws. He proposed a simple estimation method for all four parameters, called the method of moments, based on transformations of the characteristic function (CF). However, the convergence of this method to the population values depends on the choice of four estimation points, whose selection is problematic. Given an i.i.d. random sample  1 2, ,..., nX X X X of size n , the empirical characteristic function is define:     1 1ˆ = exp( ) 3.45 n i i t itxn  University of Ghana http://ugspace.ug.edu.gh 40 Since ˆ( )t is bounded by unity and all moments of ˆ( )t are finite and, for any fixed t , it is the sample average of i.i.d. random variables  exp .iitx Hence, by the law of large numbers, ˆ( )t is a consistent estimator of the characteristic function ( )t . Koutrouvelis (1980) presented a much more accurate regression-type method which starts with an initial estimate of the parameters and proceeds iteratively until some prespecified convergence criterion is satisfied. Each iterations, consists of two weighted regression runs. The number of points to be used in these regressions depends on the sample size and the starting values of . Typically not more than two or three iterations are needed and the speed of the convergence depends on the initial estimates and the convergence criterion. The regression method is based on the following observations concerning the characteristic function ( )t (Borak et al., 2010). Koutrouvelis (1980), derive the regression equation for estimating the parameters from the stable characteristic function as:      2log log ( ) log 2 log 3.46t t     The real and imaginary parts of ( )t for 1  are given by:                   exp cos tan , 3.47 2 = exp sin sign tan , 3.48 2 t t t t sign t t t t t t                                    University of Ghana http://ugspace.ug.edu.gh 41 Apart from considerations of principal values, we have:     { ( )}arctan = tan 3.49 ( ) 2 t t sign t tt        The equation (3.46) depends only on and   and, these two parameters can be estimated by regressing  2 = log log ( ) on logny t t   in the model:  , 3.50k k ky m     where kt is an appropriate set of real numbers, log(2 ),m  and k denotes an error term. Koutrouvelis (1980) proposed to use , 1, 2, ...., ;25k kt k N  with N ranging between 9 and 134 for different values of  and sample sizes. Once ˆ ˆ and   have been obtained and and   fixed at these values, estimates of and   are now obtained using (3.49). Next, the regressions are repeated with ˆ ˆˆ ˆ, , and     as the initial parameters. The iterations continue until a prespecified convergence criterion is satisfied. Koutrouvelis suggested to use Fama and Roll’s (1971) formula to estimate  and the 25% truncated mean ( X ) for initial estimates of and   , respectively. Koutrouvelis (1980) method was slower in estimating the parameters and therefore Kogon and Williams (1998) eliminated Koutrouvelis iteration procedure and simplified the regression method using McCulloch’s (1986) method with the continuous representation of the Characteristic function instead of the classical one. They used a fixed set of only ten University of Ghana http://ugspace.ug.edu.gh 42 (10) equally spaced frequency points kt and in terms of computational speed, their method was favourably faster than the original regression method (Borak et al., 2010). Borak et al. (2010) stated that the sample quantile method is fast and simple than the other two methods but it is the least accurate of all. While the maximum likelihood estimation method is the slower and complex of all but much better, in terms of accuracy and computational time. The study used a program known as STABLE with R statistical software package to estimate all the parameters of stable distribution  , under the three different estimating methods discussed in this section. The next chapter present data analysis and discussions of the study. 3.7.1 Central Limit Theorem Feller (1971) states that if the random variables 1 2, ,..., nX X X are independent and identically distributed with standard deviation   , then normalized sum nS n is approximately normal with standard deviation  , if n is sufficiently large.   1 where 3.51 n n j j S x   Dependent variables do not satisfy the conditions for the simplest Central Limit Theorem, even if they are stationary and have finite variance. This is illustrated by the extreme case where the 'X s are exact duplicates. In this situation, the normalized sums are identically zero, or grow to (plus or minus) infinity with the sample size. However, the independence assumption can be substantially relaxed while preserving the limiting normal distribution (Goldberg, Hayes, Barbieri, Dubikovsky & Gladkevich, 2009). University of Ghana http://ugspace.ug.edu.gh 43 Mathematically, the variables 1 2, ,...Y Y constitute a martingale if the expected value of tY conditional on information at time 1t  is 1tY  ;  1 1t t tE Y Y  (Hall & Heyde,1980). Let tX be the variable representing changes in value; i.e. tX is the difference of 1t tY Y  , then Jacod and Protter (2003) states the Martingale Central Limit Theorem as follows: If the random variables 1 2, ,..., nX X X satisfy     1 2 2 1 3 1 ( ) 0 ( ) 3.52 ( ) , t t t t t t i E X ii E X iii E X                   then the normalized sum nS n is approximately normal with standard deviation  if n is sufficiently large. 3.7.2 Generalized Central Limit Theorem Let 1 2, ,..., nX X X be independent identically distributed random variables with the distribution function  XF x satisfying the conditions;     1 , , X X F x cs x F x d x x        with 0  . Then there exist sequences and 0n na b  such that the distribution of the centred and normalized sum   1 1 , 3.53 n n i n in Z X ab       weakly converges to the stable distribution with parameters , 2 2, >2      and c d c d   University of Ghana http://ugspace.ug.edu.gh 44       and as ; ; , 3.54 n A Z n F x G x     where  ; ,AG x   is the distribution function of the stable distribution. 3.7.3 Infinite Variance It is commonly assumed that financial returns have finite variance. Even if this assumption holds, the squares of financial returns may have infinite variance- or equivalently, financial returns may have an infinite fourth moment. An extension of the Central Limit Theorem addresses this situation, although its conclusion differs from the previous versions. If the 'siX are identical and independent with infinite variance, there may be a limiting distribution for appropriately scaled sums, and it is the stable  distribution (Goldberg et al., 2009). 3.7.4 Infinite Variance Central Limit Theorem If the variables 1 2, ,..., nX X X are independent and identically distributed with tail index  0,2  , then the normalized sum 1nS n  is, under technical conditions, approximately   stable if n is sufficiently large (Embrechts, Kluppelberg & Mikosch, 1997). University of Ghana http://ugspace.ug.edu.gh 45 Chapter Four ANALYSIS AND DISCUSSIONS 4.0 Introduction This chapter basically discusses the return distributions of Ghanaian financial data from 2000 to 2011. The chapter used time series plots to illustrate to the progression and the volatility of four variables considered over the eleven years period. The various diagnostics tests (P-P plot, Q-Q plot, Z-Z plot and density plot) were used to graphically demonstrate goodness of fit to the Normal distribution, the Cauchy distribution and the Stable  distribution. The Goodness-of-Fit tests (Kolmogorov-Smirnov and Chi-square tests) were used to statistically test fitness of the Normal distribution, the Cauchy distribution and the Stable  distribution to the financial data. The financial data considered was modelled with Stable  distribution. 4.1 Ghana Stock Exchange All-Shares Index The sample size of 571 of GSE All-Shares index was used to perform these analysis and the data span from 2nd January, 2000 to 31st December, 2010. Due to the constant nature of the daily GSE All-Shares index, a weekly GSE All-Shares index was used. The weekly logarithm returns was computed using the formula below;        1log log 4.01t tX t Y Y  where tY is the weekly all-shares index at week t and 1tY  is the successive weekly all- shares index. University of Ghana http://ugspace.ug.edu.gh 46 Time series plot of GSE All-Shares index in Figure 1 displays a constant growth between 2000 and 2002, and then increased up to 2004 before it started decreasing till almost 2005. The process begun again and attained its maximum index at mid-way of 2008 to 2009. The rise and fall in GSE All-Shares index was as a result of high inflation rates during the general elections in 2004 and 2008. Figure 1: The GSE All-Shares index evolution from 02/01/00 to 31/12/10 Source: GSE (2000-2010) Time series plot of weekly GSE all-shares index Time G S E _ I n d e x 2000 2002 2004 2006 2008 2010 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 University of Ghana http://ugspace.ug.edu.gh 47 Times series plot of logarithm returns and returns of weekly GSE All-Shares index in Figure 2 illustrates that the Ghana stock market indices are not volatile and dynamic as emerging markets of the developed countries. The plot shows that between 2008 and 2011, the market was a little volatile. The volatility of the actual returns and the logarithm returns is the same as displayed on Figure 2. Figure 2: Logarithm returns of GSE All-Shares index Source: GSE (2000-2010) - 0 . 0 4 0 . 0 0 0 . 0 4 L o g r e t u r n s - 5 0 0 0 5 0 0 2000 2002 2004 2006 2008 2010 R e t u r n s Time Time series plot of weekly returns of GSE all-shares index University of Ghana http://ugspace.ug.edu.gh 48 Figure 3 displays a histogram plot with fitted line of the data and Normal fit to the log returns of the weekly GSE All-Shares index. The plot illustrates that, the log returns is highly picked and heavy tail than the Normal fit which conformed to the studies in the literature that, assets returns are heavy-tailed. The Figure 3 graphically demonstrates that the logarithm returns of GSE All-Shares index are not normally distributed. It is observed that the logarithm returns are in the neighbourhood of zero. Figure 3: GSE All-Shares index from 02/01/00- 31/12/10 Source: GSE (2000-2010) NB: X denote the weekly logarithm returns of GSE All-Shares index Histogram plot and Normal density curve of GSE All-Shares Index logreturns; data = green, normal fit = red x D e n s i t y -0.05 0.00 0.05 0 5 0 1 0 0 1 5 0 mean 0; sd 0; N 571 University of Ghana http://ugspace.ug.edu.gh 49 The density plots in Figure 4 demonstrates that, the  -stable distribution produces the best fit to the logarithm returns of GSE All-Shares index than the normal and Cauchy fits graphically. The - stable fit with parameters 1.005,  0.31, 0.002 and   =0.001 fit graphically best to the data than the Normal and Cauchy distributions. This displays graphically that, the weekly logarithm returns of GSE All-Shares index comes from leptokurtic distribution and heavy-tailed as proposed in the literature by Mandelbrot (1963) that the returns can be modelled by stable distributions with four parameters. Figure 4: The density plots of log returns of GSE All-Shares index Data = red line, Stable fit = green line, Normal fit = black line and Cauchy fit = blue line Source: GSE (2000-2010) NB: X denote the weekly logarithm returns of GSE All-Shares index -0.04 -0.02 0.00 0.02 0.04 0.06 0 5 0 1 0 0 1 5 0 Stable, Normal & Cauchy densities plot of logreturns of GSE all-shares index x d e n s i t y data stable fit University of Ghana http://ugspace.ug.edu.gh 50 The figures (Fig. 5, Fig. 6 & Fig. 7) are Q-Q and P-P plots of the normal distribution and the stable distribution and confirm that the weekly logarithm returns of GSE All-Shares index are not normally distributed but stable distributed with parameters 1.005 0.104, 0.31 0.135,     0.002 0.0002 and   =0.001 0.0002.  Figure 5 shows deviation of the data from normal fit at the tails, signifying that the GSE All- Shares index returns are heavy tailed. Figure 6 and Figure 7 illustrate a perfect fit of the logarithm returns of GSE All-Shares index to the - stable fit with the stable Q-Q plot displaying few outliers at the tails. The P-P and Q-Q plots of - stable fit illustrate the argument by Nolan (2003) that P-P plot allows comparison over the range of the data while Q-Q plot is not as satisfactory for comparing heavy tailed data and this is revealed by the outliers observed in Figure 6. Figure 5: The Normal Q-Q plot GSE All-Shares index Source: GSE (2000-2010) -3 -2 -1 0 1 2 3 - 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 Normal Q-Q Plot Theoretical Quantiles S a m p l e Q u a n t i l e s University of Ghana http://ugspace.ug.edu.gh 51 Figure 6: Stable Q-Q plot of GSE All-Shares index Source: GSE (2000-2010) Figure 7: The Stable P-P plot of GSE All-Shares index Source: GSE (2000-2010) -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 - 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 stable distribution with theta=(1.005,0.31,0.002,0.001) q u a n t i l e s o f d a t a qq-plot 0.0 0.2 0.4 0.6 0.8 1.0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 stable distribution with theta=(1.005,0.31,0.002,0.001) d a t a pp-plot University of Ghana http://ugspace.ug.edu.gh 52 Table 1 displays estimates of the four parameters of the stable fit and 95% confidence interval of the estimates. The index of stability (α) is approximately 1 for the maximum likelihood estimation method while the sample quantile method value of index of stability is 1.124 and empirical characteristic function method produces α as 1.174. The skewness estimates from the three methods are positive and that explains that the returns of GSE index are positively skew. The three methods approximately estimate the same value of the scale parameter (  ) to be 0.002 and different location parameter (  ). Table 1: Estimated parameters of the Stable distribution Stable Parameters Methods ˆ ˆˆ ˆ     MLE 1.005 0.104 0.31 0.135 0.002 0.0002 0.001 0.0002    S. Quantile 1.124 0.115 0.196 0.155 0.0023 0.0002 0.0007 0.0003    ECF 1.174 0.116 0.399 0.155 0.0023 0.0002 0.0009 0.0003    Source: GSE (2000-2010) Table 2 illustrates the Kolmogorov-Smirnov and Chi-Square goodness of fit tests of the maximum likelihood estimation, sample quantile and empirical characteristic function methods of estimating parameters of - stable distribution. The K-S test and Chi-square test shows that maximum likelihood estimates of the stable distribution fit perfectly well to the logarithm returns of GSE all-Shares index. The K-S test produces a p-value of 0.138 and Chi-square test p-value is 0.20, which means that, the hypothesis that logarithm returns of GSE All-Shares index is α-stable distributed with maximum likelihood estimation method is not rejected at 10% or 5% significance level. This implies that maximum likelihood estimation method produce more accurate estimates of the data. University of Ghana http://ugspace.ug.edu.gh 53 Table 2: Goodness of fit tests Statistical Tests Methods K-S Chi-Square MLE 2 0.048 3.28 p 0.138 p 0.20 D value value       Sample Quantile 2 0.084 21.36 p 0.0006 p 0.005 D value value       ECF 2 0.059 6.75 p 0.035 p 0.10 D value value       Source: GSE (2000-2010) The K-S and the Chi-square goodness of fit tests both rejected the hypothesis that the logarithm returns of GSE all-shares index is α-stable distributed with sample quantile estimates of stable parameters. Both goodness of fit test estimated an approximate p-value of 0.0, which means that α-stable fit with sample quantile estimates does not fit successfully to the GSE All-Shares index. The rejection of the sample quantile method implies that the estimates are not accurate for modelling log returns of GSE All-Shares index data. The K-S test rejected α-stable fit to the logarithm returns of GSE All-Shares index with empirical characteristic function estimation method at 5% significance level. But the Chi-square goodness of fit test says otherwise. The Chi-square test estimates a p- value of approximately 0.10, which means that the empirical characteristic function estimation method of fitting the log returns of GSE All-Shares index to α-stable fit is not significant at 5% significance level and the hypothesis is not rejected. The table 2 revealed that the MLE method produces the best and most accurate estimates and next to it is empirical characteristic function estimates and the sample quantile method produces the worst estimates as indicated in the methodology. University of Ghana http://ugspace.ug.edu.gh 54 Table 3 demonstrates the goodness of fit of the logarithm returns of GSE All-Shares index to the three distributions under study. The four tests for normality considered in this study (Kolmogorov-Smirnov, Chi-square, Anderson-Darling and Shapiro-Wilk) all rejected the claim that logarithm returns of GSE All-Shares index is normally distributed at even 1% significance level. Since Anderson-Darling and Shapiro-Wilk tests cannot be used to test goodness of fit for the Cauchy and the α-stable distributions, only K-S and Chi-square tests are employed. The K-S test and Chi-square test estimated p-values of 0.0 and 0.017 respectively for the Cauchy fit. This implies that both tests (K-S and Chi-square) rejected the claim that logarithm returns of GSE All-Shares index is Cauchy distributed. Table 3: Goodness of fit tests for Normal, Cauchy and Stable distributions Cases K-S Chi-Square Anderson-Darling Shapiro-Wilk Normal fit 2 2 0.18 408.26 A = 36.78 W = 0.78 p- 0.0 p- 0.005 p- 0.005 p- 0.0 D value value value value       Cauchy fit 2 0.486 32432 p- 0.0 p- 0.017 D value value     Stable fit 2 0.048 3.28 p- 0.138 p- 0.20 D value value     Source: GSE (2000-2010) The K-S and Chi-square goodness of fit tests consistently fail to reject the claim that logarithm returns of GSE all-shares index is α-stable distributed at 5% or 10% significance level. Both tests estimated the p-values as 0.138 for K-S test and approximately 0.20 for Chi-square test. These results conformed to the results obtained by Alfonso et al. (2011) and Xu et al. (2011) that assets returns are α-stable distributed and the two goodness of fit tests agreed with the graphical tests in Figures 4, 6 and 7. University of Ghana http://ugspace.ug.edu.gh 55 4.2 Currency Exchange Rates This section investigates the return distribution of the three major exchange rates (US dollar to Ghana cedi, Britain pounds sterling to Ghana cedi and European Euro to Ghana cedi). The sample sizes of the three exchange rates are as follows; 02/01/01 to 31/12/11 of 524 weekly data points for both US dollar and British Pounds to the Ghana cedi and 02/01/02 to 31/12/11 of 474 data points for the European Euro to the Ghana cedi. The weekly exchange rates are used due to the fact that the daily exchange rates are not volatile enough to be modelled with - stable distributions. The weekly logarithm returns was calculated using equation (4.01) in section 4.1. The figures (Fig.8 - Fig.13) display time series plots of the respective exchange rates. Figure 8 illustrates a progressive weekly USD/GHC exchange rate over the period of study and it shows that, there was steep increase between 2008 and 2009. This may be due to the 2008 general election held in December which led to high inflation indices. Figure 8: USD/GHC progression for the 11 years period Source: Bank of Ghana (2001-2011) Time series plot of weekly USD/GHC exchange rates Time w e e k l y . U S D _ G H C 2002 2004 2006 2008 2010 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 University of Ghana http://ugspace.ug.edu.gh 56 Figure 9 also demonstrates that, there were greater positive returns than negative returns over the 11 years period of study and in risk management it is good to have equal percentage of positive and negative returns. It shows that the volatility of the weekly USD/GHC exchange rates exhibits large extreme returns as compared to the developed markets exchange rates in the literature. The simple returns and the logarithm returns of USD/GHC exchange rates are almost the same in terms of volatility displayed in Figure 9. Figure 9: USD/GHC of log returns of volatility over the 11 years period Source: Bank of Ghana (2001-2011) - 0 . 0 0 5 0 . 0 0 5 L o g r e t u r n s - 0 . 0 3 - 0 . 0 1 0 . 0 1 0 . 0 3 2002 2004 2006 2008 2010 2012 R e t u r n s Time Time series plot of weekly USD/GHC exchange rates University of Ghana http://ugspace.ug.edu.gh 57 Figure 10 displays the progressive growth of the 11 years weekly British pound sterling to Ghana cedi starting from 02/01/2001-31/12/2011. The graph shows a continuous increase over the period with fluctuations than the USD/GHC exchange rate. Figure 10: GBP/GHC progression for the 11 years period Source: Bank of Ghana (2001-2011) Time series plot of weekly GBP/GHC exchange rates Time G B P w e e k l y 2002 2004 2006 2008 2010 2012 1 . 0 1 . 5 2 . 0 2 . 5 University of Ghana http://ugspace.ug.edu.gh 58 The log-returns and simple returns of the GBP/GHC weekly exchange rates are shown graphically on Figure 11 of the entire period of study; 2/01/01-31/12/11. It is observed that logarithm returns of GBP/GHC weekly exchange rates are uniformly distributed on average between -0.01 to 0.01 with exception of some few extreme returns which may be due to the inefficiency of the Ghanaian market. Figure 11: GBP/GHC of log returns of volatility over the 11 years period Source: Bank of Ghana (2001-2011) - 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 L o g r e t u r n s - 0 . 1 0 0 . 0 0 0 . 1 0 2002 2004 2006 2008 2010 2012 R e t u r n s Time Time series plot of weekly returns of GBP/GHC exchange rates University of Ghana http://ugspace.ug.edu.gh 59 Figure12 displays the evolution of the 10 years weekly European Euro to Ghana cedi. The graph shows a continuous increased with fluctuations over the period with approximately the same trend as the GBP/GHC exchange rate. Figure 12: EUR/GHC progression for the 10 years period Source: Bank of Ghana (2002-2011) Time series plot of weekly EUR/GHC exchange rates Time E U R . G H C 2002 2004 2006 2008 2010 2012 1 . 0 1 . 5 2 . 0 University of Ghana http://ugspace.ug.edu.gh 60 The weekly EUR/GHC exchange rate returns of the actual returns and log-returns are shown graphically on Figure 13 and demonstrating an average logarithm returns between -0.015 to 0.015 over the entire period of study; 2/01/02-31/12/11. The simple and logarithm returns of the euro exchange rate parades extreme returns equal to the returns of British pounds exchange rate. Figure 13: EUR/GHC log returns of volatility over the 10 years period Source: Bank of Ghana (2002-2011) - 0 . 0 3 - 0 . 0 1 0 . 0 1 0 . 0 3 L o g r e t u r n s - 0 . 1 0 0 . 0 0 0 . 1 0 2002 2004 2006 2008 2010 2012 R e t u r n s Time Time series plot of weekly returns of EUR/GHC exchange rates University of Ghana http://ugspace.ug.edu.gh 61 Figure 14 displays a histogram plot with fitted line of the data and Normal fit to the log returns of the weekly USD/GHC exchange rate. The plot illustrates that, the log returns is of USD/GHC exchange rate is more picked and heavy tail than the Normal fit and this parades that the weekly USD/GHC exchange rate log returns come from a leptokurtic distribution family. Figure 14: Histogram plot of log returns of USD/GHC Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of USD/GHC exchange rate Histogram plot and Normal density curve of USD/GHC exchange rate logreturns; data = green, normal fit = red x D e n s i t y -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 mean 0; sd 0; N 524 University of Ghana http://ugspace.ug.edu.gh 62 Figures 15 and 16 also demonstrate that, the weekly logarithm returns of GBP/GHC and EUR/GHC exchange rates do not fit to the normal distribution but come from a leptokurtic distribution family like the USD/GHC exchange rate logarithm returns. Figure 15: Histogram plot of log returns of GBP/GHC Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of GBP/GHC exchange rate Histogram plot and Normal density curve of GBP/GHC exchange rate logreturns; data = green, normal fit = red x D e n s i t y -0.04 -0.02 0.00 0.02 0.04 0 5 0 1 0 0 1 5 0 mean 0; sd 0; N 524 University of Ghana http://ugspace.ug.edu.gh 63 Figure 16: Histogram plot logarithm returns of EUR/GHC Source: Bank of Ghana (2002-2011) NB: X denote the weekly logarithm returns of EUR/GHC exchange rate The figures 17, 18 and 19 illustrate the density plots of stable, normal and Cauchy fits to the weekly logarithm returns of the three exchange rates understudy. The three density plots clearly demonstrate graphically that, the return distribution of the three exchange rates are neither normally distributed nor Cauchy distributed but are  -stable distributed. The  -stable fitted well to the three exchange rates with the following parameters; USD/GHC exchange rate estimates are: 1.073, 0.394,    0.001 and =0.0  , GBP/GHC exchange rate estimates are: 1.289, 0.173,   0.002 and =0.0  and EUR/GHC exchange rate estimates are: 1.236, 0.234,   0.002 and =0.001  . Histogram plot and Normal density curve of EUR/GHC exchange rate logreturns; data = green, normal fit = red x D e n s i t y -0.04 -0.02 0.00 0.02 0.04 0 5 0 1 0 0 1 5 0 mean 0; sd 0; N 474 University of Ghana http://ugspace.ug.edu.gh 64 Figure 17: The density plots of log returns of the data USD/GHC = red line, Stable fit = green line, Normal fit = black line and Cauchy fit = blue line; Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of USD/GHC exchange rate -0.005 0.000 0.005 0.010 0 1 0 0 2 0 0 3 0 0 4 0 0 Stable, Normal & Cauchy densities plot of logreturns of USD/GHC exchange rates x d e n s i t y data stable fit University of Ghana http://ugspace.ug.edu.gh 65 Figure 18: The density plots of log returns of the data GBP/GHC = red line, Stable fit = green line, Normal fit = black line and Cauchy fit = blue line; Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of GBP/GHC exchange rate -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0 5 0 1 0 0 1 5 0 Stable, Normal & Cauchy densities plot of logreturns of GBP/GHC exchange rates x d e n s i t y data stable fit University of Ghana http://ugspace.ug.edu.gh 66 Figure 19: The density plots of log returns of the data EUR/GHC = red line, Stable fit = green line, Normal fit = black line and Cauchy fit = blue line; Source: Bank of Ghana (2002-2011) NB: X denote the weekly logarithm returns of EUR/GHC exchange rate -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0 5 0 1 0 0 1 5 0 Stable, Normal & Cauchy densities plot of logreturns of EUR/GHC exchange rates x d e n s i t y data stable fit University of Ghana http://ugspace.ug.edu.gh 67 Figure 20 displays the normal Q-Q plot of log-returns of USD/GHC exchange rate with large deviation away from the normal distribution. The normal q-q plot shows a significant deviation away from the tails of the theoretical quantiles of the normal fit and this revealed that the USD/GHC weekly log-returns are not normally distributed. Figure 20 : Normal Q-Q plot of USD/GHC exchange rate Source: Bank of Ghana (2001-2011) -3 -2 -1 0 1 2 3 - 0 . 0 0 5 0 . 0 0 0 0 . 0 0 5 0 . 0 1 0 Normal Q-Q Plot Theoretical Quantiles S a m p l e Q u a n t i l e s University of Ghana http://ugspace.ug.edu.gh 68 Figure 21 illustrates the - stable fit diagnostic tests for log-returns of the weekly USD/GHC exchange rate. Figure 21 displays four graphs of - stable fit namely; density plot of - stable fit to the data, Q-Q plot, P-P plot and Z-Z plot. The α-stable Q-Q plot illustrates a perfect fit of the quantiles of the weekly USD/GHC log-returns to the quantiles of the theoretical α-stable fit but with few outliers at the tails. The  -stable P-P plot displays a perfect fit of the probabilities of the USD/GHC weekly log returns to the theoretical probabilities of the α-stable distribution and finally the Z-Z plot also demonstrates a perfect fit of the inverse cumulative distribution of α-stable fit of the data to the theoretical inverse of normal cumulative distribution plotted on both axes of the P- P plot. Figure 21: The diagnostics tests of USD/GHC data Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of USD/GHC exchange rate -0.005 0.005 0 1 0 0 3 0 0 stable density plot x d e n s i t y data stable fit 0.0 0.4 0.8 0 . 0 0 . 4 0 . 8 stable distribution with theta=(1.073,0.394,0.001,0) d a t a pp-plot -0.10 0.00 0.10 0.20 - 0 . 0 0 5 0 . 0 0 5 stable distribution with theta=(1.073,0.394,0.001,0) q u a n t i l e s o f d a t a qq-plot -3 -1 1 2 3 - 2 - 1 0 1 2 z scores of stable distribution with theta=(1.073,0.394,0.001,0) z s c o r e s o f d a t a zz-plot University of Ghana http://ugspace.ug.edu.gh 69 Figure 22 displays the normal Q-Q-plot for the weekly log returns of GBP/GHC exchange rate showing deviation away from the normal distribution. The normal Q-Q plot shows a significant deviation away from the tails of the theoretical quantiles of the normal fit but forming a good fit at the centre of the theoretical quantiles. This revealed that the GBP/GHC weekly log-returns deviated away from the normally distribution but the average log-return will be equal to the normal distribution estimate. Figure 22: Normal Q-Q plot of GBP/GHC exchange rate Source: Bank of Ghana (2001-2011) -3 -2 -1 0 1 2 3 - 0 . 0 4 - 0 . 0 3 - 0 . 0 2 - 0 . 0 1 0 . 0 0 0 . 0 1 0 . 0 2 Normal Q-Q Plot Theoretical Quantiles S a m p l e Q u a n t i l e s University of Ghana http://ugspace.ug.edu.gh 70 Figure 23 displays the four - stable fit diagnostic tests for the weekly log returns of GBP/GHC exchange rate. The density plot of - stable fit to the data, Q-Q plot, P-P plot and Z-Z plot attest to the fact that weekly log-returns of GBP/GHC exchange rate perfectly fit to α-stable distribution but not normal distribution. Figure 23: The diagnostics tests of GBP/GHC data Source: Bank of Ghana (2001-2011) NB: X denote the weekly logarithm returns of GBP/GHC exchange rate -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0 5 0 1 0 0 1 5 0 stable density plot x d e n s i t y data stable fit 0.0 0.4 0.8 0 . 0 0 . 4 0 . 8 stable distribution with theta=(1.289,0.173,0.002,0.001) d a t a pp-plot -0.10 0.00 0.10 - 0 . 0 4 - 0 . 0 1 0 . 0 1 stable distribution with theta=(1.289,0.173,0.002,0.001) q u a n t i l e s o f d a t a qq-plot -3 -1 1 2 3 - 2 - 1 0 1 2 z scores of stable distribution with theta=(1.289,0.173,0.002,0.001) z s c o r e s o f d a t a zz-plot University of Ghana http://ugspace.ug.edu.gh 71 Figure 24 parades the normal Q-Q plot for the weekly log returns of EUR/GHC exchange rate showing a significant deviation from the tails of the theoretical quantiles of the normal distribution. The normal Q-Q plot shows a good fit at the centre of theoretical quantiles of the normal fit and this means that the average log return will be the same as the mean estimate computed from the normal distribution. The Q-Q plot shows that the log-returns of EUR/GHC weekly exchange rates are not normally distributed. Figure 24: Normal Q-Q plot of EUR/GHC exchange rate Source: Bank of Ghana (2002-2011) -3 -2 -1 0 1 2 3 - 0 . 0 3 - 0 . 0 2 - 0 . 0 1 0 . 0 0 0 . 0 1 0 . 0 2 0 . 0 3 Normal Q-Q Plot Theoretical Quantiles S a m p l e Q u a n t i l e s University of Ghana http://ugspace.ug.edu.gh 72 Figure 25 displays the four - stable fit diagnostic tests for the weekly log-returns of EUR/GHC exchange rate. The density plot of - stable fit to the data, Q-Q plot, P-P plot and Z-Z plot attest to the fact that weekly log-returns of EUR/GHC exchange rate perfectly fit to α-stable distribution but not normal distribution. The Q-Q plot displays outliers at the tails of the theoretical quantiles of - stable distribution, showing the demerit of Q-Q plot indicated by Nolan (2003). Figure 25: The diagnostics tests of EUR/GHC data Source: Bank of Ghana (2002-2011) NB: X denote the weekly logarithm returns of EUR/GHC exchange rate -0.03 0.00 0.02 0 5 0 1 0 0 1 5 0 stable density plot x d e n s i t y data stable fit 0.0 0.4 0.8 0 . 0 0 . 4 0 . 8 stable distribution with theta=(1.236,0.234,0.002,0.001) d a t a pp-plot -0.15 0.00 0.15 - 0 . 0 3 0 . 0 0 0 . 0 2 stable distribution with theta=(1.236,0.234,0.002,0.001) q u a n t i l e s o f d a t a qq-plot -3 -1 1 2 3 - 2 - 1 0 1 2 z scores of stable distribution with theta=(1.236,0.234,0.002,0.001) z s c o r e s o f d a t a zz-plot University of Ghana http://ugspace.ug.edu.gh 73 Tables 4, 5 and 6 display estimates of the four parameters of stable fit and 95% confidence interval of the estimates using the three methods (Maximum likelihood estimation, Sample quantile and Empirical Characteristic function) methods of estimating the  -stable parameters. Table 4 displays  -stable fit estimates of USD/GHC exchange rate, Table 5 displays  -stable fit estimates of USD/GHC exchange rate and Table 6 shows  -stable fit estimates of EUR/GHC exchange rate. Table 4: Estimated parameters of the Stable distribution USD/GHC Exchange Rate Stable Parameters Methods ˆ ˆˆ ˆ     MLE 4 5 4 51.073 0.114 0.394 0.147 6.6*10 7.62*10 2.3*10 8.9*10       S. Quantile 4 5 4 51.08 0.12 0.396 0.147 6.9*10 7.87*10 1.4*10 9.3*10       ECF 4 5 4 41.26 0.13 0.205 0.186 7.8*10 8.1*10 2.5*10 1.1*10       Source: Bank of Ghana (2001-2011) Table 5: Estimated parameters of the Stable distribution GBP/GHC Exchange Rate Stable Parameters Methods ˆ ˆˆ ˆ     MLE 3 4 4 41.289 0.129 0.173 0.193 1.8*10 1.8*10 6.1*10 2.5*10       S. Quantile 3 4 4 41.38 0.13 0.071 0.22 1.9*10 1.8*10 4.3*10 2.7*10       ECF 3 4 4 41.48 0.14 0.136 0.247 2.0*10 1.8*10 7.0*10 2.9*10       Source: Bank of Ghana (2001-2011) University of Ghana http://ugspace.ug.edu.gh 74 Table 6: Estimated parameters of the Stable distribution EUR/GHC Exchange Rate Stable Parameters Methods ˆ ˆˆ ˆ     MLE 3 4 4 41.236 0.126 0.234 0.18 2.02*10 2.1*10 6.9*10 2.8*10       S. Quantile 3 4 4 41.32 0.14 0.18 0.21 1.9*10 2.1*10 5.1*10 2.9*10       ECF 3 4 4 41.404 0.139 0.425 0.218 2.23*10 2.2*10 6.4*10 3.5*10       Source: Bank of Ghana (2002-2011) Tables 7, 8 and 9 below illustrate Kolmogorov-Smirnov and Chi-Square goodness of fit tests of the Maximum likelihood estimation, Sample Quantile and Empirical Characteristic function methods of estimating parameters of stable distribution of the weekly logarithm returns of the three exchange rates understudy. Table 7, illustrates that both the K-S test and Chi-square test fail to reject the claim that the Maximum likelihood estimates of the  -stable distribution perfectly fit to the logarithm returns of USD/GHC exchange rate. The K-S goodness of fit test rejected the Sample quantile method estimates of  -stable fit while the Chi-square goodness of fit test says otherwise. The Chi-square test shows a perfect fit of the Sample quantile method estimates of α-stable fit to the USD/GHC exchange rate data. The K-S and Chi-square goodness of fit tests illustrate a poor fit of  -stable estimates of the Empirical Characteristic function method to the logarithm returns of USD/GHC exchange rate. University of Ghana http://ugspace.ug.edu.gh 75 Table 7: Goodness of fit tests for USD/GHC exchange rates Statistical Tests Methods K-S Chi-Square MLE 2 0.037 7.85 p- 0.45 p- 0.05 D value value     Sample Quantile 2 0.069 6.25 p- 0.014 p- 0.10 D value value     ECF 2 0.08 18.06 p- 0.002 p- 0.005 D value value     Source: Bank of Ghana (2001-2011) Table 8, demonstrates that the K-S goodness of fit test shows a good fit of Maximum likelihood estimates of the  -stable distribution to the logarithm returns of GBP/GHC exchange rate at any significance level while the Chi-square goodness of fit test shows a poor fit of the Maximum likelihood estimates of the α-stable distribution to the logarithm returns of GBP/GHC exchange rate at 5% significance level. Both the K-S goodness of fit test and Chi-square goodness of fit test rejected the Sample quantile method estimates of  -stable fit at any significance level. The K-S test for Empirical Characteristic function estimates of  -stable fit shows a good fit at 1% significance level and a poor fit 5% significance level to the data while the Chi-square goodness of fit test shows a poor fit to the GBP/GHC exchange rate data at any significance level. University of Ghana http://ugspace.ug.edu.gh 76 Table 8: Goodness of fits test for GBP/GHC exchange rates Statistical Tests Methods K-S Chi-Square MLE 2 0.046 11.65 p 0.21 p 0.01 D value value       Sample Quantile 2 0.086 38.60 p 0.001 p 0.005 D value value       ECF 2 0.06 14.03 p 0.042 p 0.01 D value value       Source: Bank of Ghana (2001-2011) Table 9 demonstrates that K-S goodness of fit test fail to reject the hypothesis that the three methods (MLE, S. Quantile and ECF) of estimating parameters of  -stable distribution perfectly fit to the weekly logarithm returns of EUR/GHC exchange rate and this implies that the K-S test shows a good fit of the three estimating methods to the data at 5% significance level. The Chi-square goodness of fit test in the other hand shows a good fit to the Maximum likelihood estimates and a poor fit to the Sample Quantile and Empirical Characteristic function methods of α-stable fit. Table 9: Goodness of fit tests for EUR/GHC exchange rates Statistical Tests Methods K-S Chi-Square MLE 2 0.044 7.54 p 0.32 p 0.10 D value value       Sample Quantile 2 0.061 13.38 p 0.058 p 0.01 D value value       ECF 2 0.058 11.57 p 0.082 p 0.01 D value value       Source: Bank of Ghana (2002-2011) University of Ghana http://ugspace.ug.edu.gh 77 The maximum likelihood estimation method consistently demonstrate through the two goodness of fit tests that, it is superior and better estimator than the other two estimators (quantile and ECF) in modelling the returns of GSE All-Shares index and the three exchange rates. Therefore, the Maximum likelihood estimation produces the paramount estimating method for estimating the four parameters of α-stable distribution. Tables 10-12 demonstrate the goodness of fit of the logarithm returns the three exchange rates to the three distributions under study. The four tests for normality considered in this study (Kolmogorov-Smirnov, Chi-square, Anderson-Darling and Shapiro-Wilk) all reject the claim that logarithm returns of the three exchange rates (USD/GHC, GBP/GHC and EUR/GHC) are normally distributed even at 1% significance level. This means that the return distributions of Ghana exchange rates are not normally distributed and this conformed to (McCulloch, 1996; Rachev & Mittnik, 2000) findings. The K-S and Chi- square goodness of fit tests both rejected the claim that the three exchange rates returns are Cauchy distributed. The K-S goodness of fit test parades impeccable fit of the weekly logarithm returns of all the three exchange rates to the α-stable fit and the Chi-square goodness of fit test displays a good fit of USD/GHC and EUR/GHC data to α-stable fit at 5% significance level and good fit of GBP/GHC data to  -stable fit at 1% significance level. This implies that the Chi-square goodness of fit test is very sensitive of the deviation of the data to the theoretical distribution. University of Ghana http://ugspace.ug.edu.gh 78 Table 10: Goodness of fit tests for the distributions to USD/GHC data Cases K-S Test Chi-Square Anderson-Darling Shapiro-Wilk Normal fit 2 2 0.161 273.30 A =25.99 W= 0.86 p- 0.0 p- 0.005 p- 0.005 p- 0.0 D value value value value       Cauchy fit 2 0.497 260428 p- 0.0 p- 0.0 D value value     Stable fit 2 0.037 7.85 p- 0.45 p- 0.05 D value value     Source: Bank of Ghana (2001-2011) Table 11: Goodness of fit tests for the distributions to GBP/GHC data Cases K-S Test Chi-Square Anderson-Darling Shapiro-Wilk Normal fit 2 2 0.138 212.18 A = 20.09 W=0.834 p- 0.0 p- 0.005 p- .005 p- 0.0 D value value value value       Cauchy fit 2 0.492 269860 p- 0.0 p- 0.0 D value value     Stable fit 2 0.0461 11.65 p- 0.21 p- 0.01 D value value     Source: Bank of Ghana (2001-2011) Table 12: Goodness of fit tests for distributions to the EUR/GHC Exchange data Cases K-S Test Chi-Square Anderson-Darling Shapiro-Wilk Normal fit 2 2 0.148 191.21 A =17.99 W=0.864 p- 0.0 p- 0.005 p- 0.005 p- 0.0 D value value value value       Cauchy fit 2 0.491 223254 p- 0.0 p- 0.017 D value value     Stable fit 2 0.044 7.54 p- 0.32 p- 0.10 D value value     Source: Bank of Ghana (2002-2011) University of Ghana http://ugspace.ug.edu.gh 79 Table 13 displays the estimated parameters (mean, standard error and standard deviation) of the normal distribution of GSE All-Shares index, USD/GHC, GBP/GHC and EUR/GHC exchange rates. It also shows the estimated parameters of skewness and kurtosis of the four variables considered under this study. It is observed that average logarithm returns of the GSE all-shares index and the three exchange rates is approximately zero. The return distribution of all the four financial data are asymmetry, and the GSE all-shares index and USD/GHC exchange rate are positively skewed while the other two exchange rates data (GBP/GHC and EUR/GHC) are negatively skewed. The kurtoses of the four financial data under study are higher than the normal distribution which is 3 and this conformed to the literature that asset returns are heavy tails and asymmetry. Table 13: Estimated parameters of the Normal distribution Variable Mean SE Mean Standard Dev. Skewness Kurtosis GSEI 0.00175 0.00035 0.0084 0.09 13.6 USD/GHC 0.00074 0.000096 0.0022 1.00 4.18 GBP/GHC 0.0008 0.000214 0.0049 -0.61 12.65 EUR/GHC 0.00109 0.000261 0.0057 -0.04 7.48 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) This chapter presented the findings and discussions from the analysis of the study. The progression and the volatility of the financial time series data was displayed and the graphical density plots of the logarithm returns of the financial data illustrating the distribution of the returns of the four variables study was also presented. The diagnostics test was performed to demonstrate fitness to the Gaussian or normal, Cauchy and  -stable University of Ghana http://ugspace.ug.edu.gh 80 distributions and goodness of fit tests was also used to determine the best methods of estimating the four parameters of  -stable distribution. The analysis established that the return distribution of assets and exchange rates follow  -stable distribution and does not obey the Gaussian law. The next chapter presents summary, conclusion and recommendations of the research and also further studies areas. University of Ghana http://ugspace.ug.edu.gh 81 Chapter Five SUMMARY, CONCLUSION AND RECOMMENDATIONS 5.0 Introduction This chapter presents summary of the findings from the study, and recommends rational measures for stakeholders, financial analysts and investors. It again recommends further study possibility areas for future researchers. The chapter provides the concluding statements of the research based on the findings. 5.1 Summary The main purpose of this research was to investigate and model the return distributions of financial assets and exchange rates. The study considered four different financial data namely; Ghana stock exchange all-shares index from 2000 to 2010, US dollar to Ghana cedi exchange rate from 2001 to 2011, British pound sterling to Ghana cedi from 2001 to 2011 and Euro to Ghana cedi from 2002 to 2011, and the weekly indices and exchange rates were used for this study. The study revealed that all normality tests rejected the returns distribution of the financial data being normally distributed and the Kolmogorov-Smirnov and Chi-square goodness of fit tests also rejected the data being Cauchy distributed. It was revealed that the returns of all the four financial data considered, come from a leptokurtic, heavy tails and asymmetry distribution, and the weekly logarithm returns of GSE All-Shares index, USD/GHC, GBP/GHC and EUR/GHC exchange rates are α-stable distributed with parameters , , and .    The Kolmogorov-Smirnov and Chi-square goodness of fit tests uniformly fail to reject the Maximum likelihood estimation method of estimating the four University of Ghana http://ugspace.ug.edu.gh 82 parameters of stable  distribution and produces the best fit to the financial data considered. The financial data considered were modelled with stable  distribution with the following estimates; GSE All-Shares index 1.005 0.104, = 0.31 0.135,     = 0.002 0.0002 and = 0.001 0.0002  , US dollar exchange rate 1.073 0.114,   4 5 4 5 = 0.394 0.147, = 6.6*10 7.62*10 and = 2.3*10 8.9*10 ,       British pounds sterling exchange rate 3 4 = 1.289 0.129, = 0.173 0.193, = 1.8*10 1.8*10 and       4 4 = 6.1*10 2.5*10   and Euro exchange rate 1.236 0.126, = 0.234 0.18,    3 4 4 4 = 2.02*10 2.1*10 and = 6.87*10 2.8*10 .      5.2 Conclusion The findings from the study discovered that the daily returns distribution of Ghana Stock Exchange All-Shares index, US dollar, British pounds sterling and Euro exchange rates come from leptokurtic, heavy tailed and asymmetry distribution but cannot be modelled with stable  distribution, however the weekly logarithm return distribution of Ghana Stock Exchange All-Shares index, US dollar, British pounds sterling and Euro exchange rates follows stable  distribution. It was revealed that the weekly logarithm returns of the four financial data considered (GSE All-Shares, USD/GHC, GBP/GHC and EUR/GHC) do not obey the Gaussian law but stable  law. The study demonstrated that the Ghanaian financial data is not volatile as compared to the developed countries. The study elucidated that among the three methods (MLE, Sample Quantile and Empirical Characteristic function) of estimating the four parameters of α-stable distribution, the University of Ghana http://ugspace.ug.edu.gh 83 Maximum likelihood estimation method produces the best fit and more accurate estimates of the financial data considered. The returns of Ghana Stock Exchange All-Shares index, US dollar exchange rate, British pound sterling exchange rate and Euro exchange rate are successfully modelled with stable  laws. These findings successfully help to achieve the set objectives for this study, and contradict findings of Gopikrishnan et al. (1999), Pagan, (1996), Coronel-Brizio & Hernandez-Montoya, (2005) who found out that the tail of some financial time series data have to be modelled with > 2 . The study shows that the Gaussian laws for modelling logarithm returns of Ghana financial data is wrong and that the study proposed stable laws for modelling logarithm returns of Ghana financial data. It can then be generally concluded that the weekly logarithm returns of financial data (GSE All-Shares index, currency exchange rates) of Ghana obey the laws of Stable distributions and not Gaussian laws, and therefore Stable laws should be used to manage risk and make outstanding and precise projections. 5.3 Recommendations The following are recommendations to policy makers, stakeholders, investors and financial analysts; 1. It is recommended that weekly logarithm returns of Ghana financial data should be estimated with stable  laws and avoids using Gaussian laws. 2. The bank of Ghana should encourage investment banks in currency swaps to enable greater volatility of the returns. 3. The bank of Ghana should put in place good economic and monetary policies that will make the financial market more efficient and not focus on policies that will make the market static. University of Ghana http://ugspace.ug.edu.gh 84 4. The Ghana Stock Exchange should put in place policies that will encourage buying or selling shares or assets in Forward, Futures or Options than immediate buying or selling of an asset. 5. Economists and financial analysts should not assume the return distribution of assets or currency exchange rates but explore and discover their specific return distributions. 5.4 Further Studies 1. It is recommended that future studies should be conducted on the tail decay of the return distributions of Ghanaian financial data. 2. 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University of Ghana http://ugspace.ug.edu.gh 91 APPENDIX A WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 1 737.230 0.6800 0.9714 0.6498 2 739.017 0.6800 0.9714 0.6568 3 740.063 0.6860 0.9923 0.6589 4 740.677 0.6900 1.0060 0.6633 5 742.230 0.6900 1.0155 0.6660 6 743.140 0.6900 1.0155 0.6649 7 742.103 0.6900 1.0155 0.6687 8 741.170 0.6900 1.0155 0.6696 9 740.860 0.6995 1.0155 0.6696 10 748.587 0.7100 1.0162 0.6707 11 755.410 0.7100 1.0164 0.6713 12 759.947 0.7110 1.0165 0.6720 13 761.607 0.7141 1.0192 0.6726 14 763.707 0.7141 1.0195 0.6765 15 801.000 0.7141 1.0198 0.6810 16 864.380 0.7141 1.0198 0.6838 17 866.587 0.7141 1.0193 0.6936 18 854.520 0.7141 1.0188 0.7047 19 812.413 0.7141 1.0184 0.7148 20 809.480 0.7144 1.0185 0.7229 21 810.580 0.7145 1.0183 0.7276 22 812.633 0.7145 1.0180 0.7369 23 814.463 0.7145 1.0037 0.7518 24 816.130 0.7145 1.0023 0.7708 25 818.087 0.7145 1.0030 0.7823 26 817.070 0.7020 0.9925 0.7890 27 818.920 0.7019 0.9922 0.8040 28 821.330 0.7053 0.9925 0.8082 29 821.860 0.7057 0.9925 0.8034 30 821.873 0.7057 0.9930 0.7990 31 821.403 0.7057 0.9932 0.7916 32 820.410 0.7052 0.9952 0.7916 33 819.937 0.7092 1.0079 0.7945 34 822.403 0.7092 1.0092 0.7964 35 821.680 0.7107 1.0120 0.8070 36 835.133 0.7121 1.0142 0.8092 37 848.500 0.7122 1.0233 0.8130 38 866.633 0.7123 1.0245 0.8139 39 860.067 0.7123 1.0270 0.8142 40 855.983 0.7165 1.0348 0.8139 41 855.753 0.7198 1.0348 0.8139 42 849.563 0.7200 1.0288 0.8149 43 858.993 0.7200 1.0280 0.8261 44 858.363 0.7200 1.0297 0.8406 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 92 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 45 852.173 0.7225 1.0349 0.8575 46 851.663 0.7310 1.0388 0.8587 47 857.890 0.7318 1.0374 0.8608 48 863.730 0.7321 1.0351 0.8613 49 866.490 0.7332 1.0360 0.8599 50 865.667 0.7330 1.0357 0.8624 51 859.543 0.7327 1.0355 0.8675 52 857.980 0.7328 1.0364 0.8784 53 859.193 0.7330 1.0411 0.9010 54 861.553 0.7449 1.0538 0.9175 55 861.393 0.7461 1.0559 0.9272 56 858.167 0.7495 1.0561 0.9304 57 857.633 0.7583 1.0620 0.9297 58 858.217 0.7534 1.0648 0.9306 59 857.620 0.7664 1.0665 0.9315 60 863.653 0.7684 1.0697 0.9317 61 878.717 0.7699 1.0740 0.9306 62 892.220 0.7700 1.0821 0.9294 63 893.560 0.7701 1.0822 0.9289 64 897.890 0.7762 1.0895 0.9283 65 899.290 0.7793 1.0942 0.9292 66 896.443 0.7794 1.0965 0.9381 67 897.015 0.7795 1.1112 0.9691 68 897.650 0.7816 1.1236 0.9870 69 897.650 0.7898 1.1358 0.9982 70 897.997 0.7902 1.1390 1.0436 71 897.290 0.7902 1.1420 1.0104 72 897.740 0.7906 1.1457 1.0215 73 896.143 0.7906 1.1607 1.0274 74 894.730 0.8017 1.1816 1.0162 75 895.830 0.8048 1.2010 1.0051 76 896.897 0.8076 1.2123 0.9954 77 915.890 0.8087 1.2350 0.9894 78 932.020 0.8103 1.2405 0.9866 79 933.157 0.8085 1.2418 0.9815 80 934.350 0.8099 1.2418 0.9819 81 935.603 0.8104 1.2392 1.0200 82 935.563 0.8107 1.2408 0.9757 83 995.163 0.8149 1.2415 0.9715 84 939.250 0.8167 1.2480 0.9736 85 942.917 0.8209 1.2606 0.9763 86 946.680 0.8225 1.2609 0.9821 87 949.143 0.8270 1.2689 0.9903 88 953.097 0.8283 1.2740 1.0033 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 93 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 89 950.440 0.8301 1.2772 1.0170 90 951.353 0.8311 1.2785 1.0227 91 958.480 0.8311 1.2788 1.0255 92 958.207 0.8315 1.2791 1.0266 93 958.937 0.8372 1.2892 1.0463 94 961.113 0.8477 1.3071 1.0532 95 961.017 0.8605 1.3287 1.0745 96 961.010 0.8633 1.3206 1.0768 97 959.737 0.8659 1.3332 1.0798 98 957.330 0.8675 1.3336 1.1004 99 957.930 0.8680 1.3341 1.1482 100 958.267 0.8661 1.3360 1.1662 101 958.547 0.8674 1.3408 1.1680 102 958.250 0.8687 1.3493 1.1674 103 958.890 0.8696 1.3588 1.1675 104 956.893 0.8695 1.3694 1.1704 105 955.950 0.8700 1.3847 1.1685 106 956.440 0.8706 1.3865 1.1539 107 956.440 0.8705 1.3800 1.1283 108 956.530 0.8702 1.3637 1.1228 109 958.450 0.8695 1.3611 1.1244 110 961.440 0.8695 1.3615 1.1261 111 962.597 0.8699 1.2879 1.1160 112 965.345 0.8693 1.2812 1.1060 113 970.330 0.8698 1.3506 1.0962 114 983.020 0.8695 1.3488 1.0892 115 991.857 0.8688 1.3486 1.0895 116 1006.170 0.8676 1.3570 1.0878 117 1018.373 0.8700 1.3698 1.1035 118 1019.163 0.8723 1.3788 1.1170 119 1020.923 0.8735 1.3938 1.1202 120 1026.690 0.8730 1.3976 1.1223 121 1033.417 0.8739 1.4071 1.1237 122 1041.790 0.8769 1.4296 1.1271 123 1049.967 0.8823 1.4488 1.1276 124 1061.807 0.8829 1.4541 1.1319 125 1077.037 0.8856 1.4498 1.1253 126 1123.143 0.8859 1.4204 1.1067 127 1142.837 0.8839 1.3964 1.0955 128 1172.980 0.8782 1.3941 1.1105 129 1207.800 0.8756 1.3886 1.1145 130 1217.887 0.8773 1.3890 1.1101 131 1229.737 0.8786 1.3872 1.1048 132 1237.330 0.8920 1.3841 1.1101 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 94 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 133 1246.527 0.8906 1.3836 1.1140 134 1253.023 0.8835 1.3863 1.1174 135 1259.213 0.8831 1.3880 1.1176 136 1267.137 0.8833 1.3976 1.1181 137 1272.673 0.8828 1.4174 1.1226 138 1286.560 0.8847 1.4295 1.1387 139 1302.617 0.8898 1.4371 1.1580 140 1311.857 0.8915 1.4538 1.1732 141 1307.380 0.8926 1.4706 1.1678 142 1305.650 0.8997 1.4877 1.1935 143 1307.370 0.9196 1.5159 1.2070 144 1313.653 0.9091 1.5117 1.2146 145 1327.010 0.9118 1.5318 1.2134 146 1334.010 0.9102 1.5353 1.2118 147 1335.543 0.9098 1.5402 1.2136 148 1339.573 0.9148 1.5617 1.2167 149 1343.677 0.9294 1.6220 1.2154 150 1352.543 0.9398 1.6712 1.2109 151 1359.007 0.9410 1.6755 1.2083 152 1361.767 0.9391 1.6800 1.2051 153 1367.733 0.9393 1.6892 1.2001 154 1376.367 0.9383 1.7099 1.1992 155 1379.780 0.9326 1.7159 1.2058 156 1385.300 0.9258 1.6954 1.2051 157 1397.643 0.9223 1.6623 1.2072 158 1411.407 0.9236 1.6488 1.2125 159 1420.120 0.9232 1.6614 1.1733 160 1428.007 0.9236 1.6655 1.1919 161 1433.127 0.9204 1.6616 1.1910 162 1444.333 0.9201 1.6416 1.1895 163 1459.593 0.9201 1.6275 1.1887 164 1473.573 0.9200 1.6282 1.1845 165 1485.077 0.9196 1.6271 1.1756 166 1530.267 0.9236 1.6223 1.1685 167 1586.683 0.9276 1.6387 1.1564 168 1622.983 0.9288 1.6723 1.1274 169 1635.297 0.9305 1.6790 1.1249 170 1681.570 0.9318 1.6821 1.1220 171 1714.270 0.9330 1.6883 1.1231 172 1726.283 0.9326 1.6842 1.1132 173 1751.937 0.9318 1.6906 1.1126 174 1772.723 0.9310 1.6970 1.0998 175 1791.393 0.9269 1.6893 1.0950 176 1812.383 0.9205 1.6690 1.0917 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 95 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 177 1840.710 0.9220 1.6634 1.0940 178 1864.827 0.9236 1.6672 1.1054 179 1928.853 0.9234 1.6666 1.1036 180 1931.493 0.9213 1.6488 1.1030 181 1984.703 0.9208 1.6399 1.1115 182 2058.890 0.9225 1.6393 1.1174 183 2100.243 0.9176 1.6394 1.1175 184 2148.307 0.9207 1.6392 1.1126 185 2244.227 0.9191 1.6362 1.1035 186 2294.503 0.9191 1.6300 1.1042 187 2329.770 0.9221 1.6301 1.1055 188 2386.953 0.9258 1.6535 1.1045 189 2422.477 0.9295 1.6666 1.1067 190 2484.227 0.9304 1.6880 1.1037 191 2538.497 0.9280 1.6814 1.1029 192 2586.790 0.9314 1.6980 1.1033 193 2605.733 0.9295 1.7112 1.1041 194 2599.877 0.9361 1.7322 1.1035 195 2621.957 0.9294 1.7478 1.1029 196 2647.813 0.9259 1.7470 1.1002 197 2762.980 0.9223 1.7360 1.0988 198 2854.947 0.9243 1.7329 1.1000 199 2870.857 0.9265 1.7119 1.1012 200 2900.263 0.9273 1.7310 1.1046 201 2950.187 0.9265 1.7319 1.1053 202 2988.517 0.9291 1.7335 1.1070 203 3137.087 0.9266 1.7212 1.1048 204 3297.773 0.9270 1.7225 1.1051 205 3368.923 0.9269 1.7330 1.1051 206 3402.397 0.9245 1.7372 1.1172 207 3444.800 0.9226 1.7390 1.1031 208 3500.227 0.9212 1.7424 1.1041 209 3584.003 0.9180 1.7189 1.1051 210 3605.540 0.9184 1.7140 1.1052 211 3659.873 0.9213 1.7183 1.1070 212 3721.250 0.9222 1.7255 1.1069 213 3828.590 0.9205 1.7233 1.1892 214 3927.773 0.9170 1.7180 1.1295 215 4194.928 0.9184 1.7171 1.1503 216 4501.417 0.9209 1.6974 1.1510 217 4986.640 0.9212 1.6775 1.1582 218 5153.767 0.9204 1.6646 1.1649 219 5357.340 0.9200 1.6663 1.1635 220 5562.230 0.9203 1.6655 1.1643 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 96 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 221 5792.123 0.9209 1.6664 1.1675 222 6076.400 0.9174 1.5277 1.1701 223 6208.260 0.9183 1.5400 1.1800 224 6428.373 0.9176 1.5498 1.1805 225 6599.180 0.9173 1.5893 1.1820 226 6708.090 0.9157 1.5865 1.1862 227 6774.370 0.9135 1.5914 1.1864 228 6824.763 0.9141 1.6027 1.1862 229 6852.167 0.9138 1.6161 1.1836 230 6895.140 0.9135 1.6186 1.1851 231 6942.770 0.9128 1.6283 1.1837 232 7010.110 0.9122 1.6495 1.1860 233 7046.837 0.9162 1.6475 1.1936 234 7064.020 0.9182 1.6387 1.1955 235 7098.713 0.9192 1.6158 1.2097 236 7067.203 0.9204 1.6114 1.2265 237 7120.710 0.9254 1.6100 1.2348 238 7168.853 0.9284 1.6126 1.2171 239 7325.063 0.9290 1.6216 1.2149 240 7367.020 0.9295 1.6180 1.2159 241 7440.800 0.9338 1.6170 1.2150 242 7451.830 0.9394 1.6159 1.2173 243 7157.927 0.9411 1.6159 1.2172 244 7027.073 0.9398 1.6166 1.2200 245 7102.710 0.9393 1.6208 1.2395 246 7022.017 0.9320 1.6149 1.2300 247 6998.187 0.9323 1.6064 1.2285 248 7017.753 0.9323 1.6054 1.2301 249 6998.813 0.9311 1.6090 1.2380 250 6994.560 0.9313 1.6139 1.2390 251 6951.040 0.9288 1.6160 1.2312 252 6901.127 0.9263 1.6164 1.2562 253 6894.833 0.9280 1.6124 1.2723 254 6781.413 0.9316 1.6123 1.2729 255 6744.203 0.9319 1.6132 1.2731 256 6761.120 0.9286 1.6292 1.2710 257 6778.613 0.9250 1.6050 1.2680 258 6780.657 0.9277 1.6045 1.2657 259 6789.877 0.9282 1.6054 1.3002 260 6801.280 0.9276 1.6044 1.3456 261 6840.077 0.9269 1.6011 1.3776 262 6897.343 0.9265 1.6026 1.2770 263 6897.173 0.9265 1.6293 1.2761 264 6866.293 0.9280 1.6333 1.2728 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 97 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 265 6844.920 0.9278 1.6721 1.2744 266 6777.127 0.9288 1.6964 1.2818 267 6718.353 0.9291 1.6983 1.2894 268 6686.630 0.9302 1.6913 1.3002 269 6643.060 0.9252 1.6963 1.3193 270 6526.413 0.9257 1.7157 1.3364 271 6454.680 0.9266 1.7277 1.3459 272 6386.693 0.9288 1.7306 1.3566 273 6357.540 0.9301 1.7416 1.4025 274 6329.047 0.9320 1.7392 1.4131 275 6297.570 0.9353 1.7449 1.4188 276 6102.923 0.9400 1.7569 1.4353 277 6068.277 0.9399 1.7523 1.4447 278 6051.117 0.9403 1.7529 1.4465 279 6050.810 0.9435 1.7530 1.4336 280 6050.030 0.9390 1.7537 1.4150 281 6039.580 0.9383 1.7573 1.4252 282 6033.503 0.9401 1.7655 1.4361 283 6034.013 0.9432 1.7750 1.4401 284 5952.610 0.9439 1.7781 1.4348 285 5741.635 0.9445 1.7954 1.4376 286 5393.840 0.9438 1.8131 1.4496 287 5243.450 0.9452 1.8204 1.4458 288 5196.473 0.9389 1.8134 1.4472 289 5022.617 0.9309 1.8043 1.4781 290 4912.014 0.9415 1.8091 1.5050 291 4850.644 0.9439 1.8200 1.5144 292 4838.120 0.9462 1.8330 1.5190 293 4833.322 0.9473 1.8368 1.5423 294 4831.626 0.9483 1.8393 1.5566 295 4883.036 0.9521 1.8441 1.5671 296 4896.666 0.9494 1.8461 1.5721 297 4885.570 0.9422 1.8328 1.5696 298 4880.238 0.9451 1.8234 1.5732 299 4884.254 0.9466 1.8302 1.5810 300 4890.416 0.9484 1.9209 1.5823 301 4897.504 0.9487 1.8373 1.5945 302 4898.222 0.9487 1.8556 1.5621 303 4904.164 0.9487 1.8696 1.6291 304 4906.262 0.9485 1.8695 1.6335 305 4908.400 0.9485 1.8702 1.6417 306 4813.276 0.9484 1.8688 1.6716 307 4896.116 0.9466 1.8656 1.7006 308 4801.510 0.9470 1.8685 1.6952 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 98 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 309 4792.822 0.9460 1.8812 1.6316 310 4768.078 0.9424 1.8992 1.6485 311 4766.450 0.9412 1.8978 1.6720 312 4756.530 0.9412 1.8973 1.7300 313 4713.925 0.9417 1.8908 1.6652 314 4730.990 0.9425 1.8802 1.6703 315 4738.072 0.9455 1.8795 1.6569 316 4724.248 0.9497 1.8866 1.5999 317 4743.010 0.9532 1.9012 1.5652 318 4737.626 0.9551 1.9033 1.5547 319 4737.682 0.9555 1.9162 1.5444 320 4739.660 0.9579 1.9062 1.5421 321 4743.415 0.9623 1.9394 1.5377 322 4756.830 0.9681 1.9534 1.5365 323 4769.148 0.9820 2.0097 1.5868 324 4771.890 0.9814 2.0236 1.6862 325 4776.122 0.9814 2.0177 1.6819 326 4778.923 0.9849 2.0255 1.6975 327 4780.500 0.9873 2.0271 1.7157 328 4784.884 0.9898 2.0215 1.7188 329 4792.250 0.9904 1.9983 1.7263 330 4799.654 0.9893 1.9643 1.7113 331 4783.242 0.9891 1.9558 1.7226 332 4832.000 0.9842 1.9454 1.7126 333 4856.052 0.9931 1.9392 1.7446 334 4862.084 0.9893 1.9329 1.7602 335 4863.452 0.9888 1.9336 1.7601 336 4863.336 0.9875 1.9313 1.7832 337 4853.998 0.9874 1.9297 1.8422 338 4850.872 0.9876 1.9295 1.8603 339 4854.414 0.9881 1.9530 1.8554 340 4891.098 0.9908 1.9681 1.8505 341 4909.516 0.9901 1.9578 1.8561 342 4901.078 0.9900 1.9562 1.8664 343 4892.016 0.9925 1.9594 1.8921 344 4896.448 0.9988 1.9633 1.9191 345 4915.246 1.0041 1.9710 1.9459 346 4930.876 1.0051 1.9736 1.9747 347 4928.716 1.0073 1.9616 1.9994 348 4927.993 1.0106 1.9703 2.0235 349 4975.084 1.0140 1.9822 2.0379 350 4974.462 1.0157 1.9866 2.0360 351 4963.986 1.0247 2.0078 2.0202 352 4965.932 1.0356 2.0250 2.1446 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 99 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 353 4983.204 1.0507 2.0426 2.0574 354 4982.555 1.0528 2.0564 2.0706 355 4994.376 1.0535 2.0691 2.0841 356 5004.476 1.0637 2.0978 2.0993 357 5004.510 1.0755 2.0246 2.0779 358 4994.658 1.0850 2.1409 2.0800 359 5010.188 1.1036 2.1151 2.0985 360 5013.742 1.1191 2.1153 2.0876 361 5020.324 1.1259 2.1087 2.1042 362 5026.626 1.1341 2.1006 2.1187 363 5026.770 1.1432 2.0931 2.1241 364 5027.007 1.1504 2.0996 2.1241 365 5026.150 1.1560 2.0682 2.1271 366 5028.812 1.1756 2.0125 2.1407 367 5030.690 1.1893 1.9753 2.1402 368 5032.894 1.1955 1.9539 2.1348 369 5038.386 1.1946 1.9276 2.1423 370 5049.374 1.1943 1.8480 2.1387 371 5063.360 1.1962 1.8271 2.1466 372 5071.050 1.1981 1.8327 2.1516 373 5082.640 1.2245 1.8459 2.1830 374 5082.940 1.2449 1.8747 2.1620 375 5088.622 1.2548 1.8905 2.0896 376 5104.884 1.2524 1.8608 2.0826 377 5117.102 1.2766 1.9074 2.0619 378 5140.726 1.3041 1.9224 2.0828 379 5157.146 1.3298 1.8913 2.0716 380 5159.708 1.3355 1.8781 2.0147 381 5164.122 1.3370 1.9148 2.0180 382 5171.958 1.3498 1.9345 1.9856 383 5182.258 1.3741 1.9544 1.9772 384 5192.540 1.3942 1.9559 1.9740 385 5220.700 1.3986 1.9668 1.9717 386 5251.390 1.4088 1.9639 1.9802 387 5267.604 1.4106 1.9893 1.9815 388 5293.922 1.4107 2.0002 1.9659 389 5317.148 1.4112 2.0061 1.9502 390 5333.466 1.4123 2.0317 1.9570 391 5356.258 1.4263 2.0561 1.9430 392 5378.724 1.4370 1.9897 1.9256 393 5386.062 1.4419 2.1093 1.9114 394 5379.726 1.4456 2.1448 1.8436 395 5390.082 1.4491 2.1979 1.7873 396 5445.428 1.4515 2.2694 1.7687 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 100 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 397 5514.362 1.4558 2.2905 1.7534 398 5573.592 1.4670 2.3447 1.7556 399 5575.254 1.4733 2.3765 1.7555 400 5614.320 1.4769 2.3822 1.7620 401 5662.150 1.4711 2.3666 1.7745 402 5667.778 1.4746 2.3720 1.8026 403 5704.636 1.4768 2.3881 1.8713 404 5712.084 1.4801 2.4040 1.8471 405 5786.480 1.4843 2.4353 1.8603 406 5835.432 1.4852 2.4420 1.8642 407 5842.916 1.4819 2.4196 1.8525 408 5848.176 1.4788 2.4078 1.8309 409 5857.054 1.4762 2.3869 1.8389 410 5928.002 1.4752 2.3937 1.8355 411 6229.400 1.4757 2.4116 1.8383 412 6411.582 1.4762 2.4080 1.8573 413 6443.322 1.4756 2.3943 1.8793 414 6539.596 1.4741 2.3782 1.9099 415 6589.976 1.4712 2.3632 1.9265 416 6621.834 1.4635 2.3521 1.9394 417 6663.102 1.4619 2.3615 1.9535 418 6700.224 1.4624 2.3645 2.0371 419 6716.752 1.4621 2.3749 1.9827 420 6727.530 1.4639 2.3808 1.9840 421 6812.102 1.4641 2.3836 1.9722 422 6899.978 1.4622 2.3788 1.9579 423 6967.116 1.4645 2.3779 1.9592 424 7027.712 1.4660 2.3591 1.9605 425 7135.054 1.4664 2.3446 1.9558 426 7192.010 1.4658 2.3348 1.9600 427 7754.116 1.4697 2.3257 1.9558 428 8331.630 1.4610 2.3276 1.9595 429 8599.956 1.4587 2.3408 1.9920 430 8881.186 1.4524 2.3128 2.0365 431 9112.426 1.4812 2.3305 2.0389 432 9369.790 1.4519 2.2775 2.0185 433 9529.678 1.4478 2.2725 2.0204 434 9595.432 1.4527 2.2436 2.0286 435 9744.256 1.4533 2.2191 2.0449 436 9858.340 1.4604 2.2035 2.0565 437 10008.146 1.4521 2.2049 2.0907 438 10241.822 1.4476 2.1772 2.1141 439 10327.788 1.4441 2.1730 2.1179 440 10353.372 1.4434 2.1842 2.1276 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 101 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 441 10461.464 1.4421 2.1838 2.1313 442 10559.616 1.4427 2.1841 2.1357 443 10603.694 1.4423 2.1812 2.1412 444 10614.288 1.4396 2.1297 2.1331 445 10633.870 1.4362 2.0833 2.1303 446 10565.870 1.4346 2.0709 2.1314 447 10693.092 1.4369 2.0858 2.1484 448 10802.650 1.4376 2.0873 2.1447 449 10851.866 1.4386 2.0946 2.1413 450 10878.580 1.4398 2.1084 2.1402 451 10890.534 1.4401 2.1243 2.1373 452 10902.088 1.4412 2.1480 2.1342 453 10927.400 1.4481 2.2377 2.1326 454 10853.244 1.4503 2.2076 2.1274 455 10853.660 1.4496 2.2346 2.1240 456 10851.508 1.4563 2.2412 2.1271 457 10798.546 1.4481 2.2369 2.1377 458 10684.862 1.4497 2.2348 2.1392 459 10600.990 1.4500 2.2276 2.1372 460 10568.364 1.4500 2.2194 2.1315 461 10572.818 1.4499 2.2200 2.1699 462 10507.180 1.4498 2.2254 2.1215 463 10471.862 1.4493 2.2312 2.1451 464 10460.238 1.4480 2.3109 2.1557 465 10436.144 1.4469 2.2546 2.1869 466 10430.980 1.4463 2.2547 2.2010 467 10402.152 1.4459 2.2532 2.2087 468 10314.962 1.4775 2.3139 2.2084 469 10247.070 1.4524 2.2864 2.2224 470 10152.790 1.4651 2.2904 2.1896 471 10050.888 1.4531 2.2881 2.1977 472 9935.904 1.4540 2.2698 2.2125 473 9928.408 1.4549 2.2683 2.1885 474 9823.208 1.4624 2.2677 2.1657 475 9704.390 1.4687 2.2689 476 9658.048 1.4815 2.2564 477 9367.276 1.4868 2.2837 478 9172.268 1.4883 2.3187 479 8987.622 1.4996 2.3441 480 8989.174 1.5166 2.3788 481 8937.902 1.5228 2.3875 482 8821.016 1.5150 2.3746 483 8734.506 1.5140 2.3755 484 8092.522 1.5138 2.3755 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 102 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 485 7811.080 1.5175 2.3889 486 7189.708 1.5215 2.3954 487 6700.694 1.5263 2.4137 488 6031.998 1.5289 2.4328 489 5380.196 1.5183 2.4328 490 5399.856 1.5308 2.4020 491 5508.256 1.5296 2.4190 492 5376.608 1.5281 2.4384 493 5305.258 1.5268 2.4395 494 5244.702 1.5215 2.4436 495 5208.614 1.5199 2.4375 496 5187.554 1.5176 2.4431 497 5211.516 1.5165 2.4488 498 5418.368 1.5159 2.4461 499 6286.740 1.5168 2.4424 500 6489.376 1.5174 2.4331 501 6462.884 1.5175 2.4304 502 6426.466 1.5177 2.4311 503 6255.902 1.5173 2.4336 504 6164.412 1.5152 2.4267 505 6143.640 1.5155 2.4354 506 6019.338 1.5182 2.3698 507 5538.420 1.5215 2.3954 508 5424.392 1.5233 2.4509 509 5392.424 1.5221 2.4468 510 5400.708 1.5258 2.4441 511 5396.986 1.5364 2.4428 512 5375.836 1.5558 2.4419 513 5342.636 1.5826 2.4680 514 5323.276 1.6052 2.4791 515 5418.032 1.6146 2.4941 516 5546.050 1.6107 2.4796 517 5575.472 1.6011 2.4544 518 5468.136 1.6042 2.4680 519 5546.352 1.6160 2.4976 520 5635.906 1.6210 2.5364 521 5644.200 1.6394 2.5363 522 5625.388 1.6535 2.5486 523 5632.104 1.6537 2.5519 524 5606.838 1.6559 2.5514 525 5642.906 526 5638.102 527 5625.554 528 5644.578 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 103 WEEKS GSE Index 2000-2010 USD/GHC 2001-2011 GBP/GHC 2001-2011 EUR/GHC 2002-2011 529 5569.558 530 5754.408 531 5796.866 532 5838.242 533 5901.256 534 6098.218 535 6153.242 536 6206.094 537 6466.364 538 6696.548 539 7011.168 540 7060.814 541 7130.212 542 7067.748 543 6730.306 544 6513.884 545 6698.816 546 6575.998 547 6490.928 548 6240.194 549 6261.264 550 6365.214 551 6266.136 552 6384.854 553 6537.062 554 6649.158 555 6757.654 556 6765.902 557 6769.602 558 6725.136 559 6835.102 560 6781.376 561 6789.772 562 6799.806 563 6819.670 564 6843.320 565 6822.322 566 6915.920 567 7012.594 568 7111.224 569 7191.962 570 7182.450 571 7264.597 Source: Bank of Ghana (2001-2011) and GSE (2000-2010) University of Ghana http://ugspace.ug.edu.gh 104 APPENDIX B ## Install stable 5.3 package and nortest package library(stable) library(stats) library(MASS) library(nortest) ## Load Data into R usd <- read.csv("usdweek.csv") ## example: reading USD/GHC exchange rate ## Ploting times series X <- ts(data, frequency = n, start=c(year,1)) plot.ts(X, main="title of the graph") ## Plotting Histogram and normal density curve x <- usd$logreturn mn <- mean(x) stdev <- sd(x) hist(x, seq(-0.05, 0.05, 0.002), prob=TRUE, col ="pink", freq =FALSE, main=” title") lines(density(x, bw=0.001),col="green", lwd=3,) curve(dnorm(x, mean = mn, sd= stdev), add=TRUE, col="red",lwd=2, xaxt="n") rug(x) # show the actual data points mtext(paste("mean ", round(mean(x),1), "; sd ", round(sd(x),1), "; N ", length(x),sep=""), side=1, cex=.75) University of Ghana http://ugspace.ug.edu.gh 105 ## Density plots x<- usd$logreturn theta <- stable.fit(x,1,0) # maximum likelihood estimates for stable distribution mn <- mean(x) stdev <- sd(x) stable.density.plot(x, theta, xrange = range(x), main=" title of the plot") curve(dnorm(x, mean = mn, sd= stdev), add=TRUE, col="black",lwd=3, xaxt="n") curve(dcauchy(x, location = mn, scale = stdev), add=TRUE, col="blue", lwd=3) grid() ## Diagnostic tests stable.diag(x, xrange = range(x), continuous = TRUE, main=" title" ) ppstable(x, theta, var.stabilized = FALSE, param = 0) qqstable(x, theta, param = 0, ptwise.ci = TRUE, col="blue") zzstable(x, theta, param = 0) stable.density.plot(x, theta, param=0, xrange = range(x),width=2*diff(quantile(usd,c(.25,.75)))*length(usd)^(-1/3),show.legend = TRUE, continuous = TRUE, main="stable density plot" )) qqnorm(x,col="green") # normal qq-plot qqline(x,lwd=2) grid() ## Estimating the parameters of stable distribution stable.fit(x,1,0) # MLE University of Ghana http://ugspace.ug.edu.gh 106 stable.fit(x,2,0) # Sample quantile stable.fit(x,3,0) # ECF ## Test for Normality shapiro.test(x) ks.test(x, "pnorm", mean = mean(x), sd = sqrt(var(x))) ad.test(x) chisq.test(x,dnorm(x,mean=mean(x), sd=sd(x))) ## Goodness of fit test theta <- stable.fit(x,1,0) stable.ks.gof(x,theta,0,0) # K-S goodness of fit test stable.chisq( x, theta, 0, n.classes=5) # Chi-square goodnes of fit test chisq.test(x, dcauchy(x, location = mean(x), scale=sd(x))) # Goodness of fit to Cauchy distribution ks.test(x, "pcauchy") # Goodness of fit to Cauchy distribution University of Ghana http://ugspace.ug.edu.gh