© Carl Hope Korkpoe University of Cape Coast © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library UNIVERSITY OF CAPE COAST CHARACTERISATION OF VOLATILITY OF RETURNS IN EQUITY MARKETS OF SUB-SAHARAN AFRICA BY CARL HOPE KORKPOE Thesis submitted to the Department of Statistics of the School of Physical Sciences, College of Agriculture and Natural Sciences, University of Cape Coast, in partial fulfilment of the requirement for award of Doctor of Philosophy in Statistics _._" .... ,_ .... -----------_ .......... ~ . :. CALL No., JULY 2020 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library DECLARATION Candidate's Declaration I hereby declare that this thesis is the result of my own original research and that no part of it has been presented for another degree in this university or elsewhere. Candidate's Signat r \ Date 9.?-: -::.C?7 ~. ?P ~\ Name: Carl Hope K Supervisors' Declaration We hereby declare that the preparation and presentation of the thesis were supervised in accordance with the guidelines on supervision of the thesis laid down by the University of Cape Coast. Principal Supervisor's Signature.: ..... Name: Dr. Nathaniel K. Howard . L.J . Co-Supervisor's Signature . ~7." I. : .. .................. Date ... ??:/.':'. 71 ?-O "J.-/ Name: Dr. Bismark K. Nkansah ii © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ABSTRACT Sub-Saharan Africa equity markets have been characterised in various practitioner literature as risky. Unfortunately, not much has been done in extant academic works to properly provide evidence and support to guide investor decision-making in the sub-region. Based on this argument, specifications of various regime switching GARCH models were made with various tail innovations to study the volatility of the returns of the Ghana, Kenya, Nigeria and Botswana exchanges using the daily broad market closing indices. These regime switching models were compared with the single non­ switching GARCH models using the Deviance Information Criteria (DIC) for model fit. The study established the dominance of the low volatility regime for Ghana, Kenya and Nigeria with the high volalitility periods interspersed for brief times during the sample period. The opposite was found for the Botswana exchange. The study also established the most and least volatile months of the exchanges by a process of resampling the daily into monthly data and finding the average monthly volatility ranking for the various exchanges. Findings of this study will inform investors in the sub-region equity markets on the behaviour of risk and how they can strategise their trading activities to either take advantage of the volatility or avoid losses. For policymakers, it alerts them of the presence of regimes in the markets and a reminder of when regulation and/or policy promote equity market activity in their respective countries. iii © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library GARCH Markov Chain Monte Carlo Metropolis-Hastings Price fi lters Regime switching Sub-Saharan equities KEYWORDS iv © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ACKNOWLEDGEMENTS I am most indebted to Dr. Nathaniel Howard, my principal supervisor, for a job well done. Your commitment to see this work done through the prompt feedback and guidance put me on my toes. You deserve all the thanks for seeing me through it all. My co-supervisor, Dr. Bismark Kwao Nkansah, was a key sounding board for this work. Your input and ideas throughout the work and seminars were invaluable in shaping of the work. Accept my heartfelt gratitude. And to the faculty members, your criticisms, observations and challenges during the seminars worked the magic for me. I am most grateful. v © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library DEDICATION To Regina, Natty and Shika for the love and care. vi © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library TABLE OF CONTENTS DECLARA TION ABSTRACT KEYWORDS ACKNOWLEDGEMENTS DEDICATION LIST OF ACRONYMS CHAPTER ONE: INTRODUCTION Background to the Study Statement of the Problem Objectives of the Study Research Questions Scope of the Study Ghana Stock Exchange Nairobi Stock Exchange Nigeria Stock Exchange Botswana Stock Exchange Significance of the Study Delimi tations Limitations of the Study Organisation of the Study Chapter Summary CHAPTER TWO: LITERATURE REVIEW Introduction vii Page 11 iii IV V VI XVI 2 9 9 10 10 11 13 15 18 19 21 21 22 22 24 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library The Nature of Market Volatility Classification of Markets The Nature of Frontier Economies and Stock Markets Regime-switching Models in Frontier Markets Regime Switching Versus Non-Switching Volatility Models The Case for Regime-switching in Volatility Modeling for Sub-Saharan Equity Returns Chapter Summary CHAPTER THREE: METHODOLOGY Introduction Regime-switching GARCH Models Conditional Heteroscedastic Functions The Regime-switching GARCH (1 , 1) Model The Regime-switching GJR-GARCH (1 , 1) ModelO54 027 3033 The Regime Switching Exponential GARCH Model Conditional Distributions of the Errors Normal Distribution Student-t Distribution The GED Distribution Parameter Estimation for Regime Switching Models Approach to Parameter Estimation via MCMC The Random Walk Metropolis-Hastings Algorithm Model Diagnostics Heavy-tail Distributions Chapter Summary viii 24 29 31 34 35 40 42 44 45 46 47 48 49 50 50 51 51 52 53 55 56 58 59 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER FOUR: RESULTS AND DISCUSSION Introduction Analysis Establishing Market Regimes Ghana Regime Changes in Nairobi Stock Exchange Regime Changes in Nigeria Stock Exchange Regime Changes in the Botswana Tests for Stationarity Test for GARCH Effects Model Estimation Single Regime Models Two-regime Models Backtesting the Models Model Interpretation Ghana Kenya Nigeria Botswana Dependencies and Tail Behaviour among Exchanges Estimation of the VaR of the Exchanges Remark 5 . Month-on-month Volatility Chapter Summary ix 60 61 70 70 71 72 73 73 75 76 77 79 84 88 88 91 94 97 100 105 110 110 115 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER FIVE: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Overview Summary Conclusions Recommendations REFERENCES APPENDIX A: COMPUTER CODE IN PYTHON APPENDIX B: COMPUTER CODE IN R APPENDIX C: PAPERS x 116 116 117 120 122 143 160 212 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library LIST OF TABLES Table Page 1 The market statistics of the GSE as of year ending 2018 12 2 Summary of activity on the KSE 15 3 Performance of the NSE for the period 1985-2018 17 4 Composition of the BSI up to year ending 2018 19 5 Preview of Frontier Markets according to global index providers 32 6 Summary of level of indices 61 7 Summary of the log-returns of indices 65 8 ADF and PP tests of stationarity results 74 9 Result of the Zivot-Andrews test 75 10 Result of the ARCH tests 76 11 Single-regime DICs of the GARCH models with conditional tail distributions 78 12 Two-regime DICs of the GARCH models with conditional tail distributions 80 13 Models with the minimal DICs extracted 81 14 GARCH estimates for the GSE All Share Index 82 15 GARCH estimates for the KSE All Share Index 82 16 GARCH estimates for the NSE All Share Index 83 17 GARCH estimates for the BSI All Share Index 83 18 DQ test result for both single- and double-regime models 85 19 EGARCH (1,1) with skewed GED estimates for the GSE All Share Index 88 20 Posterior mean transition matrix for the GSE 89 xi © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 21 GARCH (1,1) with GED estimates for the KSE All Share Index 91 22 Posterior mean transition matrix for the KSE 92 23 GJR-GARCH (1, 1) estimates with skewed Student's t innovations for the NSE All Share Index 95 24 Evolution of volatility over the sample period for NSE returns 96 25 GARCH (1, 1) with skewed Student's t estimates for the BSI All Share Index 98 26 Evolution of volatility over the sample period for BSI returns 99 27 Correlations among the returns of the exchanges 101 28 Number of Observations beyond +3 standard deviations 104 29 VaR of the exchanges 109 30 Hill's estimator at 99% 109 xii © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library LIST OF FIGURES Figure Page 1 Index levels of GSE over sample period 62 2 Index levels of KSE over sample period 62 3 Index levels ofNSE over sample period 63 4 Index levels ofBSI over sample period 63 5 Distribution of KSE Index Returns 66 6 Distribution of GSE Index Returns 66 7 Distribution ofNSE Index Returns 67 8 Distribution ofBSI Indices Returns 67 9 Time Series of GSE Composite Index Returns 68 10 Time Series of KSE Composite Index Returns 68 11 Time Series ofNSE Composite Index Returns 69 12 Time Series ofBSI Composite Index Returns 69 13 Regime changes in GSE returns for the sample period 70 14 Regime changes in KSE returns for the sample period 71 15 Regime changes in NSE returns for the sample period 72 16 Regime changes in BSI returns for the sample period 73 17 One-day VaR at 5% for sged E-GARCH (1,1) and 2-regime sged E-GARCH (1,1) for GSE 86 18 One-day VaR at 5% for sstd GARCH( 1,1) and 2-regime sstd GARCH( 1,1) for KSE 86 19 One-day VaR at 5% for Student-t GJR-GARCH (1, 1) and 2-regime Student-t GJR-GARCH (1,1) forNSE 87 xiii © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 20 One-day VaR at 5% for Student-t GARCH (1, 1) and 2-regime Student-t GARCH (1, 1) for BSI 87 21 Evolution of volatility over the sample period for GSE returnsO 19) 90 22 Smoothed probability plot for GSE 91 23 Evolution of volatility over the sample period for KSE returns 93 24 Smoothed probability plot for KSE 94 25 Evolution of volatility over the sample period for NSE returns 96 26 Smoothed probability for NSE 97 27 Evolution of volatility over the sample period for BSI returns 99 28 Smoothed probability plot of BSI 100 29 Extreme outcomes beyond three standard deviations for GSE Returns 102 30 Extreme outcomes beyond three standard deviations for KSE Returns 102 31 Extreme outcomes beyond three standard deviations for NSE Returns 103 32 Extreme outcomes beyond three standard deviations for BSI Returns 103 33 QQ plot for GSE Index returns 105 34 QQ plot of KSE Index returns 106 35 QQ plot ofNSE Index returns 106 36 QQ plot for BSI Index returns 107 37 12-month resampled volatility ofNSE with patched regions showin g high volatility 110 xiv © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 38 12-month resampled volatility of GSE with patched regions showing high volatility 110 39 12-month resampled volatility ofBSI with patched regions showing high volatility 111 40 12-month resampled volati Iity of KSE with patched regions showing high volatility 111 41 Average monthly volatility ranking for GSE Index returns 112 42 Average monthly volatility ranking for NSE Index returns 113 43 Average monthly volatility ranking for KSE Index returns 113 44 Average monthly volatility ranking for BSI Index returns 114 xv © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ADF AIMS ASEA BSI BSM BTCL CAGE DQ DGP DIC EAC EGARCH ETF FTSE GARCH GAX GED GEMS GFIM GJR-GARCH GoG GSE IFC IMF LIST OF ACRONYMS Augmented Dickey-Fuller Alternative Investment Market Segment African Securities Exchanges Association Botswana Stock Exchange Botswana Share Markets Botswana Telecommunication Corporation Limited Cultural, Administrative, Geographic and Economic Dynamic Quantile Data Generation Process Deviance Information Criteria East African Community Exponential GARCH Exchange-Traded Fund Financial Times Stock Exchange Generalised Autoregressive Conditional Heteroscedasticity Ghana Alternative Market Generalized Exponential Distribution Growth Enterprise Market Segment Ghana Fixed Income Markets Glosten-Jagannathan-Runkle GARCH Government of Ghana Ghana Stock Exchange International Finance Corporation International Monetary Fund xvi © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library IOSCO JSE KSE MCMC MIMS MSCI NASI NSE PP QQ RNE SAP SSE USD VaR WFE ZW International Organisation of Security Commissions Johannesburg Stock Exchange Nairobi Stock Exchange Markov Chain Monte Carlo Main Investment Market Segment Morgan Stanley Composite Index NSE All-Share Index Nigeria Stock Exchange Phillips-Perron Quantile-Quantile Relative Numerical Efficiency Structural Adjustment Programme Sustainable Stock Exchanges United State Dollars Value-at-Risk World Federation of Exchanges Zi vot -Andrews xvii © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER ONE INTRODUCTION Sub-Saharan Africa has a lot of investment potential but investors have not been able to firmly characterise the risks in the markets. Much of what is known about the equity markets is in the practitioner literature and this tends to be in-house for management and the investor community of these investment houses. There are few extant studies in the academic literature that sought to characterise the risk of investing in the sub-Saharan African markets. Unfortunately, these studies, to start with, are disparate and furthermore based on assumptions that ignore the behaviour of the data coming out of the markets. The economies of sub-Saharan African countries are buffeted by disturbances which are economic, social and political. These influence trade on the various stock exchanges and therefore the pattern of returns of financial assets. Any model that seeks to estimate risk must take these circumstances into account else they will not be measuring the real volatility behaviour in the market and serve no purpose. This chapter poses the question of whether incorporating regIme changes in the risk estimation models are the appropriate approach to use in the sub-Saharan African markets. This is important for both investors and policymakers in the sub-region. Investors in equities 'in the sub-Saharan markets are well positioned to adapt their trading strategies to profit from the particular regime in place and also manage their risks. For policymakers, knowing the regime shifts in the markets will be a pivotal moment in adjusting the monetary and fiscal policy framework to smoothen the business cycle of the respective countries. 1 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Background to the Study Volatility is an important metric In investment decision-making. Investors, and in general, finance and economic practitioners, pay particular attention to the movements or fluctuations in the trading of assets in the financial markets. The accurate characterisation of volatility of returns is essential for the pricing of financial assets, risk management and taking decisions about where and what assets to include in their portfolios. It is the quintessential that directs investors on how to optimise their resources to achieve maximum returns whiles minimising the downside risk. As a statistical artefact, however, it is not directly observable in the market. It has to be estimated from historical data or backed out from observed option prices in the markets. The financial returns for which we seek to estimate the volatility are driven by a variety of factors in the market environment. This assertion was made by Hamilton (1990) and reaffirmed by Hamilton and Susmel (1994) in their study of the performance of different volatility models in the US financial markets. Several authors have followed up since, trying to assign reasons for this phenomenon in the developed and emerging markets and among different asset classes too. In the developed markets, Ang and Timmermann (2012) and Hardy (2001) carried out studies on the financial markets in general and equity markets respectively and provided evidence of regime switching behaviour in the volatility of returns. Regime switching occurs in time series when the evolving data generation process shows a sudden or abrupt change in behaviour by means of assuming a higher or lower value from its previously 2 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library typical values. It is a phenomenon that is seen very often in financial markets where prices of assets could surge or fall with consequent decrease or increase in volatility respectively. As the regimes of the market changes, we observe a change in the statistical properties of the data as well. Regime switching and structural change come under change point problems in financial and economic time series (Kleiber, 2018; Piger, 2009; Guidolin, 2011). Barry and Hartigan (1992) and Carlin, Gelfand and Smith (1992) addressed extensively the theoretical basis of these change point problems. Basically, both problems refer to changes in the non-stationarity nature of the mean, the variance, the autocorrelation or any two or more of these characteristics of the distribution of the time series. It is to be noted that some authors use this term synonymously with structural change, a term that is reserved for the permanent switching of the parameters or properties of the data generation process (DGP). For example, Babikir, Gupta, Mwabutwa and Owusu-Sekyere (2012) studied the Johannesburg Stock Exchange (JSE) employing regime switching generalised autoregressive conditional heteroscedasticity (GARCH) models in estimating volatility but used structural changes to describe the phenomenon. Carrasco (2002) also used the terms synonymously in studying the misspecitication in macroeconomic models. In their work on house prices, Than-TIli, Dong and Chen (2019) equally referred to regime-switching as structural breaks when house prices were in effect mean-reverting. However, Piger (2009) addressed the distinction between structural change and regime switching by noting that the former involves a permanent migration of the parameters to a new level whereas the latter is characterised by mean-reverting processes. More recently, 3 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Billio, Casarin and Osuntuyi (2016) distinguished between these two terms in their work on Markov regime-switching using Gibbs sampling. For financial markets, Ang and Bekaert (2004) attributed the changes in regimes to the business cycles that are part of the natural development cycle of the economy. In the emerging markets, Assoe (1998), for example, studied market returns and found evidence of switching in vOlatility of equities. The author attributed this mainly to the uncertainties that come with the changing regulations usually associated with relatively young markets trying to find their feet. Government experimentation with economic policies, a characteristic of developing economies, also directs the attractiveness of various assets classes in the capital markets and this has huge implications for the volatility of equities (Muralidharan & Niehaus, 2017; Rydqvist, Spizman, & Strebulaev, 2014; Mosley & Singer, 2008) . For the frontier markets, especially those within sub-Saharan Africa, very little has been done in the research of these markets. Coverage by researchers and independent financial analysts is thin and sparse. Quisenberry and Griffith (2010) from the practitioner community provide some glimpses into the frontier market dynamics. Frontier markets are relatively difficult to define exactly. Samarakoon (2011) looks at frontier markets "as relatively small and illiquid markets, even by emerging market standards and include all the markets that are not classified as emerging markets". Thus, the defining characteristic of frontier markets is their size and a lack of liquidity (Zaremba, 2019). International index providers such as Morgan Stanley and Financial Times list comprehensive economic indicators on which markets are classified 4 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library as developed, emerging or frontier (MSCI Frontier Markets, 2019; FTSE, 2019). Increasingly, frontier market assets are becoming important additions to global portfolios. Miles (2005) reassures investors that in spite of the globalisation of financial markets, frontier market financial assets returns have relatively low correlations with developed and emerging market returns. This is further supported in a paper by Chen, Chen and Lee (2014) in their study of the composition of international portfolios after the global financial crisis of 2008-2009. It is instructive to note that despite the move towards globalisation of financial markets, Berger, Pukthuanthong and Yang (2011) found that frontier markets have relatively low level of integration with the rest of the world markets. This means investors can make good pickings with the potential of attractive returns. Unfortunately, for sub-Saharan Africa equity markets, there is less in the finance literature on the correct characterisation of the return generation processes that give rise to the risks observed by practitioners. Investors have had to rely on their experiences and rules of thumb from the developed and emerging markets to navigate the sometimes turbulence in the frontier markets. Admittedly, the market environment in the frontier economies is far from benign this makes accurate investment-related information critical to investment and portfolio management. It is a time­ honoured theory in finance that those who are unable to get the dynamics in the market environment right are eventually driven out (A1chian, 1950). Thus, surviving in 'new' markets away from home requires that investors are able to forecast with some degree of accuracy and confidence on the direction of the market. 5 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Unfortunately, estimation of volatility models in frontier markets has hitherto followed the GARCH framework of Engle (1982) and Bollerslev (1986) irrespective of the different environments of the markets. For frontier markets in sub-Saharan Africa, the market environment is far from benign and estimating risks with the aforementioned models is likely to lead to mispricing of assets and also the execution of bad risk management strategies. That said, assessing the correct level of volatility in frontier markets and how it evolves over time is nontrivial. This is particularly challenging due to a dearth of market information. A mUltiplicity of risk factors drive market returns in frontier markets conferring it with characteristics different from that of the emerging markets (Bekaert, Erb, Harvey, & Viskanta, 1998). These factors can be linked to the developments in the underlying economy of the frontier markets. Expectations regarding the performance of the economy in terms of government or central bank policy with respect to spending and taxation or money supply and interest rates have direct effects differefltially on fixed income and equities. Economic policies in most frontier markets which rely very much on the exports of primary commodities is largely experimental. Political upheavals in the form of civil unrests, coups, revolutions, etc. in a country normally scare investors away as they reassess country risks and the probabilities of making losses on their investments. Political risk, according to Busse and Hefeker (2007), increases the cost of doing business and therefore the expected returns demanded by investors in conflict-prone markets. Dimic, Orlov and Piljak (2015) found that political risks are relatively elevated in frontier markets compared to developed and emerging markets. 6 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Suliman and Mollick (2009) document extensively the debilitating effects of communal conflicts on investments in sub-Saharan Africa, claiming a substantial drop in foreign direct investments and subsequent exit of investors from the fragile markets. Cote d'Ivoire is an example in recent memory. By some accounts (see Klapper, Richmond and Tran (2013) and Klaas (2008) among others), the Ivorian economy slowed down substantially following the capital flight that occurred during the decade-long civil war. Foreign direct investments were in negative territory and economic growth remained at one percentage point during the civil war. Such developments in the underlying economy endow the data coming from the markets with nonlinear characteristics best described by piecewise functions with their own means, standard deviations and covariance structure and autocorrelations. Indeed, that is the position of Lamoureux and Lastrapes (1990) who argue that the persistence or near unit root observed in volatility models are due to the wrong characterisation of the data. This leads to an over- and under-estimation of risks in periods of tranquility and upsurge in volatility respectively in the markets. The transmission of such shocks to the capital markets means a more volatile and challenging environment that dIi ves market returns low as investors dump their holdings and make for the exit. Equally, there are periods of tranquility in the markets with rising valuations accompanied by decent returns for investors. How do we characterize the behaviour of volatility given these observations? Clearly, single-regime GARCH models, the workhorse of volatility modeling of financial asset returns will not suffice here. Single­ regime models lead to mis-characterisation of the data generating mechanism; 7 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library hence the wrong probability distribution of the returns. More often, this leads to loss of information and as a consequence, wrong inferences would be drawn from the data. Rather than assuming a single-modal distribution of the data, regime switching models drill down to uncover the complexity of the underlying distribution and accordingly estimate means, standard deviations and covariances based on the actual structure of the data. Given the preceding background, one is inclined to think regime switching will characterize the data generation process (DGP) in the returns of financial assets in general and equities in particular in frontier markets. These will lead directly to the use of regime switching GARCH models in the estimation of risk of financial returns. The construction of volatility measures is such as to estimate the parameters of the model piecewise in a given regime. In effect, the use and reliance of regime switching models in the estimation of volatility is motivated by the behaviour of the data as discussed in Dantelsson (2002). Market data is endogenous of the market environment. It goes therefore that any attempt at modelling volatility based on the data should take into account the environment generating the data. This is the approach adopted by this thesis going forward. This work will fill the information gap in the literature of sub-Saharan African equity markets by explaining the volatility returns based on the view that they are driven by the economic, social and political factors prevailing in a given economy. Investors, traders and policymakers are going to be guided enormously in decision-making bearing in mind the shifts in regimes in the markets. 8 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Statement of the Problem Investment and portfolio management, plicing of securities and risk management in general require fine-grained estimation of risk. Risk estimation in frontier markets hitherto relies on classical GARCH models without the incorporation of market regimes that will be typical of the market and data characteristics which are quite different from the developed and to a lesser extent the emerging markets. Frontier markets are subject to frequent upheavals and these transmit to the financial markets. Thus, any attempts at volatility estimation must incorporate regime switching in order to correctly determine the risks associated with the assets in the markets. For equity markets in sub-Saharan Africa, this is even imperative, given problems with experimentation with monetary and fiscal policies, social upheavals, a lack of quality market information for investment and financial reporting issues. This problem is widely recognized in the practitioner community but has not been given the needed attention in the academic literature. Given the increasing prominence of sub-Saharan African assets in global investment portfolios, there arises the need to address this gap in the academic literature and place the markets on the proper pedestal for both institutional and individual investor. Objectives of the Study The main objective of this study is to adequately characterise the volatility of equities in selected financial markets in sub-Saharan Africa using quantitative risk models. 9 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library The specific objectives are to: 1. Investigate the nature of evidence of regime switching III equity markets in some selected sub-Saharan African markets. 2. Determine the regime switching GARCH model which best depicts the returns in each of the selected markets. 3. Comapare the obtained regime switching models with existing single­ regime models. 4. Examine whether regime-switching induces heavy-tailed distributions in the returns for the selected equity markets. Research Questions From the foregoing, the questions addressed by this study are: 1. What is the nature of evidence of regime switching in equities in frontier sub-Saharan African markets? 2. Which regime switching GARCH models will characterise the respective returns in these markets? 3. How do regime-switching models compare with single-regime models? 4. Do regime-switching induce heavy-tailed distributions in the returns from the equity markets? Scope of the Study The study concentrates on a sample of sub-Saharan African equity markets. The classification of the markets relied on that provided by MSCI (2019) and FTSE (2019). The sub-Saharan African region has been segmented into west, east, central and south with a country each from these regions except for West Africa where Ghana and Nigeria where selected. Nigeria Stock Exchange (NSE) was selected in addition to Ghana Stock Exchange 10 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library (GSE) because it is the largest equity market in sub-Saharan Africa. The size of the NSE, number and diversity of listed firms makes it significant for investors. Nairobi Stock Exchange (KSE) and Botswana Stock Exchange (BSI) are selected from east and southern Africa regions respectively. We could not get data on markets in the central African region. Selection was loosely based on the Cultural, Administrative, Geographic and Economic (CAGE) distance principles espoused by Ghemawat (2001). The CAGE framework is based on the similarity between countries as a result of geographic proximity. By this, a country chosen to represent a geographic region will have similar economic, social and political characteristics of that region as a whole. We present an overview of the selected stock exchanges as follows: Ghana Stock Exchange The GSE was established in July, 1989 under Ghana Companies Code, 1973. It started operations in November, 1990, with initial trading through a call-over system with eleven equities, three brokers and a market capitalisation in current terms of USDO.66 million. The GSE has followed a trajectory of development as any other bourse migrating to a manual continuous auction trading system to now an automated auction market made up of three market segments with forty-one equities, seventy-six Government of Ghana (GoG) bonds and notes, corporate instruments, an exchange-traded fund (ETF) and a thriving twenty-two brokerages. It has a market capitalisation of USD 13.3 billion as of December, 2018. According to Hesse (2019), the GSE comes under the regulatory supervision of the Bank of Ghana, the Securities and 11 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Exchange Commission of Ghana, the National Pensions Regulatory Authority, and the Nationa Insurance Commission. The GSE is constituted of three market segments, namely, Main Equity Market listing the equities and ETFs, the Ghana Fixed Income Markets (GFIM) listing both government and corporate debt instruments, and the Ghana Alternative Market, an initiative to carter for the listing of small-and­ medium scale enterprises. Table 1 has the statistics of the recent performance of the exchange. Table 1: The market statistics ofthe GSE as of year ending 2018 2014 2015 2016 2017 2018 Equity market cap (USD billions) 23.83 16.32 13.17 13 .68 13.29 Domestic market cap (USD billions) 5.14 3.2 2.72 3.78 5.52 Debt market value (USD billions) 1.14 1.55 5.65 6.68 7.62 Number of listings (Main and GAX) 38 42 44 43 42 New equity securities listed 4 2 2 2 Delistings 0 0 0 3 Number of issues (GFIM) 170 125 129 129 Value traded (Equity USD millions) 128 71 61 121 143 Turnover (Debt USD millions) 2860 4192 7140 8232 Equity market returns, GSE -CIa (%) 5.4 -11.77 -15.33 52.73 -0.29 Avg. daily equity volume traded 0.83 0.99 1.02 1.3 0.81 (millions) Market cap/GDP (%) 41.4 31.66 24.5 22.91 20.34 Source: GSE, NPRA and SEC (2019) 12 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library In general, we see a decline in the stock market valuation across time compared to the other related economic indicators. The number of issues has increased but there is a corresponding decrease on the stock prices in the market as indicated by the trends in the volumes traded and the equity market capitalisation. The trend in the ratio of market capitalisation to GDP is on the decline. This indicates an increase in economic growth whereas the stock market has not shown corresponding increase. The number of issues has remained fairly stable after an initial drop but these new shares were probably less valued by the markets. The GSE faces some challenges. At a domestic equity market capitalisation to GDP and total equity market capitalisation to GDP of 8.03% and 19.35%, respectively, Hesse (2019) notes investor flight to the safety of sovereign debt instruments issued by GoG. This investor flight is mainly due to the perceived risks associated with equities, especially at the time of reforms in the financial sector of the economy and the continuing depreciation of the Cedi compared to the returns on equities. It is noteworthy that this flight to safety has the effect of increasing volatility and turnover in the equities and this is registered in the average value traded in the equities until 2018. Furthermore, the trend in market capitalisation to GOP ratio indicated a stagnated market even as GDP continue to grow. Nairobi Stock Exchange Kenya has the oldest stock exchange in the sample. The Nairobi Stock Exchange (hereafter KSE to distinguish it from the Nigeria Stock Exchange which has the same abbreviation) started in 1954 as the bourse serving Kenya, Uganda and Tanzania under what was known as the East African Community 13 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library (EAC). It became a wholly Kenyan outfit following the collapse of the EAC in 1977 and remained a purely Kenyan entity till today. Ramji, Wairimu, Mwita and Mwanyasi (2019) reported that the KSE's equities consist of the Main Investment Market Segment (MIMS), the Alternative Investment Market Segment (AIMS), and the Growth Enterprise Market Segment (GEMS). The broad market indices are made up of the NSE All-Share Index (NASI), NSE 20, FTSE NSE Kenya 15, and FTSE NSE Kenya 25. The NASI consists of the sixty-three (63) listed companies and it is the main pulse of the economy of Kenya. In addition, the KSE has a fixed­ income and exchange-trade fund together with some derivatives segments. 1n terms of numbers, the financial services sector with banks included makes about 48% of the exchange. The single market capitalisation is contributed largely by Safaricom PLe with $1 billion in traded equity out of a market capitalisation of $22.60 billion for the KSE as a whole as of July, 2019. As any other frontier market, the exchange suffers from uncertainty due to the changing regulatory landscape, low market liquity and contagion to US Federal Reserve monetary policy changes due to the number of foreign investors with an eye on US interest rates. Table 2 shows a summary of activity on the KSE for 2008-2017 periods. 14 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Table 2: Summary of activity on the KSE Year Equity Share End of Period End of Period Turnover Volume KSE Market (KShs. Bn) (Mn) 20-Share Index Cap (KShs. Bn) 2008 97.52 5856.54 3521.18 853.88 2009 38.16 3160.03 3247.44 834.17 2010 110.32 7545.79 4432.6 1166.99 2011 78.06 5721.83 3205.02 868.24 2012 86.79 5464.23 4133.02 1272 2013 155.75 7665.92 4926.97 1920.72 2014 215.73 8133.67 5112.65 2316 2015 209.38 6812.14 4040.75 2053.52 2016 147.18 5813.49 3186.21 1931.61 2017 171.61 7065.36 3711.94 2521.77 Source: KSE/Capital Markets Authority (2019) Market turnover in the KSE has fluctuated similar to market capitalisation for the period 2008-2017. The All-Share Index has similarly been choppy. Four continuous years, 2011-2014, saw a consistent increase in both turnover and the index levels. Nigeria Stock Exchange The NSE with an equity market capitalisation of $32.13 billion is the largest and most diversified exchange in the sample. It predated the colonial times when the British government used it to issue bonds for construction projects in 1946. On attaining independence, it was fonnalised into the Lagos Stock Exchange in 1960 and later transitioned into the Nigerian Stock Exchange. Following independence, the policy framework of Nigeria was toward indigenisation of businesses; hence the exchange could not expand through foreign participation to really play its role in mobilising capital for 15 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library economic expansion. Uduanu (2019) traces the history of the opening up of the bourse to foreign participation starting 1986 when world crude oil prices tumbled and Nigeria was forced to accede to economic liberalisation programme known as Structural Adjustment Programme (SAP) as a precondition for accessing International Monetary Fund (lMF) aid. The change brought by this led to a wave of listing of firms and the exchange continues to grow to this day. As of the end of 2018, the NSE listed 169 equities, 9 exchange-traded assets,S real estate investment trusts and 130 fixed-income securities. As part of its internationalisation efforts, the NSE is a member of the World Federation of Exchanges (WFE), Sustainable Stock Exchanges (SSE), the International Organisation of Security Commissions (lOSCO) and the African Securities Exchanges Association (ASEA). By far, the NSE is the most liquid and deeper of all markets among sub-Saharan Africa. Unfortunately, the NSE is more prone to turbulence whenever markets in the developed countries are gyrating as it happened in 2008 global financial crisis (Adamu, 2009; Offoing, Riman, & Godwin, 2018). This is the result of deep linkages via foreign investor participation in the NSE (Abdullahi, 2017). Compared with other frontier markets outside of sub-Saharan Africa, however, the NSE remains small with market composition largely concentrated around the financial services (33%), materials (32%) and consumer products (27%). Performance of the NSE has been mainly dictated by changing legislation, especially to do with capital controls occasioned by the crude oil prices on the world market. Table 3 shows market activity on the NSE to the year ending 2018. 16 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Table 3: Performance of the NSE for the period 1985-2018 1985 1990 1995 2000 2005 2010 2015 2018 Equity market capitalisation (USD million) 5,511 2,470 2,606 4,235 19,562 52,527 50,l30 32,l33 Value traded (USD million) 0.06 0.03 0.08 1 7.9 21.2 19.9 14.l Stock market turnover ratio 0.3 0.4 1.1 7.4 12.3 12.3 9.3 9.5 Average price-to-eamings ratio 4.3 7 6.8 7.1 12.8 10 17.8 9.3 Average dividend yield (%) 10.6 12 7.9 7.5 9.5 5.8 5.9 5.6 Number of stocks 96 l31 181 195 214 217 190 169 Source: Central Bank of Nigeria, Nigerian Stock Exchange, Nigeria Securities and Exchange Commission, and Bloomberg (2019) 17 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library The NSE has experience consistent growth largely except in the latter years where market indicators have remained almost flat. Stock turnover increased consistently from 1985-2010 which indicated market liquidity. There was a dip however in the the intervening period to 2015 perhaps coinciding with the drop in the world market price of crude oil, Nigeria's main export commodity. Botswana Stock Exchange The BSI dates 1989 with the formation of the Botswana Share Markets (BSM). The BSM eventually transformed to the BSI through legislative instruments in the latter part of 1994 and it commenced operations with its ownership consisting of brokers via proprietary rights and the government of Botswana in 1995. The bourse currently trades shares, fixed-income instruments and exchange-traded funds denominated in foreign and the local currency known as the Pula. The listed firms on BSI are engaged in a broad range of business sectors ranging from agriculture, mining, energy, financial services, security services among many; with mining representing the single largest activity of companies holding a share of 85.2% according to Bolokwe and Sedimo (2019). The market is dominated by institutional investors. Retail investors hold about 5% of the market. Table 4 shows the compositions of the listing on the BSI in the period from 2014 to 2018. 18 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Table 4: Composition of the BSI up to year ending 2018 2014 2015 2016 2017 2018 New listings 1 2 3 2 Delisting 1 4 0 2 2 Domestic companies listed 23 22 24 24 26 Foreign companies listed 12 10 10 11 9 Total 35 32 34 35 35 Source: BSI (2019) BSI is attracting the attention of some regional and international firms as well as larger privati sed companies like the Botswana Telecommunication Corporation Limited (BTCL). Like any frontier market, however, the BSI suffers from shallow liquity because institutional investors, for example, the pension funds who dominate the market are long term investors who buy and hold instead of trade assets in the market. Currently, the BSI is making efforts to broaden the financial instruments available in the market with the hope of attracting retail investors. Significance of the Study Financial markets follow the developments in their economies and thus go through phases of high and low volatility. Excessive volatility resulting from turbulent markets can be disruptive, throwing investment strategies into chaos as a country scrambles to find the right balance of both monetary and fiscal policies to contain the situation. Financial markets are the pulse of a country's economy. Every significant development in a given country, be it economic, social or political, gets transmitted through financial markets; hence regime switching should correctly capture the changing volatility in returns on 19 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library financial assets. Therefore, this study characterising the volatility of equities in the financial markets is instructive for the investment decision-making in the frontier equity markets of sub-Saharan Africa. There are lots of studies on the nature of heteroscedasticity across the selected countries (See Atoi (2014), Olowe (2009), Nortey, Asare and Mettle (2015), Adjasi (2009), Esman Nyamongo and Misati (2010), Rao and Moseki (2011), Afuecheta, Utazi, Ranganai and Nnanatu (2020), among others). However, none of these studies have incorporated regime changes into their studies. Furthermore, this study also characterises each individual market with a specific regime switchi11g GARCH model based on its data characteristics. This novelty is the point of departure from the extant literature and throws light on the nature of volatility taking cognisance of the underlying data generation process. Out of this study, the investor community will cOlTectly assess the risks associated with trading of financial assets in a multi-state heteroscedastic environment, thus being positioned to fine-tune their strategies appropriately. Also, the pricing of financial assets in these markets is improved as a result of cOlTectiy estimating the volatility in the regimes. For policy-makers, it is essential to bear in mind the changing nature of market conditions. In gyrating markets, when asset prices are falling and volatility high, monetary and fiscal policies should be accommodating to bring calm to the markets. Firms in the fmancial sector are particularly vulnerable to economic phases of high inflation and the attendant high interest rate environment. Loan defaults soar and most borrowers go bust. This affects the liquidity positions and financial health of lenders in the economy. In that case, 20 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library it is essential that policy makes capital adequacy requirements cyclical and sensitive to market regimes. Delimitations Equities of various firms are listed on the respective exchanges across the sub-Saharan Africa sub-region. As the topic suggests, the volatility of these equities is what is under investigation in this study. Given the sheer number of listed equities involved, a decision has been made to use the indices which are the composite representations of the all the list shares. Cheng (2020) and McAleer and Da Veiga (2008) are some of the examples of using stock market indices in the analyses. Limitations of the Study This study suffers essentially from two limitations. Geographically, it did not cover all the regions of sub-Sallaran Africa. It was difficult getting market data from the central region of the frontier sub-Saharan Africa. This region comprises of Cameroon, Gabon, Central African Republic, Equatorial Guinea, Republic of Congo and Chad. Schiereck, Freytag, Grimm and Bretschneider (2018) describes these equity markets as small and with a total market capitalization of nearly €178 million; hence exclusion will not have any significant effect on the findings of this study. Secondly, this study limits itself to the regime switching versions of GARCH class of models (Bollerslev, 1986; Nelson, 1991; Glosten, Jagannathan, & Runkle, 1993; Engle, 1982). Being an important but unobserved market metric, there has been a proliferation of volatility estimation models in the finance literature since Engle (1982). Among many others, we have stochastic models (Taylor, 1994b), extreme value models 21 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library (Embrechts, Kliippelberg, & Mikosch, 2013), implied volatility methods (Black & Scholes, 1973) and a hybrid of GARCH and extreme value models (McNeil & Frey, 2000). Finally, the study uses a purely reduced rather than structural model approach to investigating the nature of volatility switching in these markets without going behind the data to see what changes where are at play in the economy leading to the regime switching. Structural models as used by Merton (1974) and Black and Scholes (1973) seek the reasons behind the numbers to provide a deeper exposition of the facts behind the figures. Organisation of the Study The remainder of the thesis follows the following structure. Chapter two extends work on Chapter one by reviewing extant literature on regime switching models used for volatility estimation in financial markets. The third chapter deals with the methodology and theoretical basis of regime switching GARCH models within the context of Bayesian inference. In Chapter four, we carry out model fitting using data from selected exchanges in sub-Saharan African markets. This is extended to include model adequacy and selection criteria that leads to back-fitting for model fit. Also, the chapter explores the issues on heavy-tailed distribution of the tails and how this relates to the risk measures of value-at-risk. Chapter five concludes the thesis with observations on the chosen models and their implications for portfolio and investment management in the frontier markets of sub-Saharan Africa. Chapter Summary In this chapter, the nature of risk as it relates to the the volatility of market returns was introduced. The exchanges of four sub-Saharan African 22 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library countries have been selected in this study. The selection is based on the CAGE distance theory. The case for the evolution of volatility across these sub­ Saharan African markets is made by looking at the individual characteristics of the various exchanges in the most recent years. We enumerated the research questions and looked at the significance including the limitations of the study for both practitioners and policymakers. In the next chapter, a review of the relevant literature on volatility will be undertaken. 23 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Introduction CHAPTER TWO LITERATURE REVIEW This chapter reviews the literature on the behaviour of volatility across the developed, emerging and frontier markets. It looks at the performance of non-switching versus regime switching volatility models on the major classes of financial assets - equities, bonds, commodities - and how risk in these markets is influenced by the returns that are time-based properties of particular markets. As well, the chapter also reviews literature on the role of tail distributions in heightening risk in the market envirolill1ent tlu'ough heavy­ tails. The Nature of Market Volatility Variations in asset returns contain information vital to any investment strategy. Prolonged periods of turbulent markets signal problems in the economy. This can be the result of unfavourable monetary or fiscal conditions, political uncertainties, social upheavals and the like in the economy. On the other hand, orderly markets with low volatility in asset returns signal an optimally operating economy with few upheavals. Each of these phases may be indicative of when the investors may have to lean against the wind or reorder their strategies to take advantage of a given volatility regime depending on their risk preferences. Investor attitudes to risk is summed up in Engle (2004) that having an idea of the risks in the market enables a reordering of attitudes to it. This is at the heart of investment and portfolio management and, by and large, risks management activities in finance. 24 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Risk, modelled as a time-vatiation mettic in asset returns from their mean was first proposed in Engle's (1982) paper and later generalised by Bollerslev (1986) in which the volatility taking on a heteroscedatic model was first considered. Hitherto, asset returns volatility was simply taken as the standard deviation calculated on the historical returns (Ederington & Guan, 2006). This was oblivious of the the day-to-day, season-to-season variations actually observed in the markets during trading. However, it served investors of the day very well. Markets were relatively simple; the products were largely linear and market regulations at the time were easy to understand. Market upheavals were far and in-between. Starting 1987 when the US stock market crushed in October sending shock waves to global financial markets, the world has experienced increased volatility in returns across various markets - developed, emerging and frontier, and in all types of asset classes (Furbush, 1989; Vandewalle, Boveroux, Minguet, & Ausloos, 1998). Markets have assumed such complexity never experienced in history with financial products such as derivative and options traded with so much leverage and with it, complex market regulations to guide trading. These developments gave rise to ever more complex risk models meant to provide guidance to market agents on how to tame risk. Among many models, the exponential Generalised Autoregressive Conditional Heteroscedasticity of Nelson (1991) and the Glosten-Jagannathan-Runkle­ GARCH (GJR-GARCH) proposed by Glosten, Jagannathan and Runkle (Glosten, Jagannathan, & Runkle, 1993) capturing market leverage and asymmetry in the evolution of volatility respectively, became dominant in the financial and economic disciplines. Over time as markets swung to extremes 25 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library with increasing frequency, additional models such as extreme value theorem volatility methods were developed to accommodate those extremal outcomes (McNeil & Frey, 2000; Embrechts, Kliippelberg, & Mikosch, 2013). The stochastic volatility (SV) models proposed by Taylor (1982a) and have been used in Kim, Shephard and Chib (Kim, Shephard, & Chib, 1998) and currently used extensively in high-frequency financial returns (Barndorff­ Nielsen & Shephard, 2006; Barndorff-Nielsen & Shephard, 2004) looks at volatility as sort of a latent stochastic process in discrete time. But the complexity of SV models coupled with the difficulty in interpreting its estimates limited their use in mainsteam financial modelling (AI & Kimmel, 2007). As market became prone to increased swings at the turn of the century, one characteristic observed about asset returns distributions is the increasing presence of fat-tails and pronounced asymmetry. To capture the combined fat­ tails and asymmetry of the tails, Harvey (2013) and Creal, Koopman and Lucas (2013) put forward the Dynamic Conditional Score (DCS) models where volatility is driven by a score. Gao and Zhou (2016) and Bernardi and Catania (2016) provide evidence that these class of models do capture very closely the stylised facts of heavy tails and the asymmetry of the distributions. There is however a growing concern of the failure of these classical GARCH models to capture risks in the face of the increased turbulence in the markets. This fact is acknowledged extensively in the literature. In the equity markets, Bonilla and Sepulveda (2011) examine stocks in the emerging markets and conclude that the GARCH models find it difficult to deal with the data coming out of such markets. Failure of the classical GARCH models was 26 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library indeed pointed out in an earlier work by McMillan and Speight (2004). In that paper, it was clear the data exhibited characteristics that the model could not capture. Market data, as much as is time-varying, exhibits epochs in . its evolution depending on the changing nature of the underlying causes driving the DGP (Pincus & Kalman, 2004). Ang and Timmerman (2012) identified the underlying changes as mainly due to variations in policies in the monetary and fiscal spaces, market regulation and other secular changes. When markets are orderly, asset returns are stable for the most part. But there comes periods when volatility spikes, albeit for brief periods. Quandt (1958) and Goldfeld and Quandt (1973) formalised this observation into a linear regression specifying two regimes. The authors identified discontinuities in the data series used to estimate the regression equations and reasoned that such switchings gave rise to regimes with distinct means, variances and correlations specific to the data periods. Thus, the data generation processes in economics and finance are prone to changes which should be subject to proper analysis in order to appropriately characterise the defining statistical moments regime-by­ regime. Unfortunately, Quandt's model had two regimes specified a priori. In retrospect, this was a relatively simple but important model that captured the reality of the discontinuities in the DGP giving rise to regimes. In some instances, this discontinuity leads to a complete structural break from one state to another. These insights provided the basis for the work of Hamilton (1990) who formalised the theory behind regime changes in time series literature. Hamilton view market data as a function of the underlying dynamics in the 27 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library giving rise to it. As there are inevitably periods of rising and falling activity directly related to increasing gross domestic growth and recessions respectively, we should expect the data to follow that behaviour. This idea, as simple as it is, was revolutionary in volatility modelling. The modelling assumptions from the perspective of Harvey's model align with a nonlinear data generation process. Structural breaks and regime switching are real observations in time series and for any model to be useful, this fact has to be accommodated in the modelling assumptions. Indeed, some writers, for example, Babikir, Gupta, Mwabutwa and Owusu-Sekyere (2012), used the terms "structural break" and "regime switching" synonymously in their papers. Song (2014) provides the distinguishing difference between structural breaks and regime switching models noting that the latter denotes the situation where past states are recurrent and the former is where a change in the dynamics of the data means a movement to new permanent states in the guiding parameters of the model. In sum, with structural break, none of the previous states is capable of recurring (Maheu & Gordon, 2008; Koop & Potter, 2007). Time series models rely on stationarity for their specification. Unfortunately, the literature is mixed on the issue of stationarity for regime switching data. For the case when the data are taken as a single regime, tests for stationarity are well established with Dickey-Fuller and Kwiatkowski­ PhilIips-Schrnidt-Shin (KPSS) tests (Carrion-i-Silvestre & Sans6, 2006; Hobijn, Franses, & Ooms, 2004; Sephton, 1995). Using such tests for regime switching data, however, is known in the literature to bias the results with power gains or losses which are spurious (Busetti & Taylor, 2003). There are 28 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library well established methods for testing stationarity in structural models as seen in the works of Bai and Perron (2003a; 2003b; 1998). Zivot and Andrews (2002) provide indirect methods for assessing stationarity by establishing change points in the data. Other methods have been established based on both the frequentist and Bayesian methods in the statistical finance literature (see Erdman & Emerson (2007). Following this empirical assumption about time series returns, research 111 financial statistics, econometrics and related disciplines incorporating regime switching in time series has exploded. It is now employed in the prediction of assets returns, estimation of volatility and covariances given that there are regimes in the data. This is seen in the developed (Ang & Timmermann, 2012), emerging (Assoe, 1998) and some frontier markets (Balcilar, Demirer, & Harnmoudeh, 2013; Khalifa, Hammoudeh, & Otranto, 2014). It has also found applications in particular assets markets; crude oil markets (Wang, Wu, & Yang, 20 16; Vo, 2009), natural gas (Geng, Ji, & Fan, 2016), electricity pricing (Janczura & Weron, 2012; Lindstrom & Regland, 2012), gold markets (Bialkowski, Bohl, Stephan, & Wisniewski, 2015; Sopipan, Sattayatham, & Premanode, 2012; Baur & Glover, 2014), among others. Naturally, these markets are very volatile and market players need to estimate volatility accurately in their hedging and trading activities. Regime switching volatility models provide the fine-grained tools for achieving this important goal. Classification of Markets Global trading markets are classified into developed, emerging and frontier. A number of factors are considered in putting a country into one of 29 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library these baskets. These factors are as varied as the level of economic growth, the liquity of trading securities, how industrialised the country is, the political stability of the countries', among others. MSCI (2019), a global index provider, see developed markets as having free trade with a well-run stock market and generally a very high standard of living. On the other hand, it regards the emerging markets as those at the nascent stages of their development, having some characteristics like low sovereign and private debt, a large consuming citizel1lY and vast natural and human resource base. MSCI characterise frontier markets as less matured with much illiquidity, a huge demographic advantage and not too developed political systems. There is a fourth, less known category called stand-alone markets. Information ally, Speidell (2009) looks at this classification in tem1S of information flow in the markets and how this influences liquidity of the trading assets. A market with less information flow on the traded assets is likely to be dominated by noise traders. Perhaps, it is such a characteristic that makes the lower tier markets ripe for global investors searching for yield away from their home bases. Among the most recognised developed markets are US, Britain, France, Germany, Japan, Netherlands, Ireland and Hong Kong. The classification places Brazil, China, India, Argentina, Thailand and South Africa in the emerging category. In the frontier markets, prominent countries include Kuwait, Saudi Arabia, Nigeria, Jamaica, Kenya and Botswana. It is to be noted that a country's level of development usually proxied on the level of GDP has nothing to do with the classification of markets. This distinction is very important given the confusion investors make, sometimes synonymously 30 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library regarding countries with high GDP as developed markets and those with low GDP as frontier or in some instances as stand-alone markets. The Nature of Frontier Economies and Stock Markets Frontier markets are so-called because they are not yet patt of global finance according to the definition provided by the International Finance Corporation (lFC) in 1996. IFC's definition encompassed a hitherto unknown group of twenty-one smaller countries it tracked for investors with its frontier composite index (Speidell & Krohne, 2007). The grouping referred to as "frontier markets" has always been a loose collection of markets with potentially rapid growth in gross domestic product (GDP) and market returns which have low correlations with those of the developed and emerging markets. Marshall, Nguyen and Visaltanachoti (2013) similarly, view frontier markets in terms of their maturity compared with emerging and developed markets counterparts, claiming they lack the quality market attributes like transparency, adequate investor protection, good corporate governance and easy access to information flow on their listed firms. According to Griffin, Lystra and Quisenberry (2015), FTSE, Morgan Stanley Composite Index (MSCI), Russell and S&P have various criteria for the grouping of the world's markets. FTSE Russell's (2019) methodology for the classification of global financial markets range from accessibility to the market by investors; market quality in terms of regulations, multiplicity of trading products, procedures for custodial and selection in trading; size of the market necessary for member of global benchmarks; predictability and the consistency with which the market conducts its procedures; the endurance of mat·kets changes and stability of market regimes to and the cost of 31 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library implementing any change(s) that warrants a proposal to promote or demote the market from its present classification. A summary of the various countries and their classifications by the index providers is shown in Table 5. Table 5: Preview of Frontier Markets according to global index providers FTSE MS(I Russell s&P mE MSCI Russell S&P Argentina 01 01 01 0( M;lita 01 01 Bahrain 01 0( 01 01 Morocco i Bangladesh I 0/ 0/ 0/ Mauritius 0/ 0/ Botswilil(\ " 0/ 01 Namibia 01 Bulgcllill .( 0( 01 Nigeria 0/ 0/ .". ,,' Colombia 0/ Oman 01 01 0/ 01 Cote d'I'loire 0/ .( Paki~tal) .( 01 01 Croatia 01 .; .; l Panal11a 01 Cyprus .; 01 .( Papua New Guinea 01 Ecuador .( Oilti\f .- .( ,I .- Estonia I 01 0/ RomMlia ,f (' .; i Gabon 0/ Serbia 01 0/ I Ghana { 0/ ( Slovakia ,f 01 .; Jamaica 0/ 0/ Slovenia .; .; .; 0/ Jordan t 0/ I 0/ Sri Lanka 0/ i I i Kazakhstan .; 01 01 Trinidad and Tobago 0( 0/ Kenya i i 01 01 Tunisia 0( 01 01 01 Kuw~lt 01 I I Ukraine .; v i lebanon .; 0/ United Arab Emirates 01 .; Lithuilniil if 0/ 0/ Vietnam 0/ 0/ v M(lcl'do"i~ " Zilrllbln ./ Source: Schipke (2015) The classifications in Table 5 reflect the different methodologies used by these index providers. Botswana and Ghana, for example, are classified by three of the four index providers as frontier markets. On the other hand, Kenya and Nigeria, the bigger markets in the sample, are typical frontier markets classified by all the four index providers. These groups are actually fluid and any classification is a reflection of the dynamics prevailing at any given time. For frontier markets, Barclays (2018), a UK bank, sets out some of their characteristics as having a rapidly growing middle class with a rising 32 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library economic growth, typically young populations, countries are endowed with rich and unexplored vast natural resources, have relatively low valuations of the financial assets, their governments pursuing macroeconomic reforms including removing barriers to investor business. Dickson (2013) added that frontier markets have healthy market returns compared to their counterparts in the developed and emerging markets. These characteristics are the ingredients for economic growth and prosperity that provide future markets for firms doing business or potentially have an eye in frontier markets. In terms of size, Schipke (2015) who provides extensive work in the economies of the frontier markets beyond Asia puts the global equity holding in the region of 1-2 percent. As a group, according to Karolyi (2015) and Kearney (2012), these economies are known to be potentially politically unstable and with governance rife with corruption and markets lacking in robust regulatory frameworks - a key requirement for the smooth functioning of financial markets and an assurance of protection of investments and enforcement of contracts. The last point is underpinned by Odell and Ali (2016) who argue that the presence of legal and regulatory structures is particularly important to investors venturing into unknown markets such as in the frontier economies. Dimic, Orlov and Piljak (2015) studied the impact of political stability on the suitability of a country as a destination for investments across developed, emerging and frontier countries and reached the conclusion that instability in a country's governance has debilitating effects on its financial markets. The other concerns, especially related to financial assets, is their illiquidity. The effect of illiquidity of financial assets and how this increases risk is well 33 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library documented in literature. Marshall, Nguyen and Visaltanachoti (2013) noted that illiquidity in frontier markets is mainly the result of thin and asynchronous trading on these exchanges. These twin dangers can lead to rapid deterioration of market conditions and eventually catch investors out on their positions in the markets. Regime-switching Models in Frontier Markets Concerns about the economy, resulting mainly from monetary and fiscal policies, are a worrisome prospect for investors. In the frontier market economies, monetary and fiscal policies go through unending cycles of experimentation, effectively crimping growth and causing dislocations in the economy. The first place those signs are noticed is in the stock market. Indeed, Bj01nland and Leitemo (2009) document the interdependence between macroeconomic performance and the stock market. A soft economy prompts global portfolio investors to damp their holdings of financial assets and sparking market gyrations. In the ensuing volatile environment, a country's currency weakens as investors offload their market holdings and capital exits the economy. The situation is worsened if the country's central bank lacks the firepower with adequate reserves to stem the falling local currency. Again, most of the sub-Saharan African exchanges have so many of their listed firms in the services and commodities sectors. These firms are easily buffeted by global demand shocks beyond the control of their respective countries. The shocks in the underlying economies are transmitted directly to the stock markets as switching regimes in the level of returns on investments. Also, the activities of global rating agencies like S&P, Fitch and Moody's add to the market turbulence in sub-Saharan Africa. The investment 34 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library mandates of most of these global investing firms prohibit investments in assets with junk rating status. Also, a key requirement of investors in a country's stock market is its membership of global financial indices like the MSCI, FTSE Frontier Index, NASDAQ Emerging Markets Index, among others. As economic fortunes ebb and wane in response to economic conditions in countries, so does the risk of the economy's sovereign rating migrating over time. In response to this, investors rebalance their portfolios taking into account the risks in the market. This results in further unsettling of the financial markets. Evidence of this behaviour by global investors is provided in Della Croce, Stewart and Yermo (2011). Other researchers like Shleifer and Vishny (2011) document extensively instances where, for example, investors resorted to fire sales in situations of sovereign rating migrations to lower levels prompting a surge in volatility. In the face of all these developments, risk, being an artificial construct, should reflect fully the underlying behaviour of the economy that gives rise to the data. That is where researchers can potentially give the full narrative of the behaviour of market data. Regime Switching Versus Non-Switching Volatility Models The basis of regime switching models is relatively simple. Financial data are generated by underlying activities in an economy. Such activities undergo phases of booms, busts and everything in-between. Volatility, being central to finance, should reflect this multi-state characteristic in a fine-grained manner. Hamilton's work (1989) on the business cycles is premised on this nonlinearity. Later authors like Luo, Seco, Wang and Dash Wu (2010), Haas, Mittnik and Paolella (2004) and Cai (1994) confirmed the superiority of 35 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library regime switching over non-switching models, claiming they are more accurate in capturing the changing expectations, variances and correlations over time. Ardia, Bluteau, Boudt and Catania (2018) undertook a large-scale study of various heteroscedastic functions employing stocks, aggregate market indices and currency pairs of the developed markets and found that regime switching volatility models performed better in terms of model fit of the data by essentially capturing the nonlinear irregularities that characterises financial market data. Financial market data is a reflection of the behaviour of real people and institutions responding to monetary and fiscal policies and adjusting their decisions in the light of changes in these variables for optimum outcomes. The heterogeneity of these actions is best captured with piecewise GARCH models. Thus, the (G)ARCH models of Engle (1982) and Bollerslev (1986) are best seen as "volatility primitives", the basis of building complex and realistic models. The methodological basis for analysing market data using regIme switching models is built on sound theoretical footing (Diebold, Lee, & Weinbach, 1994). Various authors have also recommended regime switching models in place of non-switching in modeling heteroscedastic outcomes. Lamoureux and Lastrapes (1990) noted the shortfalls in volatility models especially where and when they are needed most - during financial crises when tails risks should be measured accurately. During this time, single­ regime models undershoot risks and lull investors into complacency until the risks manifest. That is the position of Danielsson (2002) who posited that financial data is endogenous to the underlying developments driving asset prices in the markets. It is an attribute missing in non-regime-switching 36 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library models which as a result lack the robustness to describe real markets and provide adequate guidance for risk-based investing and decision making. Eichengreen and Bordo (2002) observed that there is increased frequency of , crises in financial markets today. Danielsson (2008) again takes issue with the models, observing that these models fail when they are needed most. It would appear these models are built without due cognisance of the environment generating the data to carefully look at the changing stylised characteristics such as the abrupt switching of the levels of returns or the quirky but brief extreme returns that lead to heavy-tailed distributions observed in financial time series with increasing frequency. In this environment, improving the models by aligning them closely with the data generation environment for estimating the volatility models is going to be an important modelling technique to forestalling or minimising the impact of crises such as the recent global financial crisis stemming out of mispricing of risks in the US subprime mortgage sector. Non-switching volatility models adapt to changes in market conditions rather slowly. On the other hand, several studies to do with extreme risk have shown that regime switching models are responsive to the changing states of the return generation process, communicating to users the resulting rich heteroscedastic dynamics for decision making (Miao, Wu, & Su, 2013; Weron, Bierbrauer, & Truck, 2004; Kawata & Kijima, 2007). The behaviour of the stock market during a recession vis-a-vis expansion in an economy is the clearest evidence of the presence of market regimes. Standard GARCH models do not staircase risks across time, effectively discounting the evidence based on the behaviour of equity returns in periods of recessions and 37 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library expansions respectively. During recessions, nervous investors flee the equity markets to the safety of sovereign instruments. The reverse is seen in economic expansions. Thus, characterising the risks in the financial markets cannot be divorced from the underlying dynamics at play in the economy especially in measures of volatility, a pure financial construct for risk. The continuing use of the single GARCH models can no longer be justified based on the increasingly turbulent market in a nonlinear environment with many financial products trading at so much speed under a multiplicity of contracts. At best, these earlier GARCH models should be seen as primitives needed to build much more realistic market artefacts that can stand the increasingly fast and unstable real financial markets. For macroeconomics, regime switching now finds use in the policy toolbox of central banks. Alba and Wang (2017), and Duffy and Engle­ Warnick (2006) employed Markov regime switching in studying the applicability of the Taylor rule to monetary policy and how it affects GDP growth and dictates inflation outcomes in the US economy. Indeed, Hamilton (2016) makes the case for the incorporation of changes in regimes in macroeconomic planning citing instances of booms and busts as examples of regime switching mirroring underlying developments in the economy. There is evidence that this is the case in modern macroeconomic policy: inflation targeting (Claeys, 2015), exchange rate policy (Fiess & Shankar, 2009), general monetary policy (Owyang & Ramey, 2004), taxation and debt (Davig, 2004), among many studies in macroeconomics. The level of foreign exchange exerts a lot of control on the financial markets of developing countries (BoswOlth, Collins, & Reinhart, 1999). 38 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Hauser, Marcus and Yaari (1994) discussed at length the additional cost of hedging foreign currency risk by investors trading in markets prone to exchange instability. This prompted Chkili and Nguyen (2014) and Walid, Chaker, Masood and Fry (2011) to investigate the issues at the nexus of equity market returns and foreign exchange rates using multi-state heteroscedastic models to study how spikes in the foreign exchange markets spill into the stock markets in emerging economies. The authors relied on the flexibility afforded by regime switching models, taking advantage of their adaptation to nonlinear environments of evolving uncertainty in the foreign exchange markets. Thus, there is a lot in monetary and fiscal policy planning using regime switching models for policymakers. Independent of the above factors, leverage in financial markets do cause heavy-tails now and then and by extension introduce time-varying parameters of the DGP leading to observed switches in the distributions (Thurner, Farmer, & Geanakoplos, 2012). There is ample evidence in the extant literature, for example, Guidolin (2011), Diebold and Inoue (2001), and Cont (2007), that heavy-tails in financial time series induce surges in volatility leading to regime changes. These effects, usually severe in the left tails, are laden with sudden risks and had led to losses or perhaps the collapse of firms as well documented in finance (Allington, McCombie, & Pike, 2012; Stoyanov, Rachev, Racheva-Yotova, & Fabozzi, 2011; Hilal, Poon, & Tawn, 2011; Chincarini, 2007). Heavy- or fat-tailed distributions connoting distributions with far more probability mass in the tails is much more common in financial data and understanding the nature of how these is another layer of risk provides valuable insights into what causes extreme volatility in the 39 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library markets. A typical manifestation of this is the volatility clustering empirically observed in market returns. The fat-tails are a property driven largely by the extreme return outcomes and this endows the series with large kurtosis beyond values characterising normal or Gaussian distributions. It is the main property that leads to the deviations of normality observed in many financial time series (Allington, McCombie, & Pike, 2012). Frontier markets are a step away from emerging market status; thus, they share some market characteristics in terms of leverage effects (Ibragimov, Ibragimov, & Kattuman, 2013). Basically, it has been observed that leverage effects become dominant in the market when equity prices start to fall. Since a firm debt is fixed, a falling equity means debt is becoming prominent; hence risky. This sends up volatility seen as clustering in the time series of returns (Daal, Naka, & Yu, 2007). Again, heavy-tails in frontier markets could be due partly to hidden information effects related to trading of financial assets. The long memory effects observed in some frontier markets in African, namely, Botswana, Ghana, Mauritius, and Namibia by Rambaccussing (2010) could very much be a function of the latency of information flow in the assets and how this affects the frequency of trading. The halting nature of trade as investors discover information with difficulty in these markets suddenly sends equity and indeed all asset prices up or down. This shows in the form of a lower or higher volatility in the markets and registers as heavy-tails. The Case for Regime-switching in Volatility Modeling for Sub-Saharan Equity Returns For long, it has been known in the investor community that sub­ Saharan Africa frontier equity markets present investors with opportunities to 40 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library diversify their portfolios and also make rich pickings in investment. The low correlation has been attributed to lack of cross-listing of firms across these markets (Bahlous, 2013). Serkin (2015) made the point that portfolio managers who are able to navigate the information starved sub-Saharan frontier stock markets have made decent returns for their investors. Fact remains, however, that academic research has not gone beyond the single regime volatility models needed to provide guidance to investors in such a turbulent part of Africa. Across all of sub-Saharan Africa, the economies are prone regularly to bouts of economic slowdown, foreign currency problems, political uncertainty and unrest, and monetary and fiscal policy missteps. These are the underlying conditions that induce nonlinearities in financial data. The incorporation of regime switching to capture these multi-state evolving uncertainties therefore should be natural in volatility modelling. Yet there is very little in the extant literature on volatility modeling adopting this approach in sub-Saharan frontier equity markets. Outside sub-Sal1aran Africa, academics have recognised the nature of these peculiarities in their economies and the data generated; thus, volatility modeling in these regions does incorporate regimes. In the Gulf and North African region, Balcilar et al. (2013) have incorporated regimes in studying investor herd behaviour on the Gulf Arab bourses. Work done by Khalifa, Harnrnoudeh and Otranto (2014) looked at the spillover effects of the recent US financial crisis on emerging markets using regime switching. Charfeddine and Ajmi (2013) employed regime switching to study long meffiOlY effects in market returns on the Tunisia bourse. In Islamic iinance, 41 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library regime switching models have been employed in the study of the sukuk and sharia equities (Aloui, Hammoudeh, & Hamida. 2015). Unfortunately, there is a lack of regime switching model characterisation in the equity markets of the sub-Saharan African region in the academic literature. This is contrary to the evidence in the empirical data. Volatility models are limited to non-switching GARCH models whose limitations are known to researchers. Adjasi (2009), Nyamongo and Misati (2010), Chinzara and Slyper (2013), Uyaebo, Atoi and Usman (2015), Carsamer (2016) among many researchers have studied volatility of returns of equities in the sub-region employing macroeconomic variables in the context of single-regime models. None has looked at the breaks in the continuity of the data as a result of switching occasioned by the underlying developments which affect asset prices on the exchanges. This is the point where this work distinguishes itself from that of the earlier researchers. Regime switching models contrast sharply the assumptions of linearity in the DGP and clearly take advantage of the stylised facts of financial data. In short, when it comes to the trading environment, regime switching models, whether for returns or for volatility, provides a complete mapping of the developments in the underlying economy to the risks of returns by calibrating risk levels to the potential innovations or shocks arising from policy imperatives and disturbances due to economic or political changes with direct bearing on the financial markets of a country. Chapter Summary Modelling market risk is important for trading in securities. Un fornately , it is easy for the models to be misspecified and by so doing fail to properly capture 42 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library the risks associated with trading and investment decisions-making. This chapter reviewed the seminal and current thinking about risk and volatility models. The industry workhorse has always been the classical GARCH and its variants. The chapter pointed out the deficiency of these GARCH models and reviewed the regime switching models which are better at modelling the market risks based on the underlying developments driving the returns in the markets. 43 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER THREE METHODOLOGY In this chapter, the methodology of the various regime switching models including heavy-tailed distributions are reviewed. The specification adopted is the GARCH model and endow it with changes in regimes of the equity returns. Complex statistical models are known to overfit the data. We therefore look at backtesting these complex models with value-at-risk models, ensuring we have well fitted volatility functions to the data. Introduction Regime switching volatility models are able to accommodate the changing levels of heteroscedasticity found in market returns, especially where the underlying unobserved stochastic process exhibit nonlinearities in the data. It takes a piecewise approach that captures regime-by-regime, the stylized facts in financial time series unlike non-switching GARCH models that average the volatility across all time periods. Typically, sub-Saharan African equity trading is characterised by thin and asynchronous trades mirroring developments in the underlying economies. For such economies in constant flux, time-varying heteroscedastic functions with regime switching are the appropriate models that reflect the endogenous dynamics generating the data and provide a better fit to the returns. As rightly pointed in Marccuci (2005), switching models compared to single regime GARCH functions are able to adapt to the changing conditions in the market, thus offering investors a warning against excessive and disruptive market swings. 44 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library In this chapter, therefore, we specify Markov regime switching volatility models that will be used to fit the data in the next chapter. The models are considered within the univariate financial time series framework. Regime-switching GARCH Models Let {rtlI=l be a time series of sample returns where t = 1,2, ... , T. We suppose that these series exhibit all the stylised properties of financial time series observed in Cont (2001). We adopt a Bayesian inference approach instead of maximum likelihood in estimating the parameters of the model for two principal reasons. First, the problem of estimating parameters for regime switching models is prohibitively expensive or almost intractable. Gray (1996) demonstrated that for a model of k regimes involving T observations, we need to integrate over KT paths. The solution to this path-dependency problem is the Bayesian method which has been demonstrated elegantly in Haas, Mittnik and Paolella (2004) who adopted a regime-by-regime GARCH approach to modeling each regime specific parameters. Secondly, Bayesian estimation accommodates parameter uncertainty, a process that minimises model risk (Clyde & George, 2004). We adopt the specification of the model in Haas, Mittnik and Paolella (2004). In general, we state the distribution assuming regime switching governed by: (3.1) where P(f.1.k,t' qk,t' T/k) is the probability model describing the complete regime k behaviour of the returns rt with mean f.1.k,t and scale qk,t at time t. T/k is the guiding parameter(s) describing the shape of the distribution of returns. Generally /I is taken to be practically zero in the long run for the return , rk,t . 'f t then the serl'es has to be de-meaned. It must be senes or I no zero, 45 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library emphasized that the parameters in the probability model (3.1) are regIme dependent. ll1e information contained in the event history up to and including time t - 1 is denoted Ft - i . The variable St is the state space with a finite set of values k = 1,2,3, . .. , K, which evolve according to a Markov governing law given by: (3.2) The multi-state transition of the volatility from state m to state n is driven by a Kx K first-order homogenous stochastic matrix. This stochastic matrix transiting the regimes from state to state can be written out as: [ Pll JP>(St = nlSt- i = m) = : Pml Pin] . . . . .:. P~n (3.3) where Pmn is the probability of transitioning from regime m to regime n in the period t which is defined by: Pmn exp (XPmn) l+exp (XPmn) (3.4) This matrix has all the properties of a typical Markov chain such that each element Pij > 0 for all i,j E {1, 2, ... , K} and again for each row, "£J=l Pij = 1 V'i E {1, 2, ... , K}. By construction, each row of the transition matrix JP> is an approximation of a probability density functions over the given state K. Conditional Heteroscedastic Functions The variance function, qt,b in (3.1) IS a piecewise GARCH-type function which depends on a given lagged error specification, CE-l' of the residual error distribution, the past variance qt,k-l and a vector of parameters fik which are regime-dependent. This function takes the form (3.5) 46 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library The variance function qt,k C.) in most GARCH specifications is constrained such that qt,k(.) > 0 and its parameters depend on the regime Sk of the market returns. A key property of qt,kC,) is its ability to capture important empirical variance-related stylised properties of a return series. Originally, Engle's ARCH (1982), later generalized into GARCH by Bollerslev (1986) was meant to capture volatility clustering in a return series. Volatility clustering is the tendency to observe clusters in return series over time, that is, volatile periods tend to follow volatile times and vice-versa. Other important effects include asymmetry in response to bad news compared to good news (Glosten, Jagannathan, & Runkle, 1993) and the leveraged effect originally observed by Black (1976) and formalised into a heteroscedastic function by Nelson (1991). Thus, the various GARCH model specifications in the finance literature reflect some important property of financial time series (For a comprehensive review of GARCH models, see Terasvirta (2009), Poon and Granger (2005), Engle (2001) and Franses and Van Dijk (1996). For this study, consideration is given to the regime-switching versions of GARCH (1, 1), the exponential GARCH (1, 1) and the GJR-GARCH (1, 1). The choice of lower order versions of these volatility models is based on research in the econometric literature showing that higher order heteroscedastic functions do not necessarily outperform lower order models (Hansen & Lunde, 2005). The Regime-switching GARCH (1, 1) Model The regime switching GARCH (1, 1) model is specified as follows: (3.6) where 47 © University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Et -iid'.D(O, a}) (3.7) The vector of regime-dependent parameters ({Jk = {Wk' (Xk,/3d is to be estimated. We constrain the parameters as Wk > 0, ak,Pk > ° and ak + Pk < 1, guaranteeing that the variance qt,k will be positive and exist for a particular regime k . nc.) is refers to identical and independent random variables drawn from a normal, student-t, skewed student-t or some other type of distribution which describes the tail behaviour of the returns. The Regime-switching GJR-GARCH (1,1) ModelO54 027 3033 The GJR-GARCH (Glosten, Jagannathan, & Runkle, 1993) IS an asymmetric GARCH model with an extra parameter to take care of the leverage effects induced in the data when the value of equity falls relative to the market value of debt. Falling equity prices lower retUl11S and increase volatility in financial markets. Black (1976) summarised this as "a drop in the value of the firm will cause a negative return on its stock, and will usually increase the leverage of the stock r oo } That rise in the debt-equity ratio will surely mean a rise in the volatility of the stock". Christie (1982) provides additional support for this observation of how tumbling stock prices elevate volatility as a result of the rise in the debt/equity ratio in the firm's capital structure. To accommodate this behaviour, the GARCH (1, 1) in (3.6) is re- specified as: qt,k = Wk + ak EE-l + yll{Et-l