MODELLING RATES OF INFLATION IN GHANA: AN APPLICATION OF AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTIC (ARCH) TYPE MODELS BY MBEAH-BAIDEN BENEDICT (10222719) THIS THESIS IS SUBMITTED TO THE SCHOOL OF GRADUATE STUDIES, UNIVERSITY OF GHANA, LEGON IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF THE MPHIL STATISTICS DEGREE JUNE, 2013 University of Ghana http://ugspace.ug.edu.gh i DECLARATION CANDIDATE’S DECLARATION This is to certify that this thesis is the result of my own research work and that no part of it has been presented for another degree in this university or elsewhere. Signed: ............................ Date: ..................... Mbeah-Baiden Benedict (10222719) (Candidate) SUPERVISORS’ DECLARATION We hereby certify that this thesis was prepared from the candidate’s own work and supervised in accordance with guidelines on supervision of thesis laid down by the University of Ghana. Signed: ........................... Date: ........................ Dr. Julius B. Dasah (Principal Supervisor) Signed: ............................ Date: ........................ Dr. Ezekiel N. N. Nortey (Co - Supervisor) University of Ghana http://ugspace.ug.edu.gh ii ABSTRACT The research is based on financial time series modelling with special application to modelling inflation data for Ghana. In particular the theory of time series is explored and applied to the inflation data spanning from January 1965 to December 2012 which were obtained from the Ghana Statistical Service. Three Autoregressive Conditional Heteroscedastic (ARCH) family type models (traditional ARCH, Generalized ARCH (GARCH), and the Exponential GARCH (EGARCH)) models were fitted to the data. This was especially so because the data were characterized by changing mean and variance. The Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) were used to assess the performance of each of the fitted models such that the model with the minimum value of AIC and BIC was adjudged the best model. The results revealed that the ARCH – family type models, particularly, the EGARCH (2, 1) was superior in performance in forecasting Ghana’s monthly rates of inflation. The results also showed that the monthly rates on inflation were not weakly stationary and although there was the presence of asymmetric effects in the volatility in the monthly rates of inflation, there was an absence of leverage effects as positive shock increased the volatility in the monthly rate of inflation more than a negative shock of equal magnitude. The study recommends that policy makers and all interested in modelling and forecasting monthly rates of inflation in Ghana should consider using the Heteroscedastic models as it is able to properly capture the volatilities in the monthly rates of inflation. Analysis were done using MINITAB 16.0 and EVIEWS 5.0. University of Ghana http://ugspace.ug.edu.gh iii DEDICATION This work is dedicated to the Lord God Almighty for the divine wisdom and strength given me to go through this research successfully. It is also dedicated to Very Rev. Andrew Mbeah- Baiden and Mrs. Rebecca Mbeah- Baiden, my parents and also to my sweet siblings, Andrew, Christian and Gifty for their love, care and support. Lastly, this work is dedicated to Miss Deborah Esi Edzie for her gargantuan support and encouragement. University of Ghana http://ugspace.ug.edu.gh iv ACKNOWLEDGEMENT This thesis would not have been possible if it were not for the tireless guidance and support I got from my supervisors Dr. Julius B. Dasah and Dr. Ezekiel N.N. Nortey; I say thank you. To all other faculty members, I owe you my deepest gratitude for your support and assistance; you stood by me every step of the way, both during the course work and during the period of thesis write up. I am indebted to my colleagues; your presence provided a source of warmth and gave me a source of hope in challenging and difficult periods through the research. I would like to express my sincere gratitude to Victoria Agyepong for her unwavering support and encouragement. Also sincere appreciation is extended to the Carnegie Corporation of New York for funding this research under the New Generation of Academics in Africa through Office of Research, Innovation and Development (ORID), University of Ghana. Lastly, it is my pleasure to thank my cousins Antoinette, Gifty, Georgina and all those not mentioned by names but whose support made this thesis a success. May the Almighty God richly bless you all. University of Ghana http://ugspace.ug.edu.gh v TABLE OF CONTENTS Contents DECLARATION ............................................................................................................ i ABSTRACT ................................................................................................................... ii DEDICATION ............................................................................................................. iii ACKNOWLEDGEMENT ............................................................................................ iv TABLE OF CONTENTS ............................................................................................... v LIST OF TABLES ......................................................................................................... x LIST OF FIGURES ...................................................................................................... xi LIST OF ABBREVIATIONS ...................................................................................... xii CHAPTER ONE ............................................................................................................ 1 INTRODUCTION ......................................................................................................... 1 1.0 Background of the Study ................................................................................. 1 1.1 Statement of Problem ...................................................................................... 4 1.2 Objectives of the Study ................................................................................... 6 1.2.1 General Objective .................................................................................... 6 1.2.2 Specific Objectives .................................................................................. 6 1.3 Significance of the Study ................................................................................ 7 1.4 Scope and Methodology .................................................................................. 7 1.5 Organisation of the study ................................................................................ 9 CHAPTER TWO ......................................................................................................... 10 University of Ghana http://ugspace.ug.edu.gh vi LITERATURE REVIEW ............................................................................................ 10 2.0 Introduction ................................................................................................... 10 2.1 The Concept of Inflation ............................................................................... 10 2.1.1 Introduction ............................................................................................ 10 2.1.2 Consumer Price Index as a Measure of Inflation ................................... 11 2.1.3 Construction of Consumer Price Index in Ghana .................................. 12 2.1.4 The Ghanaian Experience of Inflation ....................................................... 14 2.2 Review of Related Works ............................................................................. 18 2.2.1 Review of Related Works in Ghana and Other African Countries ........ 19 2.2.2 Review of Related Works in the Rest of the World ............................... 25 CHAPTER THREE ..................................................................................................... 41 METHODOLOGY ...................................................................................................... 41 3.0 Introduction ................................................................................................... 41 3.1 Time Series and its Basic Concepts .............................................................. 41 3.1.1 Stationary and Non Stationary Processes .............................................. 43 3.2 ARCH (𝑚) MODEL ..................................................................................... 44 3.2.1 ARCH (1) Model ................................................................................... 46 3.2.2 Testing for ARCH Effects ..................................................................... 47 3.2.2.1 Ljung – Box Test .................................................................................. 47 3.2.2.2 Lagrange Multiple (𝐿𝑀) Test................................................................ 48 3.2.3 Determination of the order of ARCH (𝑚) Model .................................. 49 University of Ghana http://ugspace.ug.edu.gh vii 3.2.4 Estimation of the ARCH (𝑚) and ARCH (1) Models ........................... 49 3.2.4.1 Estimation of the ARCH (𝑚) model ..................................................... 49 3.2.4.2 Estimation of the ARCH (1) Model ...................................................... 51 3.2.5 Forecasting with the ARCH Model ....................................................... 55 3.3 The GARCH (𝑚, 𝑠) Model ............................................................................ 57 3.3.1 GARCH (1, 1) Model ............................................................................ 61 3.3.2 Estimation of GARCH (𝑚, 𝑠) model ..................................................... 63 3.3.3 Estimation of the GARCH (1, 1) ......................................................... 63 3.3.4 Forecasting with GARCH (𝑚, 𝑠) model ................................................ 66 3.4 EGARCH (m, s) Model ................................................................................. 68 3.4.1 EGARCH (1, 1) Model .......................................................................... 70 3.4.2 Forecasting with EGARCH Model ........................................................ 72 3.5 Model Selection Criteria ............................................................................... 74 3.5.1 Akaike Information Criterion (AIC) ...................................................... 75 3.5.2 Bayesian Information Criterion ............................................................. 77 3.6 Model diagnostic checks and adequacy ........................................................ 77 3.7 Model Validation ........................................................................................... 79 3.8 Assessment of Predictiveness or Forecast Accuracy of a Model .................. 79 3.9 Conclusion ..................................................................................................... 80 CHAPTER FOUR ........................................................................................................ 81 DATA ANALYSIS AND DISCUSSION OF RESULTS ........................................... 81 University of Ghana http://ugspace.ug.edu.gh viii 4.0 Introduction ................................................................................................... 81 4.1 Summary Statistics and Data Description ..................................................... 81 4.2 Preliminary Analysis ..................................................................................... 83 4.3 Model Fitting and Estimation ........................................................................ 91 4.4 Diagnostic Checks and Adequacy for estimated Models .............................. 97 4.4.1 Diagnostic Checks and Adequacy for the ARCH (2) Model ................. 97 4.4.2 Diagnostic Checks and Adequacy for the GARCH (2, 1) Model ........ 100 4.4.3 Diagnostic Checks and Adequacy for the EGARCH (2, 1) Model ..... 102 4.5 Most Appropriate Model Selection ............................................................. 105 4.6 Forecasting Evaluation and Accuracy Criteria ............................................ 106 4.7 Comparison of the EGARCH (2, 1) and ARIMA (2, 1, 1) Models ............ 107 4.8 Discussion of Results .................................................................................. 109 CHAPTER FIVE ....................................................................................................... 111 SUMMARY, CONCLUSIONS AND RECOMMEMNDTIONS ............................. 111 5.0 Introduction ................................................................................................. 111 5.1 Summary ..................................................................................................... 111 5.2 Conclusion ................................................................................................... 114 5.3 Recommendations ....................................................................................... 115 REFRENCES ............................................................................................................. 116 APPENDICES ........................................................................................................... 125 APPENDIX A ........................................................................................................ 125 University of Ghana http://ugspace.ug.edu.gh ix APPENDIX B ........................................................................................................ 137 APPENDIX C ........................................................................................................ 142 University of Ghana http://ugspace.ug.edu.gh x LIST OF TABLES Table 2.1.3: 12 Main Classes of items used in the Construction of CPI in Ghana......................................................................................................................... 13 Table 4.1.1: Descriptive Statistics of Monthly Rates of Inflation in Ghana (1965 - 2012).......................................................................................................................... 81 Table 4.2.1: Augmented Dicker Fuller (ADF) Unit Root Test for the Monthly Rates of Inflation in Ghana (1965 -2012).................................................................. 83 Table 4.2.2: Test for Heteroscedasticity (ARCH Effects) for the Monthly Rates of Inflation in Ghana (1965 - 2012)............................................................................... 85 Table 4.2.3: Test for Heteroscedasticity (ARCH Effects) for the First Differenced Monthly Rates of Inflation in Ghana (1965 - 2012).................................................. 86 Table 4.3.1: Comparison of Suggested ARCH (m) Models with fit statistics...... 91 Table 4.3.2: Model Output for ARCH (2) Model.................................................. 92 Table 4.3.3: Model Output for ARCH (3) Model.................................................. 92 Table 4.3.4: Comparison of Suggested GARCH (m, s) Models with fit statistics..................................................................................................................... 93 Table 4.3.5: Model Output for GARCH (2, 1) Model........................................... 94 Table 4.3.6: Comparison of Suggested EGARCH (m, s) Models with fit statistics..................................................................................................................... 95 Table 4.3.7: Model Output for EGARCH (2, 1) Model........................................ 95 Table 4.4.1.1: Lagrange Multiplier ARCH Test for ARCH (2) Model................... 98 Table 4.4.2.1: Lagrange Multiplier ARCH Test for GARCH (2, 1) Model.......... 101 Table 4.4.3.1: Lagrange Multiplier ARCH Test for EGARCH (2, 1) Model........ 104 Table 4.5.1: Selection Criteria Values for GARCH (2, 1) and EGARCH (2, 1) Models..................................................................................................................... 104 Table 4.6.1 Forecast performance of estimated models..................................... 106 Table 4.7.1 Comparison of performance results of ARMA (2, 1), ARIMA (2, 1, 1) and EGARCH (2, 1) Models................................................................................... 106 Table 4.7.2 One year in-sample forecast of the Monthly Inflation Rate from the EGARCH (2, 1) Model............................................................................................ 107 University of Ghana http://ugspace.ug.edu.gh xi LIST OF FIGURES Figure 4.1.1: Residual Plots of Monthly Rates of Inflation in Ghana (1965 – 2012).......................................................................................................................... 82 Figure 4.2.1: Time Series Plot of Monthly Rates of Inflation in Ghana (1965 – 2012).......................................................................................................................... 82 Figure 4.2.2: Trend Analysis Plot of Monthly Rates of Inflation Rates in Ghana (1965 – 2012)............................................................................................................ 83 Figure 4.2.3: Time Series Plot of the First Differenced series of Monthly Rates of Inflation in Ghana (1965 – 2012).............................................................................. 84 Figure 4.2.4: Autocorrelation Function (ACF) Plot of Monthly Rates of Inflation in Ghana (1965 – 2012)................................................................................................. 87 Figure 4.2.5: Partial Autocorrelation Function (PACF) Plot of Monthly Rates of Inflation in Ghana (1965 – 2012).............................................................................. 87 Figure 4.2.6: Autocorrelation Function (ACF) Plot of the First Differenced series of Monthly Rates of Inflation in Ghana (1965 – 2012)................................................. 88 Figure 4.2.7: Partial Autocorrelation Function (PACF) Plot of the First Differenced series of Monthly Rates of Inflation in Ghana (1965 – 2012)................................... 89 Figure 4.4.1.1: Time Plot of the Residuals from ARCH (2) Model.......................... 97 Figure 4.4.1.2: Normal Probability Plot of the Standardized Residuals from ARCH (2) Model................................................................................................................... 98 Figure 4.4.1.3: Histogram of the standardized Residuals from ARCH (2) Model.... 98 Figure 4.4.2.1: Time Plot of the Residuals from GARCH (2, 1) Model................... 99 Figure 4.4.2.2: Normal Probability Plot of the Standardized Residuals from GARCH (2, 1) Model............................................................................................................. 100 Figure 4.4.2.3: Histogram of the standardized Residuals from GARCH (2,1) Model....................................................................................................................... 101 Figure 4.4.3.1: Time Plot of the Residuals from EGARCH (2, 1) Model............... 102 Figure 4.4.3.2: Normal Probability Plot of the Standardized Residuals from EGARCH (2, 1) Model.......................................................................................... 103 Figure 4.4.3.3: Histogram of the standardized Residuals from EGARCH (2, 1) Model....................................................................................................................... 103 University of Ghana http://ugspace.ug.edu.gh xii LIST OF ABBREVIATIONS ACF Auto Correlation Function AIC Akaike Information Criterion APARCH Augmented Power Auto Regressive Conditional Heteroscedastic ARCH Auto Regressive Conditional Heteroscedastic ARFIMA Auto Regressive Fractionally Integrated Moving Average ARSV Auto Regressive Stochastic Volatility BIC Bayesian Information Criterion BVAR Bayesian Vector Auto Regressive COICOP Classification of Individual Consumption by Purposes CPI Consumer Price Index EGARCH Exponential Generalized Auto Regressive Conditional Heteroscedastic ERP Economic Recovery Programme GARCH Generalized Auto Regressive Conditional Heteroscedastic GARCH-M Generalized Auto Regressive Conditional Heteroscedastic in Mean GDP Gross Domestic Product GED Generalized Error Distribution GNPA Ghana National Petroleum Authority GJR Glosten Jagannathan Runkle GSS Ghana Statistical Service HQ Hannan-Quinn IGARCH Integrated Generalized Auto Regressive Conditional Heteroscedastic IT Inflation Targeting LA-VAR Lag-Augmented Vector Auto Regressive MOFEP Ministry of Finance and Economic Planning MT Monetary Targeting University of Ghana http://ugspace.ug.edu.gh xiii PACF Partial Auto Correlation Function PARCH Power Auto Regressive Conditional Heteroscedastic QTM Quantum Theory of Money RBF Radial Basis Function RW Random Walk SAP Structural Adjustment Programme SAR Simple Auto Regressive SARIMA Seasonal Auto Regressive Integrated Moving Average SVAR Structural Vector Auto Regressive SVM Support Vector Machine TAR Threshold Auto Regressive TGARCH Threshold Generalized Auto Regressive Conditional Heteroscedastic VaR Value at Risk VAR Vector Auto Regressive VAT Value Added Tax University of Ghana http://ugspace.ug.edu.gh 1 CHAPTER ONE INTRODUCTION 1.0 Background of the Study Price stability (stable inflation) is one of the main objectives of every government as it is an important economic indicator that the government, politicians, economists and other stakeholders use as their basis of argument when debating on the state of the economy (Suleman and Sarpong, 2012). In recent years, rising inflation has become one of the major economic challenges facing most countries in the world especially developing countries like Ghana. David (2001) described inflation as a major focus of economic policy worldwide. This is rightly so as inflation is the frequently used economic indicator of the performance of a country’s economy as it has a direct effect on the state of the economy. In Ghana, the debate of achieving a single digit inflation value has been the major concern for both the government and the opposition parties. While the government boasts of a stable economy with consistent single digit inflation, the opposition parties’ doubts these figures and believe that the figures had been cooked up and do not reflect the true situation in the economy. Despite the different opinions on the inflation figures, it is important to point out that, both the government and the opposition parties are concerned about the inflation (general level of prices) in the country as it affects all sectors of the economy. Webster (2000) defined inflation as the persistent increase in the level of consumer prices or a persistent decline in the purchasing power of money. Hall (1982) also expresses inflation as a situation where the demand for goods and services exceeds University of Ghana http://ugspace.ug.edu.gh 2 their supply in the economy. Inflation and its volatility entail large real costs to the economy (Moreno, 2004). Among the harmful effects of inflation volatility are the higher risk of permia for long term arrangement, unforeseen redistribution of wealth and higher costs for hedging against inflation risks (Rother, 2004). Thus inflation volatility can impede growth even if inflation on the average remains restrained (Awogbemi and Oluwaseyi, 2011) and hence monetary policy makers are more interested in containing and reducing inflation through price stability (Amos, 2010). Policy makers will be content and satisfied if they are able to understand the underlying dynamics of inflation and how it evolves. Ngailo (2011) observes that inflation dynamics and evolution can be studied using a stochastic modelling approach that captures the time dependent structure embedded in the time series inflation data. Traditional time series models assume a constant conditional variance. However, to a large extent most economic and financial series often exhibit non–constant conditional variance (Heteroscedastic) and hence traditional time series do not perform well when used to forecast such series. The heteroscedasticity affects the accuracy of forecast confidence limits and thus has to be handled by constructing appropriate non – constant variance models (Amos, 2010). According to Maddela and Rao (1996), until some fifteen years ago, the focus of statistical analysis of time series centred on the conditional first moments. The increased role played by risk and uncertainty in models of economic decision making and the finding that common measures of risks and volatility exhibit strong variation over time lead to the developments of new time series techniques for modelling time-variants in the second moments. University of Ghana http://ugspace.ug.edu.gh 3 Several models such as the Autoregressive Conditionally Heteroscedastic (ARCH) model and its variants like the Generalised ARCH (GARCH) and Exponential GARCH (EGARCH) models have therefore been developed to model the non- constant volatility of such series. The ARCH model was introduced by Engle (1982) and later it was modified by Bollerslev (1986) to a more generalized form known as the GARCH. The GARCH model has been used most widely for the specification of the ARCH. The GARCH model imposed restrictions on the parameters to assure positive variances. Nelson (1991) therefore presented an alternative to the GARCH model by modifying the GARCH to Exponential GARCH (EGARCH) model. Unlike the GARCH, the EGARCH does not need the inequality restrictions on the parameters to assume a positive variance. A practitioner has the option to use any or a combination of the models based on the performance of such a model in the particular series he or she is estimating. As a result of this, there is the need for performance evaluation to be done so that the practitioner would be able to choose the optimal model among competing class of models. Several evaluation criteria have been developed to measure the performance of these models. One criterion that is popularly used is by estimating the maximum likelihood of the models and observing which of them has the highest log-likelihood value (Shephard, 1996). In situations when the models do not have the same number of parameters, the principle of parsimony is applied and a suitable model selection criterion such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and the Hannan–Quinn (HQ) is used to choose the best model. Another model criterion is to submit estimated models to misspecification tests and observe each model perform under each scenario. University of Ghana http://ugspace.ug.edu.gh 4 Although there are several empirical approaches and findings on the relative performance of time series models that are based on the non–constant variance, these works are primarily on time series data from the developed countries such as the US and Europe. It is therefore necessary to show or exhibit more international evidence on the relative performance of these models especially in developing countries and that is the essence of this study. This study investigates and provides empirical evidence of the relative performance of the ARCH, GARCH and EGARCH models using the univariate time series analysis in which the analysis is on the present and past values of single series. That is the study seeks to investigate the time series of the monthly inflation rates in Ghana using the ARCH, GARCH and EGARCH models and determines the best among these models in forecasting the inflation rate values in Ghana. 1.1 Statement of Problem Inflation is the measure of the persistent and continuous rise in the general price levels in an economy or a country. Inflation is one economic factor that affects all other levels of the economy and every country or government aims to control inflation as a result of this. Due to the fact that inflation levels affect all other sectors of the economy especially business transactions, it is important to be able to forecast or estimate the value of inflation in the future so that such values are incorporated in decisions affecting all these other sectors. University of Ghana http://ugspace.ug.edu.gh 5 Empirical researches have been carried out in the area of inflation modelling and forecasting in Ghana. Examples include Aidoo (2010), Alnaa and Ahiakpor (2011), Suleman and Sarpong (2012), etc. All these researchers attempted to model inflation in Ghana using models that did not capture the conditional heteroscedasticity of the time series inflation data. Gujarati (2004) asserted that the underlying characteristic of most financial time series is that, in their level form they are random walks, i.e. they are non–stationary. It is been argued by Campbell, Lo and MacKinlay (1997) that it is both statistically inefficient and logically inconsistent to use models that are based on the assumption of constant variance over some period when the resulting series progress over time. In the case of financial data for example, large and small errors occur in clusters which implies that large returns are followed by more large returns and small returns are also followed by further small returns. When applied to inflation time series data, it is equivalent to saying that periods of high inflation are usually followed by further periods of high inflation while low inflation is likely to be followed by further periods of low inflation (Amos, 2010). Time series models that capture the conditional heteroscedasticity of time series inflation data had been developed to model and forecast the rates of inflation using time series analysis. These models have been used and empirical evidence on their relative performance has been given for developed economies like the US and Europe. However, limited or no studies have been done in the context of developing countries. This indicates a gap in literature or information on the relative performance of these models in the context of developing countries and poses a challenge as to which of these models is the optimal choice for modelling and forecasting economic and financial data (in particular inflation rates) for developing countries. University of Ghana http://ugspace.ug.edu.gh 6 In view of this, the study intends to model inflation in Ghana using the ARCH-type models and to choose the most appropriate model suitable for inflation modelling and forecasting in Ghana. 1.2 Objectives of the Study 1.2.1 General Objective The main objective of this study is to investigate the relative performance of three selected Autoregressive Heteroscedastic time series models (ARCH, GARCH and EGARCH). 1.2.2 Specific Objectives The specific objectives would be to; i. Fit each of the three time series models(ARCH,GARCH and EGARCH) using the inflation rates in Ghana; ii. Identify the optimal model; iii. Predict a one year out–sample forecast based on the optimal model and; iv. Identify the presence of asymmetric and leverage effects in the volatility in the monthly rates of inflation. University of Ghana http://ugspace.ug.edu.gh 7 1.3 Significance of the Study The empirical results and findings from this study would be significant to industry practitioners and policy makers such as the government, businesses and the general public as well as academics and researchers due to the following reasons: First of all by identifying the optimal model based on their relative performance, better and robust forecasts of inflation values which will be very useful in the planning activities of the government, businesses and the public in general would be obtained. Secondly, the results from this study will benefit academia and research by contributing to existing literature by closing or the elimination of the gap in literature or information on the relative performance of these models in the context of developing countries. It will also serve as a basis for further research for both academic researchers and industry practitioners. 1.4 Scope and Methodology The study was carried out in a developing country specifically Ghana. Secondary data consisting of year-on-year inflation data for each month from January 1965 to December 2012 was used in this study. The total number of data points is therefore 576. The year-on-year inflation is the percentage change in the consumer price index (CPI) over a twelve-month period which is used to measure changes over time in the general price level of goods and services that households acquire for the purpose of consumption. The monthly year-on-year inflation is collected by the Ghana Statistical Service. University of Ghana http://ugspace.ug.edu.gh 8 The data was analysed and the three selected time series models (i.e. the ARCH, GARCH and the EGARCH) for the non–constant conditional variance series were estimated using the maximum likelihood estimation process. The estimation of the model consists of four stages namely; testing for ARCH effects, identification, estimation of parameters and the diagnostic checking stages. The ARCH effects were tested using the Ljung-Box statistics Q (m) test (McLeod and Li, 1983) and the Lagrange multiplier test of Engle (1982) as this forms the basis for building ARCH-type models. The partial autocorrelation function (PACF) of the squared residuals was used to determine the order. Next the estimation of the parameters for the tentative models was carried out using the maximum likelihood estimation method. Three likelihood functions are commonly used in ARCH – typed model estimation depending on the assumption of the distribution that the residuals follow. The distributions are the normal distribution, heavy-tailed distribution such as the standardised student-t distribution and the generalised error distribution (GED). In this study, it is assumed that the residuals are normally distributed since it is the most commonly used distribution and it makes the estimation of the parameters relatively easier. Lastly, the estimated models were checked to verify if it adequately represents the series. Diagnostic checks were performed on the residuals to see the validity of the distribution assumption. In particular, the measure of skewness, kurtosis and Quantile-to Quantile plot (Q-Q plot) of the residuals was used to check for the validity of the distribution assumption. All the analyses were carried out with statistical software MINITAB 16.0 and EVIEWS 5.0. University of Ghana http://ugspace.ug.edu.gh 9 1.5 Organisation of the study The study is divided into five chapters. Chapter one introduces the research study, providing the background to the study, statement of the problem, objectives, the significance, scope and brief methodology used in the study. Chapter two focuses on the conceptual framework and review related literature that pertains to the study whilst Chapter three presents the detailed methodology used for the study. Chapter four covers the data analysis, presentation and discussion of the results. Finally, Chapter five encompasses the summary, conclusion, recommendations, and direction for future research. University of Ghana http://ugspace.ug.edu.gh 10 CHAPTER TWO LITERATURE REVIEW 2.0 Introduction This chapter reviews the relevant theories and concepts associated with inflation and related works that has been carried out by other researchers on the topic area. The chapter is divided into two main headings namely: The Concept of inflation and Review of related works. 2.1 The Concept of Inflation 2.1.1 Introduction According Webster (2000), inflation is the persistent and continuous rise in the levels of the consumer prices in an economy. Inflation can also be seen as the persistent decline in the purchasing power of money. That is, inflation means that your money can not buy today as much as what it could have bought yesterday. There are different theories that have been proposed by economists to explain the occurrence of inflationary situation. These numerous theories can be grouped into two main broad theories; the excess- demand theory and the cost- push theory. The excess-demand theory argued that the excess demand for goods and services over supply in the economy is the main source of inflation as expressed by Hall (1982). On the other hand, the cost-push theory of inflation believes that inflation can be triggered by the increase in the cost of production of firms. The increase in the cost of University of Ghana http://ugspace.ug.edu.gh 11 production will affect the profit margins of these firms and hence they will have to pass on the extra to consumers by increasing the prices of their products. The effects of inflation include among other things, people losing confidence in the currency as the real value of the currency is severely reduced. Inflation can also lead to the ‘wage-price spiral’. This is the situation in which there are higher wage demands as people try to maintain their real living standards. This leads to businesses to increase prices to maintain profits and higher prices then put further pressure on wage. Furthermore, inflation can lead to a build up of inflation expectations that can worsen the trade-off between unemployment and inflation. Lastly, the uncertainty created by the rising inflation can also disrupt business planning since budgeting becomes difficult. Bailey (1956) observed that inflation has negative effects on the economy through its cost on welfare. Furthermore he stated that the costs associated with unanticipated inflation are the distributive effects from creditors to debtors, increasing uncertainty affecting consumption, savings, borrowing and investment decisions. 2.1.2 Consumer Price Index as a Measure of Inflation Various indexes have been devised to measure inflation. These indexes include consumer price index, producer price index, cost of living index, commodity price index and the Gross Domestic Product (GDP) deflator. However, the consumer price index is the most common way of measuring inflation. The consumer price index is a measure for capturing changes over time (monthly, quarterly, yearly) in the general price level of goods and services. This is determined University of Ghana http://ugspace.ug.edu.gh 12 at a beginning period called the base period and according to a fixed pattern of consumption called weight assigned to a representative sampled basket of goods and services. The consumer price index is then used to calculate the inflation rate as shown below. Let 𝑃𝑡 be the current average price level of an economic basket of goods and services and 𝑃𝑡−1 be the average price level of the same basket a year ago, then the inflation rate 𝐼𝑡 at time 𝑡 is calculated as; 𝐼𝑡 = 𝑃𝑡 − 𝑃𝑡−1 𝑃𝑡−1 * 100% 2.1.3 Construction of Consumer Price Index in Ghana In Ghana, the consumer price index is calculated by the Ghana Statistical Service (GSS) which is a department of the Ministry of Finance and Economic Planning (MOFEP). The Ghana Living Standards Survey generates the basket of goods and services classified into 12 main classes using the classification of individual consumption by purpose (COICOP) system. These baskets of goods and services are then used in the construction of the CPI based on the weight assigned to each item in the basket. In all there are 242 items and the weight assigned to each item depends on the expenditure on that item such that high volume expenditure items carry the most weight and therefore would have the most material impact on the calculated index (GSS Newsletter, 2009). The CPI covers prices collected from a national sample of 40 markets. The markets are made up of 9 urban and 31 rural markets across the country. Prices are collected University of Ghana http://ugspace.ug.edu.gh 13 every first and third week of the month from 6 traders in the urban markets and 3 traders in the rural markets for all goods excepts those with fixed prices such as stamps. The prices are then used to construct the CPI which is in turn use to calculate the inflation rates. Currently, the construction is based on 2002 base year having been changed from 1963 firstly in 1977 and then in 1997. Table 2.1.3 shows the 12 main classes of items used in constructing the CPI with their corresponding weights and the number of items in each class. Table 2.1.3: 12 main classes of items used in constructing the CPI in Ghana CLASS WEIGHT NUMBER OF ITEMS Food and Non-alcoholic beverages 44.91 76 Alcoholic beverages, tobacco and narcotics 2.23 11 Clothing and Footwear 11.29 59 Housing, Water, Electricity, Gas and Other Fuels 6.98 10 Furnishings, Household Equipment and Routine Maintenance of the house 7.83 43 Health 4.33 9 Transport 6.21 9 Communication 0.31 3 Recreation and Culture 3.04 6 Education 1.60 2 Restaurants and Hotels 8.28 7 Miscellaneous Goods and Services 2.99 7 Total 100 242 Source: Ghana Statistical Service University of Ghana http://ugspace.ug.edu.gh 14 2.1.4 The Ghanaian Experience of Inflation This section attempts to take a look at Ghana’s inflation experience since the attainment of independence. This is relevant as a good appreciation of the need to model and forecast Ghana’s inflation certainly require an understanding of where Ghana has come from, where Ghana is currently in terms of inflation rates. Ghana has experienced high rates of inflation for several decades. However since July 2009, inflation has fallen consistently even to a single digit level being achieved since June 2010. Ocran (2007) asserts that the inflation in Ghana from independence to 2003 can be characterised as episodic, identifying four distinct episodes: the immediate post independence period which was up to 1966; immediate post Nkrumah period (1966- 1972); the deterioration period of 1972-1982 and the most recent period (1982-2003), which he termed the stabilization inflationary experience. This study would adopt Ocran (2007) episodic characterisation of inflation in Ghana and modify it slightly by adding a new episode. Hence the study would review Ghana’s inflation experience under five distinct episodes as follows: the immediate post independence period up to 1966; immediate post-Nkrumah period (1966-1972); the deterioration period (1972- 1983); recent period (1984-2000) and the most recent period of 2001-2012 which would be termed the single-digit inflationary experience. The first five years in the post independence period (1957-1962) saw inflation centring on a single-digit. This stability in prices could be attribute to the trickling effects of Ghana been a member of the West African Currency Board (WACB) which consisted of the four British Colonies of West Africa - Ghana, Nigeria, Sierra Leone and The Gambia. The currency board had no control of the discretionary monetary University of Ghana http://ugspace.ug.edu.gh 15 policy and as a result, market forces determined the money supply in the member countries. Ocran (2007) pointed out that during the currency board years, inflation was in single digits and indeed inflation rates were typically estimated at less than 1%. Between 1960 and 1962, inflation averaged 8% per annum and then increased to 23% per annum between 1964 and 1966, by which time the trickling benefits of Ghana having been a member of the WACB had been eroded. From 1966 to 1972, inflation rates were in the range of 31.2% (January 1966) to - 12.1% (July 1967), with annual averages in the range of -8.3% to 10.2%. During this period, there was a devaluation of the cedi by 30% against the US dollar and massive retrenchment exercise in the public sector (Hutchful, 2002). This led to a deflation of about 8% in 1962. It is worth noting that from December 1966 through to December 1967, the inflation was less than zero (negative). The period 1972 to 1983 arguably has been the period that inflation rates increased the most in the economic history of Ghana. According to Apaloo (2001), inflation was running at about 100% at the beginning of 1979. In the mid-1979, however, the rate dropped dramatically by about 25% following the coup-de-etat in June 1979. With the exception of the first five months in 1980, the inflation rates ranged between 40% and 88% in 1980. Throughout the period 1981to1983, inflation rates were over 100% for all the months with the rates reaching a peak of 174% at the end of June 1983 before declining to 142 % in December. In sum, the inflationary experience in most of the period 1972-1983 was largely due to expansionary fiscal and loose monetary policies and the attempt to using controls such as fixed exchange rate, import licensing and administered prices for goods and services to hold down inflation (Ocran, 2007). In particular, the high rate of inflation recorded in 1983 could be attributed to the devaluation of the cedi, the drought and famine. University of Ghana http://ugspace.ug.edu.gh 16 The recent episode of inflation started on the backdrop of the introduction of the Economic Recovery Programme (ERP) and the successive Structural Adjustment Programme (SAP) in 1983. The ERP had two stages of implementation. The first stage, ERP I (1983-1986) had a stabilization package aimed at reducing inflation and fostering external balance. The second stage, ERP II had the structural adjustment which was undertaken with the aim of removing the distortions in the incentive structure and thereby facilitating production as well as restoring broken down social and economic infrastructure (Ocran, 2007). The ERP was able to bring down inflation to an average of about 40% and subsequently lowering it to a single digit levels by the end of 1985. The success of ERP and SAP at bringing down inflation was short lived though as between 1986 and 1990, year-on-year inflation was in the range of 19% to 46%. The average inflation was between 25% and 40% per annum, far exceeding the official targets set within ERP (Apaloo, 2001). The rates of inflation fell continually from the beginning of 1991 till the beginning of 1992(the first two quarters) where single digit inflation rates were recorded. The rate of inflation was relatively kept under control till the end of 1992, largely on the account of the good harvests of the previous year and conscious efforts at monetary control (Apaloo, 2001). In January 1993, there was over 60% increase in inflation rate as the rates stood at 21.50% compared to 13.3% in December 1992. The average rate of inflation in the 1993 was 24.9% compared to the average of 10% in 1992. This increase in inflation rate was attributed to an increase in petroleum prices. The next two years (1994-1995) that followed was even worse off as there was a sharp increase in inflation rates from 22.80% in January 1994 to 70.80% in November 1995. According to Aidoo (2010), this sharp increase could be attributed to several factors. These factors included a triple year to year increase in petroleum prices in 1993, 1994 and 1995, the University of Ghana http://ugspace.ug.edu.gh 17 depreciation of the local currency (cedi) at the exchange rate level relative to the US dollar the same year, a poor performance of agriculture in 1995 and the introduction of a new tax system known as the Value Added Tax (VAT). The value of the VAT was higher than the previous sales tax and that led to an increase in general prices of commodity. Inflation fell consistently from 69.1% from the beginning of January 1996 till May 1999 except for a brief increase in March and April 1998. In May 1999, the rate was 9.4%. This drastic drop in inflation rate was due to the improvement in agriculture productivity giving credence to the fact that the food component plays a significant role in the level of inflation rates. This was short lived as the level of prices started rising again in June 1999 from 10.3% to 40.5% at the end of December 2000. This was attributed to the increase in world oil prices and a decrease in world market cocoa prices as well as reduction in agriculture performance in the year 2000 (Aidoo,2010). The most recent inflation episode started with a new government in office in January 2001. In the first quarter of 2001, inflation was still higher ranging between 40.1% and 41.9%. However, inflation rates started dropping from 39.5% in the second quarter of 2001 to 12.9% in the third quarter of 2002. In the last quarter of 2002, inflation rose again ending the year at 15.2%. Between 2003 and 2006, inflation ranged from 33.6% (August, 2003) and 10.7% (November, 2004) with an average of 29.8%, 18.2%, 15.5% and 11.7% for 2003, 2004, 2005 and 2006 respectively. Ghana adopted a monetary policy called the Inflation Targeting (IT) in 2007. This was after the Monetary Targeting (MT) framework used in the management of inflation was not effective due to the intractability of the underlying causes (Kwakye, 2004). The aim of the Bank of Ghana (BoG) was to target inflation rate and then attempt to direct actual inflation rate towards the target. The target set by the BoG was to bring University of Ghana http://ugspace.ug.edu.gh 18 inflation rate below 10%. The target of an inflation below 10% was not successful until June 2010 as the inflation figures hovered between 10.1% (October, 2007) and 20.7% (June, 2009) with an annual average of 10.7% (2007), 16.5% (2008), 19.3% (2009). Since June 2010, the inflation has since been below 10.0%, meeting the target of the inflation targeting. In conclusion, it could be seen that Ghana has had its own share of unstable and high inflation rates. However, in the last few years, the country can be seen to be wining the fight against inflation as inflation has been kept at single digits. This notwithstanding, the authorities in charge of price stability in the country should however take note of the potential threat posed by the oil production, boost in the government expenditure through the implementation of the single spine salary structure as these factors could exert both demand and cost pressures on inflation. 2.2 Review of Related Works In this section, a review of the numerous related works that has been carried out by other researchers using time series techniques and other forecasting techniques is taken into consideration. These include Vector Auto Regressive (VAR), Bayesian Vector Auto Regressive (BVAR), Structural Vector Auto Regressive (SVAR), Seasonal Auto Regressive Integrated Moving Average (SARIMA), Simple Auto Regressive (SAR), random walk (RW) and Auto Regressive Fractionally Integrated Moving Average (ARFIMA). The rest are the ARCH-typed models including the traditional Auto Regressive Conditional Heteroscedastic (ARCH) model with extensions such as Generalized ARCH (GARCH), Exponential ARCH (EGARCH), University of Ghana http://ugspace.ug.edu.gh 19 Integrated Generalized ARCH (IGARCH), Power ARCH (PARCH) and Glosten - Jagannathan Runkle GARCH (GJR - GARCH). The related works reviewed would be categorised into two: works done in Ghana and other African countries and works done in the rest of the world. 2.2.1 Review of Related Works in Ghana and Other African Countries Minkah (2007) examined the forecasting ability of three widely used time series volatility models namely, the Historical Variance, the Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model and the Risk Metrics Exponential Weighted Moving Average (EWMA). The characteristics of these volatility models were explored using data on the Standard & Poor’s (S&P) 500 Index, Dow Jones Industrial Average (DJIA), OMX Swedish Stock Exchange (OMXS30) index, Dow Jones-AIG Commodity Index (DJ-AIGCI), the 3 Months US Treasury Bill Yield, the Ghanaian Cedi and the US Dollar (CEDI/USD) exchange rates. It was observed that the complex models i.e. GARCH (1, 1) and Risk Metrics EWMA outperformed the simple Historical Variance in the In-Sample volatility forecasts. The Out-of-Sample forecasting accuracy comparisons also revealed that for shorter forecasting horizons, the GARCH (1, 1) performed better whereas at longer horizons the simple Historical Variance outperformed all in most markets. This was due to the fact that complex models have more parameters and thus add to the estimation errors and its forecasts are consistently poor in Out-of-Sample. University of Ghana http://ugspace.ug.edu.gh 20 Owusu (2010) used the ARIMA models to model inflation and forecast the monthly inflation on short-term basis. The study used different ARIMA models to model the inflation rates from 1990 – 2009. The period under consideration was split into two sub-periods: 1990 – 2000 and 2001 – 2009. The results showed that the best inflation model for the period of 1990 – 2000 was ARIMA (1, 2, 2) whilst that of the period 2001 – 2009 was ARIMA (2, 2, 1). Furthermore, the study concluded that the inflation for the period of January 2001 to December 2009 was less than that of January 1990 to December 2000. Ocran (2007) in modelling Ghana’s inflation experience sought to ascertain the key determinants of inflation in Ghana for the past 40 years. Stylized facts about Ghana’s inflation experience indicated that since Ghana’s exit from the West African Currency Board soon after independence, inflation management has been ineffective despite two decades of vigorous reforms. He used the Johansen co-integration test and an error correction model and the results identified inflation inertia, changes in money supply and changes in government Treasury bill rates, as well as changes in the exchange rate, as determinants of inflation in the short run. Of these determinants, inflation inertia was the most dominant and therefore the study suggested that to make Treasury bill rates more effective as a nominal anchor, inflationary expectations, ought to be reduced considerably. In an attempt to analyse and forecast the macroeconomic impact of oil price fluctuations in Ghana using annual data from 2000 - 2011, Abledu and Agbodah (2012) focused on the feasibility forecast using nested conditional mean (ARIMA) and conditional variance (GARCH, EGARCH and GJR) family of models as the market conditions were too volatile. The best model was the ARIMA (1, 1, 0) and it University of Ghana http://ugspace.ug.edu.gh 21 was used to predict the oil prices in Ghana National Petroleum Authority (GNPA) till the end of 2016. Using the Seasonal Autoregressive Integrated Moving Average (SARIMA) model, Aidoo (2010) examined the inflation rates in Ghana. Monthly inflation data from July 1991 to December 2009 were used. The results revealed that the ARIMA (1,1,1) × (0,0,1)12 can best represent the behaviour of inflation rates in Ghana. Suleman and Sarpong (2012) applied the Box-Jenkins approach to model monthly inflation data in Ghana. The study applied the SARIMA model to inflation rates from January 1990 to January 2012. The study concluded that the best model was the ARIMA(3,1,3) × (2,1,1)12. Alnaa and Ahiakpor (2011) also used the ARIMA approach to predict inflation in Ghana. The monthly data from June 2000 to December 2010 was used and it was found that ARIMA (6, 1, 6) was the best fitted model for forecasting inflation in Ghana. Inflation was predicted highest for the months of March, April and May to be 8.95%, 10.07% and 10.24% respectively. The researchers recommended that the appropriate measures must be put in place to prevent inflation spiral from setting in motion. Since their model suggests that, inflation has a long memory and that once the inflation spiral is set in motion, it will take at least 12 periods (months) to bring it to a stable state. Frimpong and Oteng-Abayie (2006) in studying volatility of returns on the Ghana Stock Exchange (GSE) used the random walk (RW), GARCH, EGARCH and TGARCH models. The unique ‘three days a week’ Databank Stock Index (DSI) was used to study the dynamics of the GSE volatility over a 10-year period. Their results revealed that the DSI exhibited the stylized facts such as volatility clustering, University of Ghana http://ugspace.ug.edu.gh 22 leptokurtosis and asymmetry effects associated with stock market returns on more advanced stock markets. The random walk hypothesis was also rejected and overall, the GARCH (1, 1) model outperformed the other models under the assumption that the innovation follows a normal distribution. The ARCH-type models were used by Wagala, Nassioma and Islam (2011) to model the volatility of the Nairobi Stock Exchange weekly returns. The models applied in the study included the ARCH (𝑝), standard GARCH (𝑝, 𝑞), IGARCH (𝑝, 𝑞) and TGARCH (𝑝, 𝑞). The results demonstrated that the ARCH (8) was found to be the most adequate for the NSE index, Bamburi and KQ while ARCH (9) provided the best order for the NBK series. Furthermore four different 𝑝 and 𝑞 values were tested for the GARCH (𝑝, 𝑞), EGARCH (𝑝, 𝑞) and TGARCH (𝑝, 𝑞). These were (1, 1), (1, 2), (2, 1) and (2, 2). The order (1, 1) was the best choice in all cases and it was consistent with results obtained from most GARCH research works. Comparing the diagnostics and the goodness of fit statistics, the IGARCH (1, 1) outperformed the ARCH, EGARCH and TGARCH models due to its stationarity in the strong sense. However, because the IGARCH model was unable to capture the asymmetry exhibited by the stock data, the EGARCH (1,1) and the TGARCH (1,1) provided the best options to describe the dependence in variance for all the four series since they were able to model asymmetry and parsimoniously represent a higher order ARCH (𝑝). Amos (2010) examined financial time series with special application to modelling inflation data for South Africa. The data spanned from January 1994 to December 2008. The study considered two families of time series namely the autoregressive integrated moving averages (ARIMA) with extension to the Seasonal ARIMA University of Ghana http://ugspace.ug.edu.gh 23 (SARIMA) model and the autoregressive conditional Heteroscedastic (ARCH) with extensions to the generalized ARCH (GARCH) model. The study concluded that the SARIMA (1,1,0) × (0,1,1)𝑠 was the best fitting model from the ARIMA family of models while the GARCH (1, 1) was chosen to be the best fit from the ARCH- GARCH models. Furthermore, a comparison of the two selected models based on the goodness of fit and the forecasting power of the two models was carried out. It was established that the GARCH (1, 1) model was superior to the SARIMA (1, 1, 0) × (0, 1, 1) model according to both criteria as the data was characterized by changing mean and variance. Awogbemi and Oluwaseyi (2011) described the volatility in the consumer prices of some selected commodities in the Nigerian market. The researchers examined the presence or otherwise of the volatility in their prices using ARCH and GARCH models with monthly Consumer Price Index (CPI) of five selected commodities over a period of 1997 – 2007. The results showed that ARCH and GARCH models are better models because they give lower values of AIC and BIC as compared to the conventional Box and Jenkins ARMA models. The researchers also observed that since volatility seems to persist in all the commodity items, people who expect a rise in the rate of inflation (the ‘bullish crowd’) will be highly favoured in the market of the said commodity items. Ngailo (2011) modelled financial time series with special application to modelling inflation data for Tanzania. In particular the theory of univariate non linear time series analysis was explored and applied to the inflation data spanning from January 1997 to December 2010. He fitted the ARCH and GARCH models to the data. Based on the AIC and BIC values, the results revealed that the best fit models tend to be the GARCH (1, 1) and GARCH (1, 2). However after diagnostic and forecast accuracy University of Ghana http://ugspace.ug.edu.gh 24 tests were performed, the GARCH (1, 1) model was adjudged to be the best model for forecasting. Inflation and Inflation forecasting in Uganda from 1993 to 2009 was examined by Mugume and Kasekende (2009). They employed various inflation forecasting models like Philips curve, P-star model based on Quantum Theory of Money (QTM), and the price equation and ARIMA model. They also employed M3 and the results of both short-run dynamics and long run equilibrium showed that inflation had not been a result of money growth. The long run inflation equation seemed to show that exchange rate depreciation could have had a stronger impact in driving inflation upwards than money supply, although it had no short run impact. Igogo (2010) employed the ARCH family of models to measure the effect of real exchange volatility on trade flows in Tanzania for the period of 1968 to 2007. He fitted the GARCH (1, 1) and EGARCH (1, 1) models. The results indicated that GARCH (1, 1) model violated the non-negativity conditions and hence to resolve the problem, the EGARCH (1, 1) was used. The adequacy of the EGARCH (1, 1) model to measure the real exchange rate volatility was confirmed by testing for ARCH effect after running the model. Furthermore, the study revealed that it is the real exchange rate rather than its volatility that is found to have a significant effect on trade flows although the effect is larger on exports than imports. He concluded therefore that in the short run, imports are mainly affected by the domestic income while exports are mainly affected by the real exchange rate. Ezzat (2012) studied volatility of daily stock returns listed on the Egyptian Exchange during the political turmoil of 2011. This was particular because modelling volatility during a financial crisis where massive shocks are generated presents an ideal University of Ghana http://ugspace.ug.edu.gh 25 environment for investigating the dynamics of volatility during periods of extreme fluctuations for comparison with volatility during more tranquil periods. The analysis was based on employing both GARCH and EGARCH models. Daily closing prices of four Egyptian stock market indices, the EGX 30, EGX70, EGX 100, and the EGX 20 capped were used in the analysis. The time frame was from the inception of each index to the 30th of June 2012. The sample period covers the period of pre-and post Egyptian revolution which was shaped by extreme volatile fluctuations in stock returns. The EGARCH model was the method of choice for modelling the volatility in order to investigate the long memory and the leverage effect in the volatilities of the two periods. The findings revealed higher volatility during the revolution period for all indices reflected in higher standard deviations for both daily returns and absolute returns, with the EGX 70 displaying the highest volatility. The leverage effect was more apparent during the revolution period. However, long memory was more apparent during the pre-revolution period. 2.2.2 Review of Related Works in the Rest of the World Engle (1982) studied the ARCH model and revealed that these models were designed to deal with the assumption of non-stationarity found in real life financial data. The researcher based the ARCH model on the idea that a natural way to update a variance forecast was to average the squared deviation of the rate of return from its mean just like the principle used in standard deviation. The ARCH process allowed the conditional variance to change over time as a function of past errors leaving the unconditional variance constant. Empirical evidence revealed that the ARCH model required a relatively long lag in the conditional variance equation and so to avoid the University of Ghana http://ugspace.ug.edu.gh 26 problems with negative variance parameters, a fixed lag structure was typically imposed. Bollerslev (1986) proposed a generalized ARCH (GARCH) to overcome the limitations of the traditional ARCH model of Engle (1982). The GARCH model allowed for both a longer memory and a more flexible lag structure. In the ACRH process, the conditional variance is specified as a linear function of past sample variance only whereas the GARCH process allows lagged conditional variances to enter in the model as well. Both the ARCH and GARCH models of Engle (1982) and Bollerslev (1986) could not tell how the variance of return was influenced differently by positive and negative news. Hence Nelson (1991) extended the ARCH framework in order to better describe the behaviour of return volatilities. His study broke the rigidity of the ARCH and GARCH model specification. He proposed the Exponential GARCH (EGARCH) model to test the hypothesis that variance of return was influenced differently by positive and negative excess returns. The results revealed that the hypothesis was true and also the excess returns were negatively related to stock market variance. Asri and Mohammad (2011) proposed an alternative model for modelling the volatility of the conditional variances: A (Radial Basis Function) RBF-EGARCH Neural Networks Model. Their proposed forecasting model combines a RBF neural network for the conditional mean and a parametric EGARCH model for the conditional volatility. They used the regression approach to estimate the weight and the parameters of the EGARCH model. They carried out a simulation based on a sample of Bank Rakyat Indonesia TBK stock returns and the results indicated that their proposed model is able to accurately predict 63% upward and downward University of Ghana http://ugspace.ug.edu.gh 27 movements of future predictions. They concluded that the simulation results obtained in the forecasting performances motivates further work, which will involve comparing a different method of parameters model estimation. Kunst (1997) studied the augmented ARCH models which encompasses most linear ARCH-type models. He considered the two basic ARCH variants for auto-correlated series; conditional variance lagged by errors (Engle, 1982) or conditional variance lagged by observations (Weiss, 1984). He evaluated whether the restrictions evolving from these two ARCH variants are valid in practice. Time series of stock market indexes for some major stock exchanges (Standard and Poor 500 index, Stock market index for German, French, British and Japanese) were considered. For the important US Standard & Poor 500 Index and for Japanese and German stock index, the evidence indicated more or less convincingly that fourth-moments structures in financial series may be more complicated than the traditional ARCH models. A non - parametric comparison of sample moments also supported this result. The statistical evidence presented was stronger than the weak evidence on more general structures found by Tsay (1987) in an exchange rate series. For two other countries, France and the United Kingdom, the statistical description achieved by the standard ARCH model appears to be sufficient. Su (2010) employed both GARCH and EGARCH models in studying the financial volatility in China. He applied the daily stock returns data from January 2000 to April 2010 and split the time series into two parts: before the crisis and during the crisis period. The empirical results suggested that EGARCH model fits the sample data better than GARCH model in modelling the volatility of Chinese stock returns. The result also showed that long term volatility was more volatile during the crisis period University of Ghana http://ugspace.ug.edu.gh 28 whilst Bad news produced stronger effect than good news for the Chinese stock market during the crisis. Malmsten (2004) used a unified framework for testing the adequacy of an estimated EGARCH model. The tests were Lagrange multiplier type tests and included testing an EGARCH model against a higher-order one and testing parameter constancy. Furthermore, various existing ways of testing the EGARCH model against GARCH models were also investigated as another check of model adequacy. This was done by size and power simulations. Simulations revealed that the simulated LR test is more powerful than the encompassing test and that the size of the test may be a problem in applying the pseudo-score test. Finally, the simulation results indicated that in practice, the robust versions of their tests should be preferred to non robust ones and they can be recommended as standard tools when it comes to testing the adequacy of an estimated EGARCH (𝑝, 𝑞) model. The stylized facts of financial time series using three popular models were studied by Malmsten and Terasvirta (2004). The models used were the GARCH, EGARCH and Autoregressive Stochastic Volatility (ARSV) models and they focused on how well these models are able to reproduce characteristic features (stylized facts) of financial series. Their study used stock returns as a case study of the financial series. The results showed that the GARCH model and EGARCH models were at their best when characterizing models based on time series with relatively low kurtosis and high first- order autocorrelation of squares, assuming normality of errors. However the ARSV (1) model is a better option for time series displaying a combination of high kurtosis and high autocorrelations. University of Ghana http://ugspace.ug.edu.gh 29 Blake and Kapetanios (2005) investigated the extent of the effect of neglected nonlinearity on the properties of ARCH testing procedures. They proposed and used a new ARCH testing procedures based on neural networks which are robust to the presence of neglected nonlinearity. The neural networks were used to purge the residuals of the effects of nonlinearity before applying an ARCH test. Thus they correctly sized the ARCH test while retaining good power for the ARCH test. Results based on Monte Carlo simulations showed that the new method alleviated the problem posed by the presence of neglected nonlinearity to a very large extent. Empirical evidence or results based on the application of the new test procedures to exchange rate data indicated substantial evidence of spurious rejection of the null hypothesis of no ARCH effects. There was also further evidence that exchange rates exhibited complicated, dynamic behaviour, with important nonlinearity and volatility effects. Karanasos and Kim (2003) considered the moment structure of the general ARMA (𝑟, 𝑠) -EGARCH (𝑝, 𝑞) model and compared it with the standard GARCH model and APARCH model. In particular, they derived the autocorrelation function of any positive integer power of the squared errors and also obtained the autocorrelations of the squares of the observed process and cross correlations between the levels and the squares of the observed process assuming that the error terms are drawn from either a normal, double exponential or generalised error distributions. Daily data on four East Asia stock indices – Korean Stock price index (KOSPI), Japanese Nikkei index (Nikkei) and the Taiwanese SE Weighted index (SE) for the period 1980:01 – 1997:04 and the Singaporean Straits Times price index (ST) for the period 1985:01 – 1997:04. They concluded that there were differences in the moment structure between the ARMA (𝑟, 𝑠) – EGARCH (𝑝, 𝑞) model and the standard GARCH model. The study also concluded that, to help with model identification, results of the University of Ghana http://ugspace.ug.edu.gh 30 autocorrelations of the squared deviations can be applied to the observed data and its properties compared with the theoretical properties of the models. Based on that, it was observed that the EGARCH model can more accurately reproduce the nature of the sample autocorrelations of squared returns than the GARCH models. Lee and Brorsen (1996) also studied the relative performance of the GARCH model and the EGARCH model by using a Cox-type non-nested test that used the Monte Carlo hypothesis tests. The approach used by Lee and Brorsen (1996) was similar to the approach used by Pesaran and Pesaran (1993). Whilst the approach of the Pesaran and Pesaran (1993) assumed asymptotic normality, Lee and Brorsen (1996) approach did not assume asymptotic normality. They estimated that the GARCH and EGARCH models of the daily spot prices of Deutsche Mark in terms of the United States dollars using the maximum likelihood procedure. The GARCH model was rejected whilst the EGARCH model was not rejected. The study therefore concluded that the EGARCH models were preferable to the GARCH models in modelling Deutsche mark/dollar exchange rate. The effects of good and bad news on volatility in the Indian stock markets using asymmetric ARCH models during the global financial crises of 2008-2009 was investigated by Goudarzi and Ramanaraynan (2011). The asymmetric volatility models considered were the EGARCH and TGARCH models and the BSE 500 stock index was used as a proxy to the Indian stock market. The study found out that the BSE 500 return series reacted to good news and bad news asymmetrically. That is, the BSE 500 return series reacted differently to good news and to bad news. The EGARCH (1,1) and TGARCH (1,1) models were estimated for the BSE 500 stock returns series using the robust method of Bollerslev-Wooldridge’s quasi-maximum likelihood estimation (QMLE) assuming the Gaussian standard normal distribution. University of Ghana http://ugspace.ug.edu.gh 31 The results indicated that the conditional means are significant in both estimated models. Hence the SBIC information criterion was applied to select the fittest model to the data. The TGARCH (1,1) model was selected and the study therefore concluded that the TGARCH (1,1) model can be possible representative of the asymmetric conditional volatility process for daily return series of BSE 500 as compared to the EGARCH (1,1). Jean-Philippe (2001) examined the forecasting performance of four GARCH-typed models. The comparison focused on two different aspects; the difference between symmetric GARCH model (traditional GARCH model) and asymmetric models (EGARCH, GJR and APARCH) and the difference between normal tailed symmetric, fat-tailed symmetric and fat tailed asymmetric distributions (i.e. normal distributions against student-t and skewed student-t distributions). The study concluded that noticeable improvements were made when using an asymmetric GARCH in the conditional variance and that the APARCH and GJR outperformed the EGARCH. Furthermore, non-normal distributions provided better in-sample results than Gaussian distributions. Alberg, Shalit and Yosef (2008) carried a comprehensive empirical analysis of the mean return and conditional variance of Tel Aviv Stock Exchange (TASE) indices using various GARCH models. The prediction performance of these conditional changing variance models were compared to newer asymmetric GJR and APARCH models. The results indicated that the asymmetric GARCH model with fat tailed densities improved overall estimation for measuring conditional variance. The EGARCH model using a skewed student-t distribution is the most successful for forecasting TASE indices as compared to the asymmetric GARCH, GJR and APARCH models. University of Ghana http://ugspace.ug.edu.gh 32 Angelidis, Benos and Degiannakis (2003) evaluated the performance of an extensive family of ARCH models (GARCH, TARCH and EGARCH) in modelling daily Value-at-Risk (VaR) of perfectly diversified portfolios in five stock indices using a number of distributional assumptions and sample sizes. The five perfectly diversified portfolios were the S&P 500, Nikkei 225, FTSE 100, CAC 40 and DAX 30. The different distributions were normal, student-t and generalised error distribution whilst the sample sizes were 500, 1000, 1500 and 2000. Their results show that under the evaluation framework based on the proposed quartile loss function, there was strong evidence that the combination of the student-t distribution with the simplest EGARCH models produce the most adequate VaR forecasts for the majority of the markets. Furthermore, the size of the rolling sample used in estimation turned out to be rather important since in simpler models and low confidence levels, a sample size smaller than 2000 improves probability values. In more complex models where leptokurtic distributions are used and when the confidence level is high, a small sample size led to lack of convergence in the estimation algorithms. Finally, there was no consistent relation between the sample sizes and the optimal models as there were significant differences in the VaR forecasts for the same model under the four sample sizes. Yuksel and Bayram (2005) investigated the stock market volatility in Turkish, Greek and Russian stock markets using the total return indexes based on the domestic currencies of the corresponding countries. The data set covers a period from 1994 - 2004. The study concluded that the GARCH-M (1, 1) was the best model for modelling the volatility in the stock markets in Turkey. In the case of the stock markets of Greece, the TARCH (1, 2) was the best model whilst the TARCH (1,1) was the best model for the Russian stock markets. University of Ghana http://ugspace.ug.edu.gh 33 Irfan, Irfan and Awais (2010) modelled the volatility of short term interest rates in Pakistan and India using the ARCH family models. The study used the Karachi Inter Bank Offering Rate (KIBOR) and Mumbai Inter Bank Offering Rate (MIBOR) in Pakistan and India respectively and the various time series models examined included GARCH, EGARCH,TGARCH and PARCH. The results from all the ARCH family models indicated that high volatility is present in KIBOR returns while volatility shock is moderately present in MIBOR returns. Also all the ARCH family models were compared using the within sample forecasting performance on basis of root mean squared error (RMSE) and Mean Absolute Error (MAE) and the comparison suggested that MIBOR forecasted better than KIBOR as it had minimum errors. Lastly, the TGARCH was adjudged the best model in both returns because they had all the parameters being significant whilst the PARCH (1, 1) model is selected the second best model based on the criteria of the students t-distribution. Anna (2011) examined the relationship between inflation, inflation uncertainty and output growth with evidence from the G-20 countries using several GARCH and GARCH-M models in order to generate a measure of inflation uncertainty. The study adopted two approaches to test for the impact of inflation uncertainty on inflation and vice versa. The first approach was based on the GARCH-M model that allows for simultaneous feedback between the conditional mean and variance of inflation while the second approach was based on a two-step procedure where Granger methods were employed using the conditional variance of a simple GARCH model. The results of the study suggested significant positive relationship between inflation uncertainty and inflation in most countries. These results go to support the Cukierman-Matter and Friedman-Ball hypothesis. Also the results of the study provided evidence for the Holland theory; that uncertainty lead to lower and in the case of the effect of inflation University of Ghana http://ugspace.ug.edu.gh 34 uncertainty in output growth, there was little evidence that inflation uncertainty has negative real effects. Chatfield (2000) asserted that the idea behind a GARCH model was similar to that behind the ARMA model with respect to the fact that a higher order AR or MA model may often be approximated by a mixed ARMA model with fewer parameters using a rational polynomial approximation. He described the GARCH model as an approximation to a higher-order ARCH model. He noted that the GARCH (1, 1) model has become the standard model for describing non constant variance due to its relative simplicity. Empirical evidence has revealed that often (𝛼 + 𝛽) < 1 so that the stationarity condition may be met. However if the (𝛼 + 𝛽) = 1, the process ceases to have a finite variance although it can be shown that the squared observations are stationary after taking first differences. In such a situation a better model Integrated GARCH (IGARCH) developed by Engle and Bollerslev (1986) is recommended. Rafique and Ur-Rehman (2011) compared the volatility behaviour and variance structure of high (daily) and low (weekly, monthly) frequencies of stock returns in Pakistan. The study used data from 1991 to 2008 of the KSE-100 index. By employing the EGARCH model, they found that there are significant asymmetric shocks (leverage effect) to volatility in the three series but the intensity of the shock were not equal for all the series. Furthermore, it was concluded that the variance structure of high frequencies (daily) data is dissimilar from the low frequencies (weekly, monthly) data. Karanasos, Karanassou and Fountas (2004) also examined the relationship between inflation and inflation uncertainty in the US using a GARCH model that allows for simultaneous feedback between the conditional mean and variance of inflation. The University of Ghana http://ugspace.ug.edu.gh 35 results showed that there was a strong positive bi - directional relationship between inflation and inflation uncertainty. The results are also in agreement with the predictions of economic theory expressed by the Friedman-Ball and Cukierman- Meltzer hypothesis; however, it was in conflict with existing empirical evidence. The study also compared the properties of the observed time series with the theoretical properties of GARCH models to illustrate how theoretical results on correlation structure can facilitate model identification. The results showed that the AR-GARCH- M-L model can approximate reality well. Ling and Li (1997) considered fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity, which combined the popular generalised autoregressive conditional Heteroscedastic (GARCH) and fractional ARIMA models. Drost and Klassen (1997) constructed adoptive and hence efficient estimators in a general GARCH –M in mean type context including integrated GARCH models. A time lag between a change in money supply and the inflation rate response was examined by Jehovanes (2007). He employed a modified GARCH model to monthly inflation data for the period 1994 to 2006. The maximum likelihood estimation technique was used to estimate the parameters of the model and to determine significance of the lagged values. Results showed that the GARCH model was a better fit and indicated that a change in supply of money would affect inflation rate considerably in seven months ahead. Brooks (2008) studied the stochastic volatility models and found that most time series models such as GARCH will have forecasts that tend towards the unconditional variance of the series as the prediction horizon increases. This implies that if they are University of Ghana http://ugspace.ug.edu.gh 36 at a low level relative to their historic average they will have a tendency to rise back towards the average and this feature is accounted for in GARCH volatility forecasting models. Mushtaq, Shah and Ur-Rehman (2011) examining the relationship between stock exchange market volatility and macroeconomic variables volatility with respect to Pakistan. To measure this time series relationship for Pakistan, exponential generalized autoregressive conditional heteroscedasticity (EGARCH) and lag- augmented vector auto regression (LA-VAR) models were used. It was found that there is a positive relationship of consumer price index (CPI) and foreign direct investment (FDI) with stock market; however, exchange rate (ER) and T-bill rate (TBR) are inversely related to stock market volatility. On the other hand, they found strong evidence that there is a bilateral relationship of FDI and ER with stock prices, while a unidirectional relationship was found between TBR and stock market prices, with the direction from stock prices to treasury bills interest rates. However, a significant causal relationship was not found between CPI and stock prices. The analysis of this study reveals that the stock market of Pakistan is relatively less efficient as compared to US and other developed economies of the world. Nakajima (2008) proposed the EGARCH model with jumps and heavy-tailed errors, and studied the empirical performance of different models including the stochastic volatility models with leverage, jumps and heavy-tailed errors for daily stock returns. In the framework of a Bayesian inference, the Markov Chain Monte Carlo estimation methods for these models were illustrated using a simulation study. The model comparison based on the marginal likelihood estimation was carried out with data on the U.S. stock index. Based on the estimates of the marginal likelihood, the study found that the jumps and heavy-tails raise the marginal likelihood of the EGARCH University of Ghana http://ugspace.ug.edu.gh 37 model. The EGARCH model with jumps and heavy-tails and the SV model with heavy-tails and leverage fit to the data better than other competing models for their dataset. Ou and Wang (2010) used a probabilistic method called the Relevance Vector Machine (RVM) to predict GARCH, EGARCH and GJR based volatilities of the Hang Seng Index (HSI) for two stage out-of-sample forecasts. The RVM is a powerful tool for prediction problems as it uses a Bayesian approach whose functional form is identical to a well-known Support Vector Machine (SVM). Their goal was to compare the model with an SVM approach and classical GARCH, EGARCH and GJR models. The experimental results suggested that the proposed models can capture two different asymmetric effects of news impacts, and hence outperforms the other models; particularly, the RVM based GJR generated a best ability for first stage forecast and the RVM based EGARCH was superior for the second stage forecast of HSI volatility, in terms of the evaluation metrics: RMSE, MSE, MAD, NMSE, and linear regression R squared. Duan, Gauthier, Simonato and Sasseville (2006) extended the analytical approach to pricing European options in the GARCH framework developed earlier in Duan, Gauthier and Simonato (1999). They extended the approximation to two other popular GARCH specifications namely the GJR-GARCH and EGARCH using the cumulative asset return as their data set. The study provided the corresponding formula and also examined their numerical performance. In each case, the resulting formula was the Black-Scholes formulae plus adjustment terms accounting for skewness and kurtosis. Also their results suggested that the approximations were adequate, particularly for shorter-maturity options. The results also revealed that their analytical approximation University of Ghana http://ugspace.ug.edu.gh 38 formula can be useful for a large-scale GARCH option pricing model where computation time can be a serious concern. Ramasamy and Munisamy (2012) compared three simulated exchange rates of Malaysian Ringgit with actual exchange rates using GARCH, GJR and EGARCH models. For testing the forecasting effectiveness of GARCH, GJR and EGARCH the daily exchange rates of four currencies - Australian Dollar, Singapore Dollar, Thailand Bhat and Philippine Peso - were used. The forecasted rates, using Gaussian random numbers, were compared with the actual exchange rates of year 2011 to estimate errors. Both the forecasted and actual rates were then plotted to observe the synchronisation and validation. The results showed more volatile exchange rates are predicted well by the GARCH models efficiently than the hard currency exchange rates which are less volatile. Among the three models the effective model was indeterminable as these models forecast the exchange rates in different number of iterations for different currencies. The leverage effect incorporated in GJR and EGARCH models did not improve the results much. Shamiri and Hassan (2005) examined and estimated the three GARCH(1,1) models (GARCH, EGARCH and GJR-GARCH) using the daily price data of two Asian stock indices, Strait Times Index in Singapore (STI) and Kuala Luampur Composite Index in Malaysia (KLCI) over a 14- years period. The competing models GARCH, EGARCH and GJR-GARCH were developed based on three different distributions, Gaussian normal, Student-t, Generalized Error Distribution. The estimation results showed that the forecasting performance of asymmetric GARCH Models (GJR- GARCH and EGARCH), especially when fat-tailed asymmetric densities are taken into account in the conditional volatility, was better than symmetric GARCH. Moreover, it was found that the AR (1)-GJR model provided the best out-of-sample University of Ghana http://ugspace.ug.edu.gh 39 forecast for the Malaysian stock market, while AR(1)-EGARCH provided a better estimation for the Singaporean stock market. Jiang (2011) examined the relationship between inflation and inflation uncertainty in China. He believed that it was worthy to investigate the inflation and inflation uncertainty relationship in China as it is commonly believed that one possible channel that inflation imposes significant economic costs is through its effect on inflation uncertainty. Jiang (2011) addressed the relationship of inflation and its uncertainty in China’s urban and rural areas separately given the huge urban-rural gaps. The GARCH(1,1) and E-GARCH(1,1) models were used to generate the measure of inflation uncertainty and then Granger causality tests were performed to test for the causality between inflation and inflation uncertainty. GARCH (1, 1)-M models were also employed to further investigate the inflation-uncertainty nexus. The results provided strong statistical supportive evidence that higher inflation raises inflation uncertainty. On the other hand, the evidence on the effect of inflation uncertainty on inflation was mixed and depended on the sample period and areas examined. Hassan, Moud and Ekonomi (2006) explored the varying volatility dynamic of inflation rates in Malaysia for the period from August 1980 to December 2004. The GARCH and EGARCH models were used to capture the stochastic variation and asymmetries in the economic instruments. Also, an in-sample evaluation of the sub- periods volatility was done using both models. The results indicated that, the EGARCH model gave better estimates of sub-periods volatility as compared to the GARCH model. Berument, Kivilcim and Neyapti (2001) used the EGARCH to model inflation uncertainty in Turkey. Their study used the monthly CPI inflation covering the period University of Ghana http://ugspace.ug.edu.gh 40 from 1986 to 2000. Their study gave further contribution to literature due to the inclusion of seasonal terms in the conditional variance equation. The results of the study provided evidence to show that in Turkey, the effect on inflation uncertainty of positive shocks to inflation are greater than that of negative shocks to inflation. Also, when monthly dummies were used in modelling both inflation and inflation uncertainty, the effect of lagged inflation on inflation uncertainty disappeared. They concluded that there is no significant lagged effect of inflation on inflation uncertainty. Lastly, there was evidence of significant seasonal effects of inflation on conditional variability. Alam and Rahman (2012) explored the application of GARCH type models such as GARCH; EGARCH; TARCH; and PARCH; to modelling the BDT/USD exchange rate using the daily foreign exchange rate series fixed up by Bangladesh Bank. The BDT/USD time series from July 03, 2006 to April 30, 2012 were used for the study purpose out of which in-sample and out-of-sample date set covered from July 03, 2006 to May 13, 2010 and May 14, 2010 to April 30, 2012 respectively. They benchmarked their results with AR and ARMA models. They found that all GARCH type models demonstrated that past volatility of exchange rate significantly influenced current volatility. Both the AR and ARMA models were found as the best model as per in-sample statistical performance results, whereas according to out-of-sample, GARCH model was the best model with transaction costs and the TARCH model was nominated as the best model without transaction costs. The EGARCH and TARCH models outperform all the other models as per to in-sample and out-of-sample trading performance outcomes respectively including transaction costs. University of Ghana http://ugspace.ug.edu.gh 41 CHAPTER THREE METHODOLOGY 3.0 Introduction This chapter deals with the methodology for the study and it has been sub divided into five sections aside the introductory section. Section one looks briefly at time series and its basic concepts like stationarity; Sections two to four will give a detailed description and explanation of the theory and concept of the ARCH-type models (i.e. ARCH, GARCH and EGARCH models) that would be used in the chapter four to analyse the data. The final section would be the conclusion. 3.1 Time Series and its Basic Concepts Chatfield (2000) defines time series as a series or sequence {𝑥𝑡} of data points measured typically at successive times. The data points are commonly spaced equal in time. Time series analysis comprises methods that attempt to understand the underlying generation process of the data points and constructs a mathematical model to represent the process. The constructed model is then used to forecast future events based on known past events. Time series often makes use of the natural one-way ordering of time so that values in a series for a given time will be expressed as being derived from past values rather than future values. A time series model usually reflects the fact that observations close together in time domain are more correlated as compared to observations further apart. That is, there is ‘volatility clusters’- small (large) shocks are again followed by small (large) shocks. University of Ghana http://ugspace.ug.edu.gh 42 Original time series data are made up of various patterns that are derived on casual factors which are identified by time series analysis methods. The four patterns that characterize economic and business series are the long-run development known as the trend, cyclical or periodic component, seasonal component and the error or residual component. The trend component deals with the general and overall pattern of the time series; the cyclical component refers to the variation in the series which arise out of the phenomenon of business cycles. It usually spans within periods of more than one year. The seasonal variations refers to the periodic and repetitive ups and downs in the series that occur within a year and lastly the error term is the component that contains all moments which neither belong to the trend nor to the cycle nor to the seasonal component. The models for time series data can have many forms and represents different stochastic processes which could be linear or non-linear. Among the linear models include autoregressive (AR) model of order (𝑝), moving average (MA) of order (𝑞) and autoregressive moving average (ARMA) model of order(𝑝, 𝑞). A combination of the above models produce the autoregressive integrated moving average (ARIMA) model with a generalized model known as the autoregressive fractionally integrated moving average (ARFIMA) model. The non-linear time series model represent or reflect the changes of variance along with time known as heteroscedasticity. With these models, changes in variability are related to and/or predicted by recent past values of the observed series. The wide variety of non-linear models include the symmetric models such as Autoregressive Conditional Heteroscedastic (ARCH) model with order (𝑝) and Generalized ARCH (GARCH) model with order(𝑝, 𝑞). Other asymmetric models are the Power ARCH University of Ghana http://ugspace.ug.edu.gh 43 (PARCH), Threshold GARCH (TGARCH), Exponential GARCH (EGACRH), Integrated GARCH (IGARCH), etc. All these asymmetric models have order (𝑝, 𝑞). The above mentioned non-linear models form part of a large family of the ARCH- type models. In this study, three of such models – ARCH, GARCH and EGARCH- would be fitted to the data set. The theory and concepts of these models are explained in detail in later sections of this chapter. Other forms of non-linear models include the bilinear model, threshold autoregressive (TAR), state-dependent model, markov switching models, etc. 3.1.1 Stationary and Non Stationary Processes The foundation of time series analysis is stationarity. That is, before time series analysis is carried out, one needs to verify whether the series is stationary or otherwise. However, an assumption of stationarity is usually made. In this section, we define and describe stationarity (non-stationarity). A series is said to be stationary if the mean and auto covariances of the series do not depend on time. There are two forms of stationarity - strict stationarity and weak stationarity. Under strict stationarity, the common distribution function of the stochastic process does not change by a shift in time. That is, a time series {𝑥𝑡} is said to be strictly stationary if the joint distribution of (𝑥1, ⋯ , 𝑥𝑘) is identical to that of (𝑥1+𝑡 , ⋯ , 𝑥𝑘+ 𝑡) for all 𝑡, where 𝑘 is an arbitrary positive integer and (1,⋯ , 𝑘) is a collection of 𝑘 positive integers. The shifting of the time origin by 𝑡 has no effect on the joint distribution which depends only on the intervals between the two sets of points given by 𝑡 which is called a lag. University of Ghana http://ugspace.ug.edu.gh 44 The concept of strict stationarity is difficult to apply in practice and hence weak stationarity or stationarity in the second moment is often assumed. A time series {𝑥𝑡} is weakly stationary if both the mean of 𝑥𝑡 and the covariance between 𝑥𝑡 and 𝑥𝑠 are time-invariant. More specifically, {𝑥𝑡} is weakly stationary if: (a) 𝔼 (𝑥𝑡) = µ, which is a constant, and (b) 𝑐𝑜𝑣 (𝑥𝑡, 𝑥𝑠) = γ which is only a function of the time distance between the two random variables and does not depend on the actual points in time 𝑡. 3.2 ARCH (𝒎) MODEL An ARCH process is a mechanism that includes past variance in the explanation of future variances (Engle, 2004). The ARCH model was developed by Engle (1982) and it provides a systematic framework for volatility modelling. ARCH models specifically take the dependence of the conditional second moments in consideration when modelling. Let {𝑥𝑡} be the mean-corrected return, 𝜀𝑡 be the Gaussian white noise with zero mean and unit variance and 𝐼𝑡 be the information set at time 𝑡 given by 𝐼𝑡 = {𝑥1, 𝑥2, … , 𝑥𝑡−1}. Then the ARCH (𝑚) model is specified as: 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.1a) 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 + .... + 𝛼𝑚𝑥 2 𝑡−𝑚 (3.1b) where 𝛼0 > 0 and 𝛼𝑖 ≥ 0 , 𝑖 = 1, … ,𝑚 University of Ghana http://ugspace.ug.edu.gh 45 and 𝔼(𝑥𝑡|𝐼𝑡) = 𝔼[𝔼(𝑥𝑡|𝐼𝑡)]= 𝔼[𝜎𝑡𝔼(𝜀𝑡)] = 0 (3.2a) 𝑉(𝑥𝑡|𝐼𝑡) = 𝔼(𝑥 2 𝑡) = 𝜎 2 𝑡 = 𝛼0+ ∑ 𝛼𝑖𝑥 2 𝑡−𝑖 𝑚 𝑖=1 (3.2b) and the error term 𝜀𝑡 is such that 𝔼(𝜀𝑡|𝐼𝑡) = 0 (3.3a) and 𝑉(𝜀𝑡|𝐼𝑡) = 1 (3.3b) From equations (3.3a) and (3.3b), it can be seen that the error term 𝜀𝑡 is a conditional standardised martingale difference. A stochastic series {𝑥𝑡} is said to be a martingale difference if its expectation with respect to past values of another stochastic series {𝑦𝑖} is zero (Amos, 2010). That is 𝔼(𝑥𝑡+𝑖|𝑦𝑖,𝑦𝑖−1,…) = 0 for 𝑖 = 1,2, … (3.4) From the structure of the model, it can be seen that the dependence of the present volatility {𝑥𝑡} is a simple quadratic function of its lagged values. The coefficients 𝛼𝑖 , 𝑖 = 0,… ,𝑚 can consistently be estimated by regressing {𝑥 2 𝑡} on 𝑥 2 𝑡−1, 𝑥 2 𝑡−2 , ... , 𝑥2𝑡−𝑚. To ensure that the conditional variance 𝜎 2 𝑡 is always positive for all 𝑡, it is required that 𝛼0 > 0 and 𝛼𝑖 ≥ 0, 𝑖 = 1,… ,𝑚. From equations (3.1a) and (3.1b) it follows that large past squared values {𝑥2𝑡−𝑖}, 𝑖 = 1,… ,𝑚 imply a large conditional variance 𝜎2𝑡 for the present volatility{𝑥𝑡}. Consequently, {𝑥𝑡} tends to assume a large value in absolute value. Hence under the ARCH framework, large shocks tend to be University of Ghana http://ugspace.ug.edu.gh 46 followed by another large shock. We would take a particular case of the ARCH (𝑚) model where 𝑚 = 1, ARCH(1) to help understand the ARCH(𝑚) better. 3.2.1 ARCH (𝟏) Model The ARCH (1) model is a special case of the general ARCH (m) model. Let {𝑥𝑡} be the mean-corrected return, 𝜀𝑡 be the Gaussian white noise with zero mean and unit variance. If 𝐼𝑡 is the information set available at time 𝑡 given by 𝐼𝑡 = {𝑥1, 𝑥2, … , 𝑥𝑡−1}, then the process {𝑥𝑡} is ARCH (1) where 𝑚 = 1, if 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.5a) 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 (3.5b) where 𝛼0 and 𝛼1 are unknown parameters. The process {𝑥𝑡} can be stated conditionally in terms of 𝐼𝑡 similar to the variance 𝜎 2 𝑡 under the normality assumption of the error term 𝜀𝑡. Again to ensure that the conditional variance is always positive, the constraints 𝛼0 > 0 and 𝛼𝑖 ≥ 0, 𝑖 = 1,… ,𝑚 is required. Since the ARCH (1) is a special case of ARCH (𝑚), whatever applies to the ARCH (𝑚) model also applies for the ARCH (1). Hence it can be concluded from equations (3.5a) and (3.5b) that a large past squared mean-corrected return {𝑥2𝑡−𝑖}, 𝑖 = 1,… ,𝑚 implies a large conditional variance(𝜎2 𝑡 ), resulting in 𝑥𝑡 being large in absolute value. For the ARCH (𝑚) models to be valid, the presence of ARCH effects should be statistically significant and hence the presence of the ARCH effects should be tested for. University of Ghana http://ugspace.ug.edu.gh 47 3.2.2 Testing for ARCH Effects The presence of conditional heteroscedasticity is referred to as the ARCH effects. To determine the presence of the ARCH effect, a formal statistical test is required. Two tests are available. These are Ljung-Box Statistics 𝑄 (𝑚) test and Lagrange Multiplier (𝐿𝑀) test, which would be discussed in detail in the next two subsections. Let 𝑥𝑡 = 𝑟𝑡 - 𝑢𝑡 be the mean corrected return, where 𝑟𝑡 is the return of an asset, 𝑢𝑡 is the conditional mean of 𝑟𝑡. The squared series {𝑟 2 𝑡 } is then used to check for the presence of ARCH effects. 3.2.2.1 Ljung – Box Test The null hypothesis (𝐻0) for this test is that the first 𝑚 lags of the autocorrelation function of the series {𝑟2 𝑡 } is zero against the alternative hypothesis (𝐻1) that not all the first 𝑚 lags of the autocorrelation function of the series is zero. The test statistic is given as; 𝑄 = 𝑇 ∑ 𝑝(𝑖)̂ 2𝑚𝑖=1 (3.6a) where 𝑝(𝑖)̂ is the consistent estimator of the autocorrelation function and 𝑇 is the sample size. Under the null hypothesis, 𝑄 is asymptotically distributed as chi-square with 𝑚 degrees of freedom. For small samples, the test statistic is given as; 𝑄∗ = 𝑇(𝑇 + 2)∑ 𝑝(𝑖)2̂ 𝑇−𝑖 𝑚 𝑖=1 (3.6b) University of Ghana http://ugspace.ug.edu.gh 48 𝑄∗ is also asymptotically distributed as chi-square with 𝑚 degrees of freedom under the null hypothesis. The decision rule is to reject the null hypothesis of non- autocorrelation of the residuals if 𝑄 or 𝑄∗ are too large than the corresponding critical value of the distribution with 𝑚 degrees of freedom for a specified significance level (𝛼) or if the p value of 𝑄 or 𝑄∗ is less than the significance level (𝛼). 3.2.2.2 Lagrange Multiple (𝑳𝑴) Test This test is equivalent to the usual F statistic for testing 𝛼𝑖 = 0, (𝑖 = 1,… ,𝑚) in the linear regression 𝑥2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 + .... + 𝛼𝑚𝑥 2 𝑡−𝑚 + 𝜖𝑡, 𝑡 = 𝑚 + 1, . . . , 𝑇 (3.7) where 𝜖𝑡 denotes the error term, 𝑚 is the pre-specified positive integer and 𝑇 is the sample size. Specifically the null hypothesis is 𝐻0 = 𝛼1 = … = 𝛼𝑚 = 0 with the test statistic given by 𝐹 = (𝑆𝑆𝑅0 − 𝑆𝑆𝑅1) 𝑚⁄ 𝑆𝑆𝑅1 (𝑇−2𝑚−1)⁄ where 𝑆𝑆𝑅0 = ∑ (𝑥 2 𝑡 − ?̅? ) 2𝑇 𝑡=𝑚+1 , ?̅? = ∑ 𝑥2𝑡 𝑇 𝑡=1 𝑇 is the sample mean of 𝑥2𝑡 and 𝑆𝑆𝑅1 = ∑ 𝜖2?̂? 𝑇 𝑡=𝑚+1 , 𝜖?̂? is the least square residual of the prior linear regression. Under the null hypothesis, 𝐹 is asymptotically distributed as chi-squared distribution with 𝑚 degrees of freedom. The decision rule is to reject the null hypothesis if the 𝐹 is greater than the corresponding critical value of the chi-square distribution with 𝑚 degrees of University of Ghana http://ugspace.ug.edu.gh 49 freedom for a specified significance level (𝛼) or if the p-value of 𝐹 is less than the significance level (𝛼). 3.2.3 Determination of the order of ARCH (𝒎) Model If the presence of the ARCH effect is significantly established, the ARCH model is valid and can be used to model the series. However to model the ARCH (𝑚), the order 𝑚 should be determined. The partial autocorrelation function (PACF) of the 𝑥2𝑡 is used to determine the order 𝑚, of the ARCH (𝑚) model. Given that 𝑥𝑡= 𝜎𝑡𝜀𝑡 and 𝜎 2 𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 + .... + 𝛼𝑚𝑥 2 𝑡−𝑚 as shown by equations (3.1a) and (3.1b), for a given sample, 𝑥2𝑡 is an unbiased estimate of 𝜎 2 𝑡 and hence 𝑥2𝑡 is expected to be linearly related to 𝑥 2 𝑡 −1, ..., 𝑥 2 𝑡−𝑚 similar to that of an autoregressive model of order m. It should be noted that a single 𝑥2𝑡 is generally not an efficient estimate of 𝜎2𝑡. However, it serves as an approximate value that could be informative in specifying the order 𝑚. 3.2.4 Estimation of the ARCH (𝒎) and ARCH (𝟏) Models 3.2.4.1 Estimation of the ARCH (𝒎) model There are three likelihood functions that are commonly used in ARCH (𝑚) estimation depending on the distributional assumption made on the error term 𝜀𝑡. The three common distributions are the normal distribution, standardized student-t distribution University of Ghana http://ugspace.ug.edu.gh 50 which is a heavy tailed distribution and the generalised error distribution (GED). This study assumes that the error term 𝜀𝑡 is normally distributed. Based on the assumption of normality made on the error term 𝜀𝑡, the likelihood function of an ARCH (𝑚) model is given as: 𝑓(𝑥1, ⋯ , 𝑥𝑡|𝜃) = 𝑓(𝑥𝑡|𝑥𝑡−1) 𝑓(𝑥𝑡−1|𝑥𝑡−2)⋯ 𝑓(𝑥𝑚+1|𝑥𝑚) 𝑓(𝑥1, ⋯ , 𝑥𝑚|𝜃) = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=𝑚+1 𝑓(𝑥1, ⋯ , 𝑥𝑚|𝜃) (3.8) where 𝜃 = (𝛼0, 𝛼1, ⋯ , 𝛼𝑚) ′ and 𝑓(𝑥1, ⋯ , 𝑥𝑚|𝜃) is the joint probability density function of 𝑥1, ⋯ , 𝑥𝑚. Since the exact form of 𝑓(𝑥1, ⋯ , 𝑥𝑚|𝜃) is complicated and difficult to obtain, it is commonly dropped from the prior likelihood function, especially when the sample size is sufficiently large. Rather it is practically easier to condition on the first 𝑥1, ⋯ , 𝑥𝑚 since they are usually known and equal to its observed values. This results in the conditional likelihood function being: 𝑓(𝑥1, ⋯ , 𝑥𝑡|𝜃; 𝑥1, ⋯ , 𝑥𝑚) = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=𝑚+1 (3.9) where 𝜎2𝑡 can be evaluated recursively. Under the normality assumption, the estimates 𝛼0̂, 𝛼1̂, ⋯, 𝛼?̂?, are obtained by maximising the prior likelihood function called the conditional maximum likelihood estimates (MLE) (Tsay, 2002). Maximising the conditional likelihood function can be difficult to handle. An equivalent way which is easier to handle is to maximise the logarithm of the conditional likelihood function. Accordingly, the conditional log likelihood function is given as University of Ghana http://ugspace.ug.edu.gh 51 ℓ(𝑥𝑚+1, ⋯ , 𝑥𝑡|𝜃; 𝑥1, ⋯ , 𝑥𝑚) = ∑ (− 1 2 ln 2𝜋 − 1 2 ln 𝜎2𝑡 − 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=𝑚+1 = − ∑ ( 1 2 ln 𝜎2𝑡 + 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=𝑚+1 + 𝐾 (3.10) where 𝐾 = −( 𝑇−𝑚) 2 ln(2𝜋) since the first term 1 2 ln 2𝜋 does not involve any parameter and hence its exclusion has no effect on the estimation process. Again 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 +.... + 𝛼𝑚𝑥 2 𝑡−𝑚 can be evaluated recursively. 3.2.4.2 Estimation of the ARCH (1) Model Given that the order (𝑚) is 𝑚 = 1, the ARCH (1) can be estimated. Based on the assumption of normality made on the error term, 𝜀𝑡, the maximum likelihood estimation is used. Let {𝑥𝑡} be a realization from an ARCH (1) process. Then the likelihood of the data is written as a product of the conditionals as 𝑓(𝑥1, ⋯ , 𝑥𝑡|𝜃) = 𝑓(𝑥𝑡|𝑥𝑡−1) 𝑓(𝑥𝑡−1|𝑥𝑡−2)⋯ 𝑓(𝑥2|𝑥1) 𝑓(𝑥1|𝜃) = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 𝑓(𝑥1|𝜃) (3.11) Where 𝜃 = (𝛼0, 𝛼1) ′. Since it is complicated and difficult to obtain the exact form of 𝑓(𝑥1|𝜃), it is more practical and easier to condition on the first 𝑥1 since 𝑥1 is usually known and equal to its observed value. This result in the conditional likelihood function being: 𝑓(𝑥1, ⋯ , 𝑥𝑡|𝜃; 𝑥1) = 𝑓(𝑥𝑡|𝑥𝑡−1) 𝑓(𝑥𝑡−1|𝑥𝑡−2)⋯ 𝑓(𝑥2|𝑥1) 𝑓(𝑥1|𝜃; 𝑥1) University of Ghana http://ugspace.ug.edu.gh 52 = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 (3.12) Since 𝑥𝑡|𝐼𝑡 ~ 𝑁 (0, 𝜎 2 𝑡) with a probability density function of 𝑓(𝑥𝑡|𝐼𝑡) = 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 ) where 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1. The conditional log likelihood function is expressed as ℓ(𝑥2, ⋯ , 𝑥𝑡|𝜃; 𝑥1) = ∑ (− 1 2 ln 2𝜋 − 1 2 ln 𝜎2𝑡 − 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 = − ∑ ( 1 2 ln 𝜎2𝑡 + 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 (3.13) since the first term 1 2 ln 2𝜋 does not involve any parameter and hence its exclusion has no effect on the estimation process and 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 can be evaluated recursively. The maximum likelihood estimates are obtained by maximising this function with respect to 𝛼0 and 𝛼1 . Since the function is non-linear in these parameters, its maximisation must be done using appropriate non-linear optimization routine. Let {𝑥𝑡} , 𝑡 = 1,… , 𝑇 be a series generated by an ARCH (1) process, where 𝑇 is the sample size. By conditioning on the initial observation (𝑥1), the joint probability density function is written as 𝑓(𝑥) = ∏ 𝑓(𝑥𝑡|𝐼𝑡) 𝑇 𝑡= 2 = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 as in equation (3.12) with the conditional log likelihood function expressed as ℓ(𝑥2, ⋯ , 𝑥𝑡|𝜃; 𝑥1) = ∑ (− 1 2 ln 2𝜋 − 1 2 ln 𝜎2𝑡 − 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 University of Ghana http://ugspace.ug.edu.gh 53 = − ∑ ( 1 2 ln 𝜎2𝑡 + 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=2 as in equation (3.13) The conditional maximum likelihood estimates of 𝛼0 and 𝛼1 are obtained by taking the derivatives of the conditional log likelihood with respect 𝛼0 and 𝛼1 to as given below: 𝜕ℓ 𝜕𝛼0 = 1 2𝜎2𝑡 ( 𝑥2𝑡 𝜎2𝑡 − 1) 𝜕ℓ 𝜕𝛼0 = 1 2𝜎2𝑡 ( 𝑥2𝑡 𝜎2𝑡 − 1) 𝜕ℓ 𝜕𝜎2𝑡 × 𝜕𝜎2𝑡 𝜕𝛼0 (3.14a) and 𝜕ℓ 𝜕𝛼1 = 1 2𝜎2𝑡 ( 𝑥2𝑡 𝜎2𝑡 − 1) 𝜕ℓ 𝜕𝛼1 = 1 2𝜎2𝑡 ( 𝑥2𝑡 𝜎2𝑡 − 1) 𝜕ℓ 𝜕𝜎2𝑡 × 𝜕𝜎2𝑡 𝜕𝛼1 (3.14b) Generally, the partial derivative of ℓ is 𝜕ℓ 𝜕𝜃 = ∑ ( 𝜕ℓ 𝜕𝜎2𝑡 × 𝜕𝜎2𝑡 𝜕𝜃 )𝑇𝑡=2 = − 1 2 ∑ ( 1 𝜎2𝑡 − 𝑥2𝑡 𝜎4𝑡 ) ( 1 𝑥2𝑡−1 )𝑇𝑡=2 = 1 2 ∑ ( 𝑥2𝑡 𝜎2𝑡 − 1) 1 𝜎2𝑡 ( 1 𝑥2𝑡−1 )𝑇𝑡=2 (3.15) recalling that 𝜎2𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡−1. The Hessian (ℍ) is then given by 𝜕2ℓ 𝜕𝜃𝜕𝜃′ = ∑ ( 𝜕2ℓ 𝜕𝜎4𝑡 × 𝜕𝜎2𝑡 𝜕𝜃 × 𝜕𝜎2𝑡 𝜕𝜃′ )𝑇𝑡=2 University of Ghana http://ugspace.ug.edu.gh 54 = − 1 2 ∑ ( 𝑥2𝑡 (𝜎2𝑡)3 + ( 𝑥2𝑡 𝜎2𝑡 − 1) 1 𝜎4𝑡 ) ( 1 𝑥2𝑡 𝑥2𝑡 𝑥 4 𝑡 )𝑇𝑡=2 (3.16) Since 𝜕2𝜎2𝑡 𝜕𝜃𝜕𝜃′ = 0 The Fisher information matrix defined as the negative expected value of the Hessian and denoted usually by 𝑔 is given as 𝑔 = −𝔼 ( 𝜕2ℓ 𝜕𝜃𝜕𝜃′ ) Now since 𝔼𝑥𝑡|𝐼𝑡 {( 𝑥2𝑡 𝜎2𝑡 − 1) 1 𝜎4𝑡 ( 1 𝑥2𝑡 𝑥2𝑡 𝑥 4 𝑡 )} = 0 and 𝔼𝑥𝑡|𝐼𝑡 { 𝑥2𝑡 (𝜎2𝑡)3 } = { 𝔼𝑥𝑡|𝐼𝑡(𝑥 2 𝑡) (𝜎2𝑡)3 } = 1 𝜎4𝑡 , It follows then that 𝑔 = 1 2 ∑ ( 1 𝜎4𝑡 ) ( 1 𝑥2𝑡 𝑥2𝑡 𝑥 4 𝑡 )𝑇𝑡=2 as in Engle (1982). (3.17) Non-linear optimization routines are iterative, thus if 𝜃𝑖 denotes the parameter estimates after the 𝑖𝑡ℎ iterations, then 𝜃𝑖+1 has the form 𝜃𝑖+1 = 𝜃𝑖 + 𝜆𝑀−1 { 𝜕ℓ 𝜕𝜃 } (3.18) Where 𝜆 is a step-length chosen to maximise the likelihood function in the direction of 𝜕ℓ 𝜕𝜃 . For the Newton Raphson based routines 𝜆 = 1 and = 𝜕2ℓ 𝜕𝜃𝜕𝜃′ , and for the Fisher scoring method 𝜆 = 1 and 𝑀 = 𝑔 (Mills, 1994 and Engle 1982). University of Ghana http://ugspace.ug.edu.gh 55 3.2.5 Forecasting with the ARCH Model One important aim of developing a time series model is to estimate future values before they are realized. The ARCH model is no exception. The ARCH model provides good estimates of the series before it is realized. The theory of forecasting with the ARCH models is presented and discussed in detail in this section. Let 𝑥1, 𝑥2, … , 𝑥𝑡 be an observed time series, then the 𝜁 − step ahead forecast for 𝜁= 1, 2,⋯ at the origin, denoted as 𝑥𝑡 (𝜁) is taken to be the minimum mean squared error predictor. That is the value of 𝜁 that minimizes the function 𝔼(𝑥𝑡+ 𝜁 − 𝑓(𝑥)) 𝟐 (3.19) where 𝑓(𝑥) is a function of the observation. Then 𝑥𝑡 (𝜁) = 𝔼(𝑥𝑡+ 𝜁|𝑥1, 𝑥2, … , 𝑥𝑡) Tsay (2002). (3.20) For the ARCH (1) model 𝑥𝑡 (𝜁) = 𝔼(𝑥𝑡+ 𝜁|𝑥1, 𝑥2, … , 𝑥𝑡) = 0 according to Shepard (1996). It is important to note that the forecast for the 𝑥𝑡 series provide no much helpful information and therefore it is imperative to look at the squared returns 𝑥2𝑡 given as 𝑥2𝑡 (𝜁) = 𝔼(𝑥 2 𝑡 + 𝜁|𝑥 2 1, 𝑥 2 2, … , 𝑥 2 𝑡). (3.21) For the ARCH (𝑚) model, the 1- step forecast for 𝑥2𝑡 at the origin t is given by 𝑥2𝑡 (1) = 𝛼0̂ + 𝛼1̂𝑥 2 𝑡 + ⋯ + 𝛼?̂?𝑥 2 𝑡+1− 𝑚 = 𝜎2𝑡 (1) (3.22) University of Ghana http://ugspace.ug.edu.gh 56 where 𝛼?̂?, 𝑖 = 0, 1, … ,𝑚 are the conditional maximum likelihood estimates of 𝛼𝑖 ,𝑖 = 1, … ,𝑚. The 2-step ahead forecast for 𝑥2𝑡 is given as 𝑥2𝑡 (1) = 𝛼0̂ + 𝛼1̂𝑥 2 𝑡(1) + 𝛼2̂𝑥 2 𝑡 ⋯ + 𝛼?̂?𝑥 2 𝑡+2− 𝑚 = 𝛼0̂ + 𝛼1̂𝜎 2 𝑡 (1) + 𝛼2̂𝑥 2 𝑡 ⋯ + 𝛼?̂?𝑥 2 𝑡+2− 𝑚 (3.23) and in general, the 𝜁- step ahead forecast for 𝑥2𝑡 is given as 𝑥2𝑡 (𝜁) = 𝜎 2 𝑡 (𝜁) = 𝛼0̂ + ∑ 𝛼?̂?𝜎 2 𝑡 (𝜁 − 𝑖) 𝑚 𝑖=1 where 𝜎2𝑡 (𝜁 − 𝑖) = 𝑥 2 𝑡+ 𝜁−𝑖 if 𝜁 − 𝑖 ≤ 0 , Tsay (2002). (3.24) In the special case of the ARCH (1) model the 1- step forecast for 𝑥2𝑡 at the origin 𝑡 is given by 𝑥2𝑡 (1) = 𝔼(𝑥 2 𝑡+1 |𝑥𝑡) = 𝛼0̂ + 𝛼1̂𝑥 2 𝑡 = 𝜎2𝑡 (1) (3.25) Since 𝜎2𝑡 (1) = 𝔼(𝜎 2 𝑡+1 |𝑥𝑡) = 𝛼0̂ + 𝛼1̂𝑥 2 𝑡 where 𝛼0 ̂ and 𝛼1̂ are the conditional maximum likelihood estimates of 𝛼0 and 𝛼1. Similarly a 2-step ahead forecast for 𝑥2𝑡 is given as 𝑥2𝑡 (2) = 𝔼(𝑥 2 𝑡+2 |𝑥𝑡) = 𝔼(𝜎2𝑡+2 |𝑥𝑡) University of Ghana http://ugspace.ug.edu.gh 57 = 𝛼0̂ + 𝛼1̂𝔼(𝑥 2 𝑡+1 |𝑥𝑡) = 𝛼0̂ + 𝛼1̂(𝛼0̂ + 𝛼1̂𝑥 2 𝑡) = 𝛼0̂ (1 + 𝛼1̂) +𝛼1̂ 2 𝑥2𝑡 = 𝜎 2 𝑡 (2) (3.26) and the 𝜁- step ahead forecast for 𝑥2𝑡 is given as 𝑥2𝑡 (𝜁) = 𝔼(𝑥 2 𝑡+𝜁 |𝑥𝑡) = 𝔼(𝜎2𝑡+𝜁 |𝑥𝑡) = 𝛼0̂ (1 + 𝛼1̂ + 𝛼1̂ 2 + ⋯ +𝛼1̂ 𝜁−1 ) +𝛼1̂ 𝜁 𝑥2𝑡 = 𝜎 2 𝑡 (𝜁) (3.27) Despite the advantages of the ARCH models, there are problems in using the ARCH models. First of all, the ARCH models assume that positive and negative shocks have the same effects on volatility because it depends on the square of the previous shocks, which is not the case in practice. Also the ARCH formulation can lead to complexity if the order of the model is higher. This necessitated the introduction of the GARCH model as an extension of the ARCH models (Tsay, 2002). 3.3 The GARCH (𝒎, 𝒔) Model The Generalized ARCH (GARCH) model was developed by Bollerslev (1986) as an extension of the ARCH model in the same way the ARMA process is an extension of the AR process. The principle of parsimony may be violated when a model has a large number of parameters resulting in difficulties in using the model to adequately University of Ghana http://ugspace.ug.edu.gh 58 describe the data. In particular, although the ARCH model is simple, it may require many parameters as there might be a need for a large value of lag 𝑞 and hence the principle of parsimony would be violated in such a case. An ARMA model may have fewer parameters compared to the AR model and similarly, a GARCH model may contain fewer parameters when compared to an ARCH model. Thus a GARCH model may be preferred to an ARCH model using the principle of parsimony. Let 𝑥𝑡 = 𝑟𝑡 - 𝑢𝑡 be the mean corrected return, where 𝑟𝑡 is the return of an asset, 𝑢𝑡 is the conditional mean of 𝑥𝑡. Then 𝑥𝑡 follows a GARCH (𝑚, 𝑠) model if 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.28a) 𝜎2𝑡 = 𝛼0 + ∑ 𝛼𝑖 𝑚 𝑖=1 𝑥 2 𝑡 −𝑖 + ∑ 𝛽𝑗 𝑠 𝑗=1 𝜎 2 𝑡 −𝑗 (3.29b) where {𝜀𝑡} is a sequence of independent, identically distributed random variables with mean zero and unit variance and the parameters of the model are 𝛼𝑖 , 𝑖 = 𝑜,⋯ ,𝑚 and 𝛽𝑗 ,𝑗 = 1,⋯ , 𝑠 such that 𝛼𝑖 ≥ 0 and 𝛽𝑗 ≥ 0 ; ∑ (𝛼𝑖 + 𝛽𝑖) < 1 𝑣 𝑖=1 , where 𝑣 = max(𝑚, 𝑠) and 𝛼𝑖 = 0 for 𝑖 > 𝑚 and 𝛽𝑗 = 0 for 𝑗 > 𝑠. The constraints on 𝛼𝑖 + 𝛽𝑖 implies that the unconditional variance of 𝑥𝑡 is finite, whereas its conditional variance 𝜎2𝑡 evolves over time. From the equations (3.28a) and (3.28b), it is seen that the GARCH (𝑚, 𝑠) model employs the same equation (3.1a) for the mean corrected return 𝑥𝑡 as in the ARCH (𝑚) but the equation for the volatility includes 𝑠 new terms. Therefore equations (3.28a) and (3.28b) reduces to a pure ARCH (𝑚) model if 𝑠 = 0. Thus the GARCH model generalizes the ARCH model by introducing values of 𝜎2𝑡−1, 𝜎 2 𝑡 − 2,⋯ . The parameters 𝛼𝑖 and 𝛽𝑗 are respectively referred to as the ARCH and GARCH parameters. The GARCH (𝑚, 𝑠) model can be stated differently. Let 𝜂𝑡 = 𝑥 2 𝑡 − 𝜎 2 𝑡 so that University of Ghana http://ugspace.ug.edu.gh 59 𝜎2𝑡 = 𝑥 2 𝑡 − 𝜂𝑡. By substituting 𝜎 2 𝑡−𝑖 = 𝑥 2 𝑡 −𝑖 − 𝜂𝑡−𝑖 , (𝑖 = 𝑜,⋯ ,𝑚 ) into equation (3.28b), the GARCH (𝑚, 𝑠) can be written as 𝑥2𝑡 = 𝛼0 + ∑ (𝛼𝑖 + 𝛽𝑖) 𝑣 𝑖=1 𝑥 2 𝑡 −𝑖 + 𝜂𝑡 − ∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗 (3.29) where 𝑣 = max(𝑚, 𝑠), 𝛼𝑖 = 0 for 𝑖 > 𝑚 and 𝛽𝑗 = 0 for 𝑗 > 𝑠. Thus the equation of 𝜎2𝑡 has an ARMA (𝑚, 𝑠) representation and it can be seen that {𝜂𝑡} is a martingale difference series (i.e.𝔼(𝜂𝑡) = 0 and 𝑐𝑜𝑣 (𝜂𝑡, 𝜂𝑡−𝑗) for 𝑗 ≥ 1). However, the {𝜂𝑡} is not an independent, identically distributed random sequence. In order to find the GARCH (m, s) process, we solve for 𝛼0 in the equation (3.29) by letting the variance of 𝑥𝑡 be 𝜎 2 𝑡. This yields 𝛼0 = 𝜎 2 𝑡 (1 − ∑ 𝛼𝑖 𝑚 𝑖=1 − ∑ 𝛽𝑗 𝑠 𝑗=1 ) (3.30) And substituting the value of 𝛼0 as given by equation (3.29) into equation (3.30) gives 𝑥2𝑡 = 𝜎 2 𝑡[1 − ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 ] + [∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 ]𝑥 2 𝑡 −𝑖 − ∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗 + 𝜂𝑡 = 𝜎2𝑡 + ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 (𝑥 2 𝑡 −𝑖 − 𝜎 2 𝑡) − ∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗 + 𝜂𝑡 (3.31) Therefore 𝑥2𝑡 − 𝜎 2 𝑡 = ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 (𝑥 2 𝑡 −𝑖 − 𝜎 2 𝑡) − ∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗 + 𝜂𝑡 (3.32) Multiplying both sides of equation (3.32) by (𝑥2𝑡−𝑘 − 𝜎 2 𝑡) results in (𝑥2𝑡−𝑘 − 𝜎 2 𝑡)(𝑥 2 𝑡 − 𝜎 2 𝑡) = ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 (𝑥 2 𝑡 −𝑖 − 𝜎 2 𝑡)(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡) − ∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡) + 𝜂𝑡(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡) (3.33) And taking expectations of equation (3.33), we have University of Ghana http://ugspace.ug.edu.gh 60 𝔼[(𝑥2𝑡−𝑘 − 𝜎 2 𝑡)(𝑥 2 𝑡 − 𝜎 2 𝑡) ] = 𝔼[∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 (𝑥 2 𝑡 −𝑖 − 𝜎 2 𝑡)(𝑥 2 𝑡−𝑘 − 𝜎2𝑡)] − 𝔼[∑ 𝛽𝑗 𝑠 𝑗=1 𝜂𝑡−𝑗(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)] + 𝔼 [𝜂𝑡(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)] (3.34) But 𝔼 [𝜂𝑡(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)] = 𝔼 [(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)𝔼(𝜂𝑡|𝑥𝑡)] = 0 since 𝜂𝑡 is a martingale difference and also 𝔼[𝛽𝑗𝜂𝑡−𝑗(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)] = 𝔼 [(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)𝔼(𝜂𝑡−𝑗|𝑥𝑡−𝑗)] = 0 for k < j. Thus the autocovariance of the squared returns for the GARCH (𝑚, 𝑠) model is given by 𝑐𝑜𝑣 (𝑥2𝑡, 𝑥 2 𝑡−𝑘) = 𝔼[∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 (𝑥 2 𝑡 −𝑖 − 𝜎 2 𝑡)(𝑥 2 𝑡−𝑘 − 𝜎 2 𝑡)] = ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 𝑐𝑜𝑣 (𝑥 2 𝑡, 𝑥 2 𝑡−𝑘+𝑖) (3.35) Dividing both sides of equation (3.35) by 𝜎2𝑡 gives the autocorrelation function at lag 𝑘 as 𝜌𝑘 = ∑ (𝛼𝑖 + 𝛽𝑗) 𝑣 𝑖,𝑗=1 𝜌𝑘−𝑖 , for 𝑘 ≥ (𝑚 + 1) (3.36) This result is analogous to the Yule-Walker equations for an AR process. Hence the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the squared returns in a GARCH process has the same pattern as those of an ARMA process. The ACF and PACF are useful in determining the orders 𝑚 and 𝑠 of the GARCH (𝑚, 𝑠) process. Also the ACF is used in checking model accuracy; in which case, the ACF’s of the residuals indicates the presence of a white noise if the model is adequate. The parameters 𝛼0, 𝛼1, ⋯ , 𝛼𝑚; 𝛽1, 𝛽2,⋯ , 𝛽𝑠 affect the autocorrelation but given the 𝜌𝑘 , ⋯ , 𝜌𝑚+1− 𝑣, the autocorrelation at higher lags are determined uniquely by the University of Ghana http://ugspace.ug.edu.gh 61 expression in equation (3.33) (Bollerslev,1986) as cited in Ngailo (2011). Denoting the 𝑣𝑡ℎ partial autocorrelation for 𝑥2𝑡 by ∅𝑣𝑣 then 𝜌𝑘 = ∑ ∅𝑣𝑣 𝑣 𝑖,𝑗=1 𝜌𝑘−𝑖 , 𝑘 = 1,⋯ , 𝑣 = max(𝑚, 𝑠) (3.37) It can be seen from equation (3.33) that, there are cut offs after lag 𝑚 for an ARCH (𝑚) process such that ∅𝑣𝑣 ≠ 0 for 𝑘 ≤ 𝑚 and ∅𝑣𝑣 = 0 for 𝑘 > 𝑚 and it is similar to the AR (𝑚) process and decays exponentially (Bollerslev, 1986). To understand the theory and concepts of the GARCH model, we would focus on the special case of the GARCH (1, 1) model. 3.3.1 GARCH (1, 1) Model The GARCH (1,1) model is a particular case of the GARCH (1,1) model where the orders 𝑚 and 𝑠 are both equal to one (i.e. 𝑚 = 𝑠 = 1). Let {𝑥𝑡} be the mean corrected return, 𝜀𝑡 be a Gaussian white noise with mean zero and unit variance. If 𝐼𝑡 is the information set available at time t given by 𝐼𝑡 = {𝑥1, 𝑥2, ⋯ , 𝑥𝑡−1 ; 𝜎 2 1, 𝜎 2 2, ⋯ , 𝜎 2 𝑡−1}, then the process {𝑥𝑡} follows a GARCH (1, 1) model if 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.38a) 𝜎2𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛽1𝜎 2 𝑡 −1 (3.38b) where 𝛼0, 𝛼1 and 𝛽1 are the parameters of the model such that 𝛼0 ≥ 0,𝛼1 ≥ 0 , 𝛽1 ≥ 0 and (𝛼𝑖 + 𝛽𝑖) < 1. The constraints on the parameters are to ensure that the conditional variance 𝜎2𝑡 is positive. Clearly from (3.38a) and (3.38b), it is evidenced University of Ghana http://ugspace.ug.edu.gh 62 that large past mean corrected return 𝑥2𝑡−1 or past conditional variance 𝜎 2 𝑡 −1 give rise to large values of 𝜎2𝑡 (Tsay,2002). It can be seen that {𝑥𝑡} is martingale difference as the conditional mean is zero (i.e. 𝔼 (𝑥𝑡|𝐼𝑡) = 0). Taking 𝜂𝑡 = 𝑥 2 𝑡 − 𝜎 2 𝑡 so that 𝜎 2 𝑡 = 𝑥 2 𝑡 − 𝜂𝑡, the GARCH (1,1) can be represented differently. By substituting 𝜎2𝑡−1 = 𝑥 2 𝑡 −1 − 𝜂𝑡−1 , into equation (3.38b), the GARCH (1,1) can be written as 𝑥2𝑡 = 𝛼0 + (𝛼1 + 𝛽1)𝑥 2 𝑡 −1 + 𝜂𝑡 − 𝛽1𝜂𝑡−1 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛽1(𝑥 2 𝑡 −1 − 𝜂𝑡−1 ) + 𝜂𝑡 (3.39) Again it can be seen that {𝜂𝑡} is a martingale difference series as 𝔼 (𝜂𝑡 |𝐼𝑡) = 0 (i.e. 𝔼(𝜂𝑡) = 0 and 𝑐𝑜𝑣 (𝜂𝑡, 𝜂𝑡−𝑗) for 𝑗 ≥ 1) and {𝜂𝑡} is an uncorrelated sequence. This implies from equation (3.39) that 𝔼(𝑥2𝑡 ) = 𝜎 2 𝑡 = 𝛼0 + (𝛼1 + 𝛽1) 𝔼(𝑥 2 𝑡−1 ) ⇒ 𝜎2𝑡 = 𝛼0 + (𝛼1 + 𝛽1) 𝔼(𝜎 2 𝑡 𝜀 2 𝑡) ⇒ 𝜎2𝑡 = 𝛼0 + (𝛼1 + 𝛽1) 𝜎 2 𝑡𝔼(𝜀 2 𝑡) ⇒ 𝜎2𝑡 = 𝛼0 + (𝛼1 + 𝛽1) 𝜎 2 𝑡 , since 𝔼(𝜀 2 𝑡) = 𝑉𝑎𝑟 (𝜀 2 𝑡) = 1 ⇒ 𝛼0 = (1 − 𝛼1 − 𝛽1 )𝜎 2 𝑡 ⇒ 𝜎2𝑡 = 𝛼0 [1− (𝛼1 + 𝛽1)] , provided |𝛼1 + 𝛽1| < 1 (3.40) University of Ghana http://ugspace.ug.edu.gh 63 3.3.2 Estimation of GARCH (𝒎,𝒔) model Once the orders m and s have been identified, the parameters 𝛼0, 𝛼1, ⋯ , 𝛼𝑚; 𝛽1, 𝛽2, ⋯ , 𝛽𝑠 of the GARCH (𝑚, 𝑠) model can then be estimated. The maximum likelihood estimation is used to estimate the parameters of the model. The initial values of both the squared returns and past conditional variances are needed in estimating the parameters of the model. Bollerslev (1986) and Tsay (2002) suggest that the unconditional variance given in equation (3.28b) or the past sample variance of the returns may be used as initial values. Therefore assuming 𝑥1, 𝑥2, ⋯ , 𝑥𝑚 ; 𝜎 2 1, 𝜎 2 2, ⋯ , 𝜎 2 𝑠 are known, the conditional log-likelihood is given by ℓ(𝑥𝑚+1, ⋯ , 𝑥𝑡; 𝜎2𝑠+1 , ⋯ ,𝜎2𝑡|𝜃; 𝑥1, 𝑥2, ⋯ , 𝑥𝑚 ; 𝜎 2 1, 𝜎 2 2, ⋯ , 𝜎 2 𝑠) = ∑ (− 1 2 ln 2𝜋 − 1 2 ln 𝜎2𝑡 − 𝑥2𝑡 2𝜎2𝑡 )𝑇𝑡=𝑣+1 (3.41) where 𝜃 = (𝛼0, 𝛼1, ⋯ , 𝛼𝑚; 𝛽1, 𝛽2,⋯ , 𝛽𝑠) and 𝑣 = max(𝑚, 𝑠) It follows that the conditional maximum likelihood estimates are obtained by maximizing the conditional log-likelihood function given by equation (3.41) 3.3.3 Estimation of the GARCH (1, 1) The estimation of the GARCH (1, 1) model is done in the same way as in the ARCH (1) model. The initial value of the past conditional variance (𝜎2 1 ) is needed since the conditional variance of the GARCH (1, 1) model depends also on the past conditional variance. The unconditional variance of 𝑥𝑡 can be taken as an initial value for this University of Ghana http://ugspace.ug.edu.gh 64 variance. That is, it can be taken as 𝛼0 [1− (𝛼1 + 𝛽1) ] . In some cases, the sample variance of the return series can be taken to be the initial value of the past conditional variance ( 𝜎21). Let 𝑥1, 𝑥2, ⋯ , 𝑥𝑛 be a sample of log-returns. The distribution of 𝑥𝑡 conditional on 𝛼0,𝛼1 , 𝛽1and 𝑥𝑡−1 is normal with zero mean and variance 𝜎 2 𝑡. Thus 𝑥𝑡|𝛼0,𝛼1 , 𝛽1, 𝑥𝑡−1 is normal such that 𝔼(𝑥𝑡|𝛼0,𝛼1 , 𝛽1, 𝑥𝑡−1) = 0 (3.42a) and 𝑉𝑎𝑟 (𝑥𝑡|𝛼0,𝛼1 , 𝛽1, 𝑥𝑡−1) = 𝜎 2 𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛽1𝜎 2 𝑡 −1 (3.42b) The likelihood function is given by 𝑓(𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) = ∏ 1 √2𝜋𝜎2𝑡 exp( −𝑥2𝑡 2𝜎2𝑡 )𝑛𝑡=1 (3.43) Such that 𝜎2𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛽1𝜎 2 𝑡 −1. The log-likelihood function of 𝛼0,𝛼1 , 𝛽1 is given as ℓ(𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) = ∑ (− 1 2 ln 2𝜋 − 1 2 ln 𝜎2𝑡 − 𝑥2𝑡 2𝜎2𝑡 )𝑛𝑡=1 (3.44) To obtain the estimates 𝛼0̂, 𝛼1 ̂ and 𝛽1̂ of 𝛼0, 𝛼1 and 𝛽1, maximize the log likelihood function by taking the partial derivatives of ℓ(𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) with respect to 𝛼0, 𝛼1 and 𝛽1. Respectively the partial derivatives of 𝛼0, 𝛼1 and 𝛽1 are 𝜕ℓ( 𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) 𝜕𝛼0 = − 1 2 ∑ 1 𝜎2𝑡 𝑛 𝑡=1 + 1 2 ∑ 𝑥2𝑡 𝜎4𝑡 𝑛 𝑡=1 (3.45a) University of Ghana http://ugspace.ug.edu.gh 65 𝜕ℓ( 𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) 𝜕𝛼1 = − 1 2 ∑ 𝑥2𝑡−1 𝜎2𝑡 𝑛 𝑡=1 + 1 2 ∑ 𝑥2𝑡 𝑥 2 𝑡−1 𝜎4𝑡 𝑛 𝑡=1 (3.45b) 𝜕ℓ( 𝑥1, ⋯ , 𝑥𝑛|𝛼0, 𝛼1 , 𝛽1) 𝜕𝛽1 = − 1 2 ∑ 𝜎2𝑡−1 𝜎2𝑡 𝑛 𝑡=1 + 1 2 ∑ 𝑥2𝑡 𝜎 2 𝑡−1 𝜎4𝑡 𝑛 𝑡=1 (3.45c) Recalling that 𝜎2𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛽1𝜎 2 𝑡 −1 and equating equations (3.45a), (3.45b) and (3.45c) to zero, three (3) systems of equations with three (3) unknowns are obtained as below: ∑ ( 1 𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1 ) =𝑛𝑡=1 ∑ ( 𝑥2𝑡 (𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1) 2) 𝑛 𝑡=1 (3.46a) ∑ ( 𝑥2𝑡−1 𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1 ) =𝑛𝑡=1 ∑ ( 𝑥2𝑡 𝑥 2 𝑡−1 (𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1) 2) 𝑛 𝑡=1 (3.46b) ∑ ( 𝛼0̂ + 𝛼1̂𝑥 2 𝑡 −2 + 𝛽1̂𝜎 2 𝑡 −2 𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1 ) =𝑛𝑡=1 ∑ ( 𝑥2𝑡(𝛼0̂ + 𝛼1̂𝑥 2 𝑡 −2 + 𝛽1̂𝜎 2 𝑡 −2) (𝛼0̂ + 𝛼1̂𝑥2𝑡 −1 + 𝛽1̂𝜎2𝑡 −1) 2 ) 𝑛 𝑡=1 (3.46c) where 𝜎2𝑡 −1 and 𝜎 2 𝑡 −2 can be expressed in terms of the log returns only, given that some initial values of 𝜎20 and 𝜎 2 1 are known. The maximum likelihood estimator of 𝜎2𝑡 is given as 𝜎2𝑡 ̂ = ∑ 𝑥2𝑖 𝑛 𝑖=1 𝑛 . Alternatively, the above sums can start from 𝑡 = 2 and 𝑡 = 3. Numerical methods are used in solving for the estimates 𝛼0̂, 𝛼1 ̂ and 𝛽1̂ and it can be verified that the Hessian matrix evaluated at 𝛼0 = 𝛼0̂, 𝛼1 = 𝛼1 ̂ and 𝛽1 = 𝛽1̂ defined by University of Ghana http://ugspace.ug.edu.gh 66 ℍ = [ 𝜕2ℓ 𝜕𝛼02 𝜕ℓ 𝜕𝛼0𝛼1 𝜕ℓ 𝜕𝛼0𝛽1 𝜕ℓ 𝜕𝛼1 𝛼0 𝜕ℓ 𝜕𝛼1 2 𝜕ℓ 𝜕𝛼1 𝛽1 𝜕ℓ 𝜕𝛽1𝛼0 𝜕ℓ 𝜕𝛽1𝛼1 𝜕ℓ 𝜕𝛽1 2 ] (3.47) is a negative matrix and so 𝛼0̂, 𝛼1 ̂ and 𝛽1̂ are the maximum likelihood estimates of 𝛼0, 𝛼1 and 𝛽1. Stated earlier, since the conditional variance of the GARCH (1, 1) model depends on the past conditional variance, an initial value of the past conditional variance (𝜎2 1 ) is needed. Usually, the unconditional variance of 𝑥𝑡 is used as an initial value for the past conditional variance. That is 𝜎21 is taken to be 𝛼0 [1− (𝛼1 + 𝛽1) ] . However, the sample variance of the return series can be taken to be the initial value. 3.3.4 Forecasting with GARCH (𝒎, 𝒔) model Forecasting of a GARCH model can be obtained using methods similar to those of an ARMA model. Thus the conditional variance of {𝑥𝑡} is obtained by taking the conditional expectation of the squared mean corrected returns. Consider the GARCH (𝑚, 𝑠) model as stated in equations (3.28a) and (3.28b). Assuming a forecasting origin of 𝑡, then the 𝜁-step ahead volatility forecast is given by 𝑥2𝑡 (𝜁) = 𝔼(𝑥 2 𝑡 + 𝜁|𝑥𝑡) = 𝛼0 + ∑ (𝛼𝑖 + 𝛽𝑖) 𝑚 𝑖=1 𝔼(𝑥 2 𝑡+ 𝜁 −𝑖|𝑥𝑡) + 𝜂𝑡 − ∑ 𝛽𝑗 𝑠 𝑗=1 𝔼(𝜂𝑡+𝜁−𝑗|𝑥𝑡) = 𝜎2𝑡 (𝜁); (3.48) where 𝑥2𝑡, ⋯ , 𝑥 2 𝑡+ 1 − 𝑚 ; 𝜎 2 𝑡, ⋯ , 𝜎 2 𝑡 + 1− 𝑠 are assumed known at time 𝑡 and the true parameters 𝛼𝑖; (𝑖 = 1,⋯ ,𝑚) and 𝛽𝑗 ; (𝑗 = 1,⋯ , 𝑠) values are replaced by their University of Ghana http://ugspace.ug.edu.gh 67 estimates. Furthermore, 𝔼(𝑥2𝑡 + 𝜁|𝑥𝑡) for 𝑖 < 𝜁 can be obtained recursively. For 𝑗 ≥ 𝜁 𝔼(𝜂𝑡+𝜁−𝑗|𝑥𝑡) = 0 and for < 𝜁 , 𝔼(𝜂𝑡+𝜁−𝑗|𝑥𝑡) = 𝜂𝑡+𝜁−𝑗. Considering the special case of GARCH (1, 1) model in equations (3.38a) and (3.38b) and assuming that the forecast origin of 𝑡, the 1-step ahead volatility forecast is given by 𝜎2𝑡 (1) = 𝑥 2 𝑡 (1) = 𝛼0 + 𝛼1𝑥 2 𝑡 + 𝛽1𝜎 2 𝑡 (3.49) where 𝑥𝑡 and 𝜎 2 𝑡 are known at the time index 𝑡. For a multi - step ahead forecast, we use 𝑥2𝑡 = 𝜎 2 𝑡 𝜀 2 𝑡 and rewrite the volatility equation in (3.38b) as 𝜎2𝑡+1 = 𝛼0 + (𝛼1 + 𝛽1) 𝜎 2 𝑡 + 𝛼1 𝜎 2 𝑡 ( 𝜀 2 𝑡 − 1) (3.50) when 𝑡 = ℎ + 1, then equation (3.50) becomes 𝜎2ℎ +2 = 𝛼0 + (𝛼1 + 𝛽1) 𝜎 2 ℎ+1 + 𝛼1 𝜎 2 ℎ+1 ( 𝜀 2 ℎ+1 − 1) (3.51) A 2-step ahead volatility forecast at the forecast origin 𝑡 is given as 𝜎2𝑡 (2) = 𝑥 2 𝑡 (2) = 𝛼0 +(𝛼1 + 𝛽1)𝜎 2 𝑡 (1) (3.52) and in general, the 𝜁- step ahead forecast is given as 𝜎2𝑡 (𝜁) = 𝑥 2 𝑡 (𝜁) = 𝛼0 +(𝛼1 + 𝛽1)𝜎 2 𝑡 (𝜁 − 1), 𝜁 > 1 (3.53) This result is exactly the same as that of an ARMA (1, 1) model with AR polynomial 1− (𝛼1 + 𝛽1)ℬ. By repeated substitutions in equation (3.49), the 𝜁- step ahead volatility forecast can be written as 𝜎2𝑡 (𝜁) = 𝛼0[1− (𝛼1+ 𝛽1) 𝜁−1] [1− (𝛼1 + 𝛽1)] + (𝛼1 + 𝛽1) 𝜁−1𝜎2𝑡 (1) (3.54) University of Ghana http://ugspace.ug.edu.gh 68 Therefore 𝜎2𝑡 (𝜁) → 𝛼0 [1− (𝛼1 + 𝛽1)] as 𝜁 → ∞ provided that (𝛼1 + 𝛽1) < 1. Consequently, the multi-step ahead volatility forecast of a GARCH (1, 1) model converges to the unconditional variance of 𝑥𝑡 , as the forecast horizon increase to infinity provided that the variance of 𝑥𝑡 (𝜎 2 𝑡) exists (Tsay,2002). Despite the added advantage that the GARCH model brought to the ARCH – type models, the GARCH model had the same weakness as the ARCH model. It also assumes that the return volatilities (conditional variance) respond equally to positive and negative shocks. That is the GARCH model is a symmetric model and does not capture the asymmetry effect that is inherent in most real life financial data (Frimpong and Oteng - Abayie, 2006). To circumvent this problem of asymmetric effects on the conditional variance, Nelson (1991) extended the ARCH framework by proposing the Exponential GARCH (EGARCH) model. 3.4 EGARCH (m, s) Model The Exponential GARCH model was proposed by Nelson (1991) to overcome some weakness of the GARCH model in dealing with financial time series. In particular, the EGARCH model is used to allow for asymmetric effects between positive and negative asset returns. Nelson (1991) considered the weighted innovation (error term) 𝑔(𝜀𝑡) = 𝜃𝜀𝑡 + 𝛾[|𝜀𝑡| − 𝔼(|𝜀𝑡|)] (3.55) University of Ghana http://ugspace.ug.edu.gh 69 where 𝜃 and 𝛾 are real coefficients such that 𝔼[𝑔(𝜀𝑡) ] = 0 since both |𝜀𝑡| and 𝔼(|𝜀𝑡|) are identical and independently distributed sequence with continuous distributions with zero mean. The asymmetry of 𝑔(𝜀𝑡) can be easily seen by rewriting 𝑔(𝜀𝑡) as 𝑔(𝜀𝑡) = { (𝜃 + 𝛾 )𝜀𝑡 − 𝛾𝔼(|𝜀𝑡|) , 𝜀𝑡 ≥ 0 (𝜃 − 𝛾 )𝜀𝑡 − 𝛾𝔼(|𝜀𝑡|), 𝜀𝑡 < 0 (3.56) As stated earlier in section 3.2.4.1, the error term (innovation) 𝜀𝑡 is assumed to be a standard normal (Gaussian), standardised student-t distribution or generalized error distribution (GED). For a standard normal random variable 𝜀𝑡, 𝔼(|𝜀𝑡|) = √2 𝜋⁄ and for the standardised student –t distribution, we have 𝔼(|𝜀𝑡|) = 2 √𝜈−2 Γ [(𝜈+1) 2⁄ ] (𝜈−1)Γ(𝜈 2⁄ )√𝜋 where 𝜈 is the degrees of freedom. Let 𝑥𝑡 = 𝑟𝑡 - 𝑢𝑡 be the mean corrected return, where 𝑟𝑡 is the return of an asset, 𝑢𝑡 is the conditional mean of 𝑥𝑡. Then 𝑥𝑡 follows an EGARCH (𝑚, 𝑠) model if 𝑥𝑡 = 𝜎𝑡𝜀𝑡 and (3.57a) ln(𝜎2𝑡) = 𝛼0 + 1 + 𝛽1ℬ+ ⋯+𝛽𝑠−1ℬ 𝑠−1 1 − 𝛼1ℬ+ ⋯+𝛼𝑚ℬ𝑚 . 𝑔(𝜀𝑡−1) (3.57b) where 𝛼0 is a constant, ℬ is the lag (back-shift) operator such that ℬ𝑔(𝜀𝑡) = 𝑔(𝜀𝑡−1), 1 + 𝛽1ℬ + ⋯+ 𝛽𝑠−1ℬ 𝑠−1 and 1 − 𝛼1ℬ + ⋯+ 𝛼𝑚ℬ 𝑚 are polynomials with zeros outside the unit circle and have no common factors. By zeros outside the circle we mean the absolute values of the zeros are greater than 1. From equations (3.57a) and (3.57b), it is seen that the EGARCH (m,s) model uses the usual ARMA parameterization to describe the evolution of the conditional variance of 𝑥𝑡 and hence some of the properties of the EGARCH model can be obtained in a similar manner as those of the GARCH model. An example of such properties is that University of Ghana http://ugspace.ug.edu.gh 70 the unconditional mean of ln(𝜎2𝑡) = 𝛼0. However, unlike the GARCH (m,s) model where there are positivity constraints made on the model constraints to ensure that the conditional variance (𝜎2𝑡) is positive, the EGARCH (m,s) model relaxes the positivity constraints on the model parameters. This is due to the fact that the EGARCH (m,s) models the logarithm of the conditional variance ln(𝜎2𝑡) instead of the conditional variance (𝜎2𝑡) itself. Also, the use of the ln(𝜎 2 𝑡) enables the model to respond asymmetrically to positive and negative lagged values of 𝑥𝑡. The EGARCH (m,s) model can be stated alternatively as 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.58a) ln(𝜎2𝑡) = 𝛼0 + ∑ 𝛼𝑖|𝑥𝑡−𝑖|+ 𝛾𝑖𝑥𝑡−𝑖 𝑠 𝑖=1 𝜎𝑡−𝑖 + ∑ 𝛽𝑗 𝑚 𝑗=1 ln(𝜎 2 𝑡−𝑗) (3.58b) A positive 𝑥𝑡−𝑖 contributes 𝛼𝑖(1 + 𝛾𝑖 )|𝜀𝑡−𝑖|to the log volatility, whereas a negative 𝑥𝑡−𝑖 contributes 𝛼𝑖(1 − 𝛾𝑖 )|𝜀𝑡−𝑖|, where 𝜀𝑡−𝑖 = 𝑥𝑡−𝑖 𝜎𝑡−𝑖⁄ . The parameter 𝛾 signifies the leverage effect and is expected to be negative. A simple EGARCH model of order (1,1) is considered to help better understand the theory and concept of the EGACRCH model. 3.4.1 EGARCH (1, 1) Model The EGARCH (1, 1) model is a particular case of the EGARCH (𝑚, 𝑠) model with order (1, 1) (i.e. 𝑚 = 𝑠 = 1). Let {𝑥𝑡} be the mean corrected return, 𝜀𝑡 be an identical and independently distributed standard normal white noise. Then the process follow an EGARCH (1, 1) model if University of Ghana http://ugspace.ug.edu.gh 71 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.59a) (1 − 𝛼𝛽) ln(𝜎2𝑡) = (1 − 𝛼)𝛼0 + 𝑔(𝜀𝑡−1) (3.59b) Given that 𝜀𝑡 be an identical and independently distributed standard normal with mean 𝔼(|𝜀𝑡|) = √2 𝜋⁄ , the model for ln(𝜎 2 𝑡) becomes (1 − 𝛼𝛽) ln(𝜎2𝑡) = { 𝛼∗ + (𝛾 + 𝜃 )𝜀𝑡−1, 𝜀𝑡−1 ≥ 0, 𝛼∗ + (𝛾 − 𝜃 )(−𝜀𝑡−1), 𝜀𝑡−1 < 0. (3.60) where 𝛼∗ = (1 − 𝛼)𝛼0 − 𝛾√2 𝜋⁄ Equation (3.60) is a nonlinear function and hence for this simple EGARCH model, the conditional variance (𝜎2𝑡) evolves in a nonlinear manner depending on the sign of 𝑥𝑡−1. Specifically, we have 𝜎2𝑡 = 𝜎 2𝛼 𝑡−1. exp(𝛼∗) { [(𝛾 + 𝜃 ) 𝑥𝑡−1 𝜎𝑡−1 ] , 𝑖𝑓 𝑥𝑡−1 ≥ 0, [(𝛾 − 𝜃 ) |𝑥𝑡−1| 𝜎𝑡−1 ] , 𝑖𝑓 𝑥𝑡−1 ≥ 0. (3.61) The coefficients (𝛾 + 𝜃 )and (𝛾 − 𝜃 ) show the asymmetric response to positive and negative lagged returns (𝑥𝑡−1). The model is therefore nonlinear if 𝜃 ≠ 0. 𝜃 is expected to be negative since negative shocks tend to have larger impacts and for higher order EGARCH models, the nonlinearity becomes much more complicated. Using the alternative form of the EGARCH (𝑚, 𝑠) model, the EGARCH (1, 1) can be written as 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (3.62a) ln(𝜎2𝑡) = 𝛼0 + 𝛼1|𝑥𝑡−1|+ 𝛾1𝑥𝑡−1 𝜎𝑡−1 +𝛽1 ln(𝜎 2 𝑡−1) (3.62b) University of Ghana http://ugspace.ug.edu.gh 72 3.4.2 Forecasting with EGARCH Model The forecasting using an EGARCH model is done using methods similar to those of an ARMA model since the EGARCH model uses the usual ARMA parameterization to describe the evolution of the conditional variance of 𝑥𝑡 . The simple EGARCH (1, 1) model would be used to illustrate the multi - step ahead forecasts of the EGACRH models. Assuming that the model parameters are known and the distributional assumption made on the error term is the standard Gaussian, the EGACRH (m, s) model is given as ln(𝜎2𝑡) = (1 − 𝛼1)𝛼0 + 𝛼1ln(𝜎 2 𝑡−1) + 𝑔(𝜀𝑡−1) (3.63) where 𝑔(𝜀𝑡−1) = 𝜃𝜀𝑡−1 + 𝛾[|𝜀𝑡−1| − 𝔼(|𝜀𝑡−1|)] = 𝜃𝜀𝑡−1 + 𝛾 [|𝜀𝑡−1| − √2 𝜋⁄ ]. Assuming a forecasting origin of 𝑡, then the 1- step ahead volatility forecast is given by 𝜎2𝑡 (1) = 𝜎 2 𝑡+1 = 𝜎 2𝛼1 𝑡 . 𝑒𝑥𝑝[(1 − 𝛼1)𝛼0]𝑒𝑥𝑝[𝑔(𝜀𝑡)] (3.64) where all of the quantities on the right-hand side are known. For a 2 - step ahead volatility forecast, at the forecast origin 𝑡, equation (3.36) gives 𝜎2𝑡 (2) = 𝜎 2 𝑡 +2 = 𝜎 2𝛼1 𝑡+1 . 𝑒𝑥𝑝[(1 − 𝛼1)𝛼0]𝑒𝑥𝑝[𝑔(𝜀𝑡+1)] (3.65) Taking conditional expectation on equation (3.65) at time 𝑡, we have 𝜎2𝑡 (2) = 𝜎 2 𝑡 +2 = 𝜎 2𝛼1 𝑡+1 . 𝑒𝑥𝑝[(1 − 𝛼1)𝛼0]𝔼𝑡{𝑒𝑥𝑝[𝑔(𝜀𝑡+1)]} University of Ghana http://ugspace.ug.edu.gh 73 where 𝔼𝑡 denotes a conditional expectation taken at the time origin 𝑡. The prior expectation can be obtained as follows 𝔼{𝑒𝑥𝑝[𝑔(𝜀)]} = ∫ 𝑒𝑥𝑝 [𝜃𝜀 + 𝛾 (|𝜀| − √2 𝜋⁄ )] 𝑓(𝜀)𝑑𝜀 ∞ − ∞ = 𝑒𝑥𝑝 (− 𝛾√2 𝜋⁄ ) [∫ 𝑒(𝜃+𝛾 )𝜀 ∞ 0 . 1 √2𝜋 . 𝑒− 𝜀2 2 𝑑𝜀 + ∫ 𝑒(𝜃−𝛾 )𝜀 0 −∞ . 1 √2𝜋 . 𝑒− 𝜀2 2 𝑑𝜀] = 𝑒𝑥𝑝 (− 𝛾√2 𝜋⁄ ) [𝑒 (𝜃+𝛾 )2 2 Φ(𝜃 + 𝛾 ) + 𝑒 (𝜃−𝛾 )2 2 Φ(𝛾 − 𝜃) ] (3.66) where 𝑓(𝜀) and Φ(𝑥) are the probability density function and cumulative density function of the standard normal distribution respectively. Consequently, the 2-step ahead volatility forecast is 𝜎2𝑡 (2) = 𝜎 2 𝑡 +2 = 𝜎 2𝛼1 𝑡 . 𝑒𝑥𝑝 [(1 − 𝛼1)𝛼0 − √2 𝜋⁄ ] × {𝑒𝑥𝑝 [ (𝜃+𝛾 )2 2 ]Φ(𝜃 + 𝛾 ) + 𝑒𝑥𝑝 [ (𝜃−𝛾 )2 2 ]Φ(𝛾 − 𝜃 ) } (3.67) Repeating the previous procedure, a recursive formula for the 𝜁- step ahead forecast is obtained as 𝜎2𝑡 (𝜁) = 𝜎 2𝛼1 𝑡 (𝜁 − 1) exp(𝜔) × {𝑒𝑥𝑝 [ (𝜃+𝛾 )2 2 ]Φ(𝜃 + 𝛾 ) + 𝑒𝑥𝑝 [ (𝜃−𝛾 )2 2 ]Φ(𝛾 − 𝜃 ) } (3.68) where 𝜔 = (1 − 𝛼1)𝛼0 − √2 𝜋⁄ . University of Ghana http://ugspace.ug.edu.gh 74 The values of Φ(𝜃 + 𝛾 ) and Φ(𝜃 − 𝛾 ) can be obtained from the following approximation. The cumulative density function of Φ(𝑥) where 𝑥 is a standard normal random variable can be approximated by Φ(𝑥) = { 1 − 𝑓(𝑥)[𝑐1𝑘 + 𝑐2𝑘 2 + 𝑐3𝑘 3 + 𝑐4𝑘 4 + 𝑐5𝑘 5] 𝑖𝑓 𝑥 ≥ 0, 1 − Φ(−𝑥) 𝑖𝑓 𝑥 < 0. (3.69) where (𝑥) = exp(−𝑥2 2⁄ ) √2𝜋 ; 𝑘 = 1 (1+0.2316419𝑥) , 𝑐1 = 0.319381530, 𝑐2 = − 0.356563782, 𝑐3 = 1.781477937, 𝑐1 = −1.821255978 and 𝑐5 = 1.330274429 Further, with the advancement of technology, it possible for the value of Φ(𝜃 + 𝛾 ) and Φ(𝜃 − 𝛾 ) to be obtained from most statistical packages (Tsay, 2002). 3.5 Model Selection Criteria The ACF and PACF assist in determining the order of the model but this is just a suggestion of where the model can be built from and it is imperative to build the model around the suggested model order (Aidoo, 2010). Several models with different orders can be considered and the ultimate (most suitable) model be selected from the family of candidate models that characterize the ordering data. The information criteria have been widely used in time series analysis to determine the appropriate order of a model. The idea behind the information criteria is to provide a measure of information in terms of the order of the model, which strikes a balance between the measure of goodness of fit and parsimonious specification of the model. The information criteria make use of the Kullback-Leibler effect in determining the suitable model. The Kullback-Leibler quantity of information contained in a model is University of Ghana http://ugspace.ug.edu.gh 75 the distance from the ‘true’ model and is measured by the log likelihood function (Aidoo, 2010). Several selection criteria have been proposed to aid in selecting the most appropriate model. Among others, we have the Akaike Information criteria (AIC) by Akaike (1974), Bayesian Information criterion (BIC) by Schwartz (1978), Hannan-Quinn (HQ) by Hannan and Quinn (1979), the coefficient of determination (𝑅2), etc. The several competing models are ranked according to their AIC, BIC or HQ values with the model having the lowest information criterion value being adjudged the best. If two or more competing models have the same or similar AIC, BIC or HQ values, then the principle of parsimony is applied to select the most appropriate model. The principle of parsimony states that a model with fewer parameters is usually better than a complex model. Alternatively to the use of the principle of parsimony, forecast accuracy tests between the competing models can be used (Aidoo, 2010). In general, the model selected as the most appropriate model by two different criteria may differ and thus it should be noted that the selection of an ARCH-type model depends on the selection criteria used (Talke, 2003). 3.5.1 Akaike Information Criterion (AIC) The Akaike Information Criterion (AIC) was introduced by Hirotogu Akaike in 1973. It was the first model selection criterion to gain widespread acceptance. The AIC was an extension to the maximum principle and consequently the maximum likelihood principle is applied to estimate the parameters of the model once the structure of the model has been specified. The AIC is defined as University of Ghana http://ugspace.ug.edu.gh 76 AIC = 2(𝑁) − 2(𝑙𝑜𝑔𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑) (3.70) where 𝑁 denotes the number of parameters in the model. Given a family of competing models of various structures, the maximum likelihood estimation is used to fit the model and the AIC is computed based on each model fit. The selection of the most appropriate model is then made by considering the model with the minimum AIC. Akaike’s idea was to combine estimation and structural determination into a single procedure. The first term of the AIC in equation (3.70) measures the goodness of fit of the model whereas the second term is called the penalty function of the criterion since it penalizes a candidate model by the number of parameters used. The main advantage of the AIC is that it is useful for both in-sample and out-of- sample forecasting performance of a model. In-sample forecasting indicates how the chosen model fits the data in a given sample while out-of-sample forecasting is concerned with determining how a fitted model forecast future values of the regressed given the values of the regressors. Secondly, the AIC is useful for both nested and non-nested models. Despite the advantages of the AIC such as mentioned above, the AIC has been criticised because of its inconsistency and tendency to over-fit a model. This inconsistency was shown by Shibata (1976) for autoregressive models (AR) and Hannan (1982) for ARMA models as cited in Shittu and Asemota (2009). To overcome this problem especially that of inconsistency, Schwartz (1978) proposed the Bayesian Information Criterion. University of Ghana http://ugspace.ug.edu.gh 77 3.5.2 Bayesian Information Criterion The Bayesian Information criterion (BIC) is related to the Bayes factor and is useful for selecting the most appropriate model out of a candidate of families of models. The BIC is obtained by replacing the non-negative factor 2(𝑁) in equation (3.70) by 𝑘 ln(𝑛). Hence, the BIC is defined as BIC = 𝑘 ln(𝑛) − 2(𝑙𝑜𝑔𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑) (3.71) Where k denotes the number of parameters in the model, n is the length of the time series or the sample size. Again, the maximum likelihood estimation is used to fit the model and the BIC is computed for each of the models in a family of competing models and the fitted model with the minimum BIC is considered to be most appropriate model. Comparing equations (3.70) and (3.71), it is can be seen that the BIC imposes a harsher penalty than the AIC especially for models with many parameters (i.e. complex models). The advantages of the Bayesian information criterion is that for a wide range of statistical problems, it is order consistent (i.e. when the sample size goes to infinity, the probability of choosing the right model converges to unity) leading to more parsimonious model. Also, like the AIC, the BIC can be used to compare in-sample or out-of-sample forecasting performance of a model. 3.6 Model diagnostic checks and adequacy The model diagnostic checks are performed to determine the adequacy or goodness of fit of a chosen model. The model diagnostic checks are performed on residuals and University of Ghana http://ugspace.ug.edu.gh 78 more specifically on the standardized residuals (Talke, 2003). The residuals are assumed to be independently and identically distributed following a normal distribution (Tsay, 2002). Plots of the residuals such as the histogram, the normal probability plot and the time plot of residuals can be used. If the model fits the data well the histogram of residuals should be approximately symmetric. The normal probability plot should be a straight line while the time plot should exhibit random variation (Bowerman and O’Connell, 1997). The ACF and the PACF of the standardized residuals are used for checking the adequacy of the conditional variance model. The Lagrange multiplier and the Ljung Box Q-test (given in section 3.2) are used to check the validity of the ARCH effects as well as test for autocorrelation in the data. To test the presence of ARCH effects, the null hypothesis of no ARCH effects is rejected if the significance probability value (p-value) is less than specified level of significance. In case of testing for the presence of autocorrelation, the null hypothesis of no autocorrelation is rejected if the Ljung –Box (Q) statistics (as defined under section 3.2.2.1 by equations 3.6a and 3.6b) of some of the lags are significant. Thus if the probability value of Ljung –Box (Q) statistics of some of the lags are less than the specified level of significance, then the null hypothesis of no autocorrelation is rejected. Once the estimated model satisfies all these model assumptions, it can be seen as an appropriate representation of the data. Having established that the model fits the data well, the model can then be used to compute forecasts of the series under consideration. University of Ghana http://ugspace.ug.edu.gh 79 3.7 Model Validation The data set was divided into two parts; an initialization or training set and a verification or test set. The training set was used to estimate the model parameters whilst the test set was used to validate the model. This validation process is necessary to evaluate the model for how accurate it is in forecasting. If a chosen model is able to describe the testing set well, then the model is considered valid and adequate and hence it can be used in forecasting the series under consideration. 3.8 Assessment of Predictiveness or Forecast Accuracy of a Model As pointed earlier in section 3.5, forecast accuracy test can be used as criteria for selecting the best model. Several measures for assessing the forecast accuracy of ARCH-type models have been proposed. Some of these measures are the mean square error (MSE), mean absolute error (MAE) and Theil’s U – statistic. The MSE is defined as the average of the squared difference between the actual variance and the volatility forecast (𝜎2𝑡). In the absence of the observed true variance, the squared time series observation 𝑥2𝑡is used. The MSE is given by MSE = ∑ (𝑥2𝑡 − 𝜎2?̂?) 2𝑇 𝑡=1 𝑇 (3.72) where 𝜎2?̂?, 𝑡 = 1,⋯ , 𝑇 is the estimated conditional variance obtained from fitting the ARCH-type model. The MSE is criticised because 𝑥2𝑡 is noisy and unstable although the 𝑥2𝑡 is a consistent estimator of 𝜎 2 𝑡 (Tsay, 2002). Alternatively, other University of Ghana http://ugspace.ug.edu.gh 80 measures have been proposed. The mean absolute error (MAE) was proposed by Lopez (1999) and defined as MAE = ∑ |𝑥2𝑡 − 𝜎2?̂?| 2𝑇 𝑡=1 𝑇 (3.73) The last but not the least criterion is the Theil’s U-statistic which is used to test the accuracy of the future predictions. The Theil’s U-statistic is defined as 𝑈 = √ ∑ (𝐹𝑃𝐸𝑡+1− 𝐴𝑃𝐸𝑡+1)2 𝑇−1 𝑡=1 ∑ (𝐴𝑃𝐸𝑡+1)2 𝑇−1 𝑡=1 (3.74) where 𝐹𝑃𝐸𝑡+1 = (𝑥𝑡+1̂− 𝑥𝑡) 𝑥𝑡 is the forecasted relative change, and 𝐴𝑃𝐸𝑡+1 (𝑥𝑡+1−𝑥𝑡 ) 𝑥𝑡 is the actual relative change. If the forecasts are good then U should be close to zero. A U-statistic of one implies that the model under consideration and the benchmark model are equal. The models could equal in terms of accuracy or inaccuracy. A U- statistic of less than one implies that the model is superior to the benchmark while a U-statistic of greater than one implies the model is inferior to the benchmark model. 3.9 Conclusion The chapter has provided an overview of time series and its basic concepts as well as a detailed description – order determination, estimation and forecasting – of the three main autoregressive Heteroscedastic models that were used in the study. Furthermore, various selection criteria that help to select the best fit model among a class of competing models were discussed. Finally, there was an exposition on model validation as well as assessing the predictiveness or forecast accuracy of a model. The empirical findings based on the methods discussed under this chapter follow in the next chapter. University of Ghana http://ugspace.ug.edu.gh 81 CHAPTER FOUR DATA ANALYSIS AND DISCUSSION OF RESULTS 4.0 Introduction This chapter presents the analysis and the discussion of the results obtained from the study. The chapter is further organised into five sub sections excluding the introductory section. A description of the data with respect to basic statistics is done under section 4.1. Section 4.2 deals with the preliminary analysis of the data. Model estimation and fitting as well as model evaluation and diagnostics are presented under sections 4.3 and 4.4 respectively. The last section, section 4.5 focused on forecasting of monthly inflation rates based on the model selected as the most appropriate model under section 4.4. The analysis was carried out using both MINITAB 16 and EVIEWS 5.0 statistical software. The MINITAB 16 was used to obtain the various graphs due to its pictorial clarity whilst the EVIEWS 5.0 was used for the rest of the analysis such as descriptive statistics, estimation, etc. 4.1 Summary Statistics and Data Description The sample data consists of Five Hundred and Seventy Six (576) observations of the monthly rates of inflation in Ghana. It covers a forty - eight (48) year period spanning from January 1965 to December 2012. The data was divided into two parts; a training set consisting of the monthly inflation rates from January 1965 to December 2011 which was used to estimate the model parameters and a test set consisting of the monthly rates on inflation from January 2012 to December 2012 which was used to University of Ghana http://ugspace.ug.edu.gh 82 validate the chosen model. The data were obtained from the Ghana Statistical Service (GSS) as published on their official website www.gss.gov.org. The GSS is the official government institution mandated to provide the rate of inflation in Ghana on periodic bases such as monthly rates of inflation. Table 4.1.1 shows the descriptive statistics of the monthly rates of inflation. Table 4.1.1: Descriptive Statistics of Monthly Rates of Inflation in Ghana (1965 - 2012) Statistic Value Statistic Value Mean 29.82 Range 186.20 S.E Mean 1.28 Skewness 2.14 Std. deviation 30.73 Kurtosis 7.58 Median 20.20 Jarque-Bera 931.27 Maximum 174.10 Probability 0.0000 Minimum -12.10 Sample 576 Source: Researcher’s Calculation based on sampled data From Table 4.1.1, the results show that the mean of the monthly inflation rate is 29.82 with a standard error of 1.28 and a standard deviation of 30.73. The maximum rate of inflation was 174.10 whilst the minimum rate of inflation was -12.10. Thus the range of the rate of inflation over the 28 years period under consideration was 186.20. Also the rates of inflation were centred on a median of 20.20. Further, the data had a positive skewness of 2.14 implying that the distribution of the data has a long right tail and a kurtosis of 7.58 (i.e. a high excess kurtosis of 4.58) indicating that the distribution of the monthly rate of inflation was leptokurtic. The Jarque-Bera statistic of 931.27 is statistically significant at 1% level of significance. Thus it can be concluded that the monthly rates of inflation has a non-normal distribution. These confirm the non-normality and positive skewness of the monthly rates of inflation as University of Ghana http://ugspace.ug.edu.gh 83 revealed by Figure 4.1.1. The histogram of the residuals has a long right tail indicating positive skewness whilst the normal probability (QQ) plot of the residuals shows the data in a curvilinear form implying a deviation from normality. Figure 4.1.1: Residual Plots of the Monthly Rates of inflation in Ghana (1965 - 2012) 4.2 Preliminary Analysis The plot of the monthly rates of inflation for the period January 1965 to December 2012 is given by Figure 4.2.1. From Figure 4.2.1, it is evident that both the mean and variance are changing over time. That is, the monthly rate of inflation series is characterised by a non-constant mean and an unstable variance. The changing mean and variance over time is an indication of the non-stationarity of the monthly rates of inflation. Moreover, the trend analysis as shown in Figure 4.2.2 reveals a decreasing trend. 1000-100 99.99 99 90 50 10 1 0.01 Residual P e r c e n t 353025 150 100 50 0 -50 Fitted Value R e s i d u a l 1209060300-30 200 150 100 50 0 Residual F r e q u e n c y 550500450400350300250200150100501 150 100 50 0 -50 Observation Order R e s i d u a l Normal Probability Plot Versus Fits Histogram Versus Order University of Ghana http://ugspace.ug.edu.gh 84 Year Month 200519971989198119731965 JanJanJanJanJanJan 200 150 100 50 0 I N F L A T I O N Figure 4.2.1: Time Series plot of the Monthly rates of Inflation in Ghana (1965 - 2012) Year Month 200519971989198119731965 JanJanJanJanJanJan 200 150 100 50 0 I N F L A T I O N MAPE 176.271 MAD 20.390 MSD 924.269 Accuracy Measures Actual Fits Variable Linear Trend Model Yt = 37.29 - 0.025919*t Figure 4.2.2: Trend Analysis Plot for Monthly rates of inflation in Ghana (1965 - 2012) To confirm the presence of stationarity, the Augmented Dickey-Fuller (ADF) test was performed. The test fails to reject the null hypothesis of unit root at 5% level of significance and thus it can be concluded that the rate of inflation is not stationary University of Ghana http://ugspace.ug.edu.gh 85 over the period January 1965 to December 2012. The result of the Augmented Dickey-Fuller (ADF) test is shown on Table 4.2.1. Table 4.2.1: Augmented Dickey-Fuller (ADF) Unit Root Test for the monthly Rates of Inflation in Ghana (1965 - 2012) Model Type Test Statistic Critical Value P-value Constant -2.65 -2.87 0.08 Constant + Trend -2.74 -3.42 0.22 None -1.77 -1.94 0.07 Source: Researcher’s Calculation based on sampled data A transformation was carried on the data to bring it to stationarity, which is a desirable characteristic feature in most time series models. There are several transformations that are usually used to achieve the desired characteristics in a time series data. Examples of such transformations include ordinary differencing, seasonal differencing, taking of natural logarithms, taking square roots, etc. The ordinary differencing was preferred because of the following reasons. First of all, some of the monthly rate of inflation had negative values and as such taking the natural logarithm or square roots of the data would have resulted in missing values. Secondly, the data does not reveal any form of seasonality (see Figure 2B and Figure 3B in Appendix B) and hence seasonal differencing was not necessary. Thus the choice of the ordinary difference was used. Figure 4.2.3 gives the plot of the first ordinary difference of the monthly rates of inflation. University of Ghana http://ugspace.ug.edu.gh 86 Year Month 200519971989198119731965 JanJanJanJanJanJan 40 30 20 10 0 -10 -20 -30 -40 1 S T D I F F Figure 4.2.3: Time series plot of the first difference of the monthly rates of inflation in Ghana (1965 - 2012) The plot in Figure 4.2.3 reveals that the first ordinary difference monthly rate of inflation series appears to be stable in both the mean and variance over time implying there is stationarity in the first ordinary difference monthly rate of inflation series. This is also confirmed by the residual plot and trend analysis of the first ordinary difference monthly rate of inflation series (see Figure 4B and Figure 5B respectively in Appendix B). From Figure 4.2.1, the monthly rate of inflation series exhibits heteroscedasticity (changing variance over time). A formal test for heteroscedasticity was however carried out to confirm the presence of heteroscedasticity (ARCH effects). The Ljung- Box (Q) test was performed and the results for some selected lags are shown in Table 4.2.2 (the results of all lags are given in Table 1B in Appendix B). From the results, the p-values are less than 5% level of significance indicating that, the Ljung-Box test statistic is significant at all lags giving an evidence of the presence of heteroscedasticity (ARCH effects) in the monthly rate of inflation series. University of Ghana http://ugspace.ug.edu.gh 87 Table 4.2.2: Test for Heteroscedasticity (ARCH effects) in monthly rates of inflation in Ghana (1965 - 2012) Lag Q – statistic P-value 1 551.52 0.0000 6 2425.30 0.0000 12 3073.10 0.0000 18 3263.20 0.0000 24 3608.50 0.0000 30 4120.20 0.0000 36 4499.30 0.0000 Source: Researcher’s Calculation based on sampled data The test for heteroscedasticity (ARCH effects) was also performed for the first difference monthly rate of inflation series and the results as shown by Table 4.2.3 revealed that there was significant evidence of heteroscedasticity (ARCH effects) although it has been reduced as compared to the case of the original monthly rate of inflation series. Table 4.2.3: Test for Heteroscedasticity (ARCH effects) in the first difference monthly rates of inflation in Ghana (1965 -2012) Lag Q - statistic P-value 1 130.74 0.0000 6 225.81 0.0000 12 387.17 0.0000 18 533.88 0.0000 24 537.62 0.0000 30 570.25 0.0000 36 584.07 0.0000 Source: Researcher’s Calculation based on sampled data University of Ghana http://ugspace.ug.edu.gh 88 Furthermore, the test for serial correlation (autocorrelation) in the monthly rate of inflation series was also performed. This was done by obtaining the Autocorrelation function (ACF) and Partial Autocorrelation function (ACF) plots of the monthly rate of inflation series as given by Figures 4.2.4 and 4.2.5 respectively. Significant spikes at lags 1 and 12 of the PACF given by Figure 4.2.5 may be an indication of seasonal variation. However, from the seasonal analysis done (see Figures 2B and 3B in Appendix B), the data does not reveal any form of seasonality and hence the spikes could be attributed to random effects. Furthermore, from the plots of the ACF and PACF shown by Figure 4.2.4 and Figure 4.2.5 respectively, there was an indication of correlation in the monthly rates of inflation. Engle (1982) asserts that any autocorrelation in a time series has to be removed before any ARCH-family models is constructed. To eliminate the autocorrelation, the first difference transformation of the monthly rates of inflation was obtained. Figures 4.2.6 and Figure 4.2.7 gives the ACF and PACF of the first difference of the monthly rates of inflation. 35302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Lag A u t o c o r r e l a t i o n Figure 4.2.4: Autocorrelation function (ACF) plots of the monthly rates of inflation (1965 - 2012) University of Ghana http://ugspace.ug.edu.gh 89 35302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Lag P a r t i a l A u t o c o r r e l a t i o n Figure 4.2.5: Partial Autocorrelation Function (ACF) plots of the monthly rates of inflation (1965 -2012) 35302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Lag A u t o c o r r e l a t i o n Figure 4.2.6: Autocorrelation Function (ACF) plots for the first difference of Monthly Rates of Inflation (1965 -2012) University of Ghana http://ugspace.ug.edu.gh 90 35302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 Lag P a r t i a l A u t o c o r r e l a t i o n Figure 4.2.7: Partial Autocorrelation Function (PACF) plots for the first difference of Monthly Rates of Inflation (1965 - 2012) The ACF dies in a sine wave form whilst the PACF shows significant number of spikes also dying down in a sine wave fashion. This indicates that there is no significant correlation in the first difference monthly rates of inflation. From the fore going analysis, it can be concluded that the first difference monthly rates of inflation satisfy all the data assumption or characteristics for a volatility model such as the ARCH-family models. Hence in subsequent analysis, the first differenced monthly rates of inflation was used or considered. The next step in time series model building procedure after all the assumptions or properties of the series has been satisfied is the determination of the order of the model. The Autocorrelation Function (ACF) and Partial Autocorrelation Function (ACF) plots of the series were used to determine the order. From the ACF and PACF plots of the first difference in Figure 4.2.6 and Figure 4.2.7 respectively, the ACF tails of at lag 2 whilst the PACF spike at lag 1. This suggests that the order 𝑚 = 2 and 𝑠 = 1 and hence therefore an AR (2) and MA (1) models were suspected combining to give an ARMA (2, 1) model. University of Ghana http://ugspace.ug.edu.gh 91 In conclusion, it is has been observed that the monthly rates of inflation had a unit root or was non-stationary and the first difference transformation brought stationarity. It was also observed that the first difference monthly rates of inflation have both significant heteroscedasticity (ARCH effects) and autocorrelation present in the series. Lastly from the ACF and PACF plots of the first difference of the monthly rates of inflation, the AR (2) and MA (1) were suspected giving an indication of an ARMA (2, 1) model around which to build the ARCH-family models. 4.3 Model Fitting and Estimation After the determination of the order of the model and consequently the model identification has been done, the parameters of the model can now be estimated. The method used to estimate the parameters is the maximum likelihood method. The maximum likelihood function that was used in the estimation was based on the distributional assumption of normality made on the error term or residuals as discussed under section 3.2.4.1. This distributional assumption of normality made on the error term or residuals was confirmed by histogram of the residuals of the first difference of the monthly rates of inflation as shown by Figure 6B in Appendix B. Since the order determined is usually a suggestion of the order around which the most appropriate model is found, several models of different order that lies close to the suggested model of ARMA (2, 1) was fitted and the most appropriate model was selected based on the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), Log Likelihood, R-square and the significance tests. The criteria is that the smaller the AIC, BIC and larger the R-square the better. Also the idea is to University of Ghana http://ugspace.ug.edu.gh 92 have a parsimonious model that captures as much variation in the data as possible. The EVIEWS 5.0 software was used to perform the trial and error modelling to determine the best fitting model. Table 4.3.1 gives the various suggested models for the ARCH model with their respective fit statistics. Table 4.3.1: Comparison of suggested ARCH (m) models with fit statistics Model AIC BIC ARCH(1) 5.36 5.41 ARCH(2) 5.31 5.36 ARCH(3) 5.31 5.37 Source: Researcher’s Calculation based on the sampled data From Table 4.3.1, both ARCH (2) and ARCH (3) have the same AIC and almost the same BIC value and hence any of these models might be appropriate. Therefore the most adequate model is specified by 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (4.1a) for 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1 + 𝛼2𝑥 2 𝑡−2 (4.1b) in the case of ARCH (2) or 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (4.1c) for 𝜎2𝑡 = 𝛼0+ 𝛼1𝑥 2 𝑡−1+ 𝛼2𝑥 2 𝑡−2+ 𝛼3𝑥 2 𝑡−3 (4.1d) in the case of ARCH (3). University of Ghana http://ugspace.ug.edu.gh 93 Assuming that the residuals are normally distributed, the Table 4.3.2 and Table 4.3.3 present the model outputs respectively for ARCH (2) and ARCH (3). Table 4.3.2: Model output for ARCH (2) Coefficient Std. Error z-Statistic Prob. C -0.154409 Mean Equation 0.100296 -1.539531 0.1237 AR(1) 0.038025 0.080133 0.474520 0.6351 AR(2) 0.247293 0.038061 6.497345 0.0000 MA(1) 0.236626 0.091648 2.581898 0.0098 Variance Equation 𝛼0 2.581578 0.169496 15.23090 0.0000 𝛼1 1.071298 0.093814 11.41942 0.0000 𝛼2 0.598491 0.061488 9.733461 0.0000 Source: Researcher’s Calculation based on sampled data Table 4.3.3: Model output for ARCH (3) Coefficient Std. Error z-Statistic Prob. C -0.212855 Mean Equation 0.113236 -1.879746 0.0601 AR(1) 0.892542 0.093720 9.523500 0.0000 AR(2) -0.110620 0.045480 -2.432313 0.0150 MA(1) -0.627033 0.080582 -7.781275 0.0000 Variance Equation 𝛼0 2.533706 0.169868 14.91570 0.0000 𝛼1 1.111763 0.104060 10.68385 0.0000 𝛼2 0.644038 0.101071 6.372114 0.0000 𝛼3 -0.001810 0.015143 -0.119515 0.9049 Source: Researcher’s Calculation based on sampled data From Table 4.3.2, the estimates of the coefficients of ARCH (2) as seen in the variance equation are all significant whilst although the estimates of the coefficients in Table 4.3.3 as seen in the variance equation meet the general requirement of an ARCH (3) model, the estimate of 𝛼3 is not statistically significant at the 5% level of significance. Thus the ARCH (3) parameter adds little explanatory power to model. Therefore, the model can be simplified by dropping the non significant parameter. University of Ghana http://ugspace.ug.edu.gh 94 Hence it can be concluded that the ARCH (2) model is the most appropriate among the ARCH (m) models and hence from Table 4.3.2 equations (4.1a) and (4.1b) can now be written as 𝑥𝑡 = 𝜎𝑡𝜀𝑡 𝜎2𝑡 = 2.5816 + 1.0713𝑥 2 𝑡−1 + 0.5985𝑥 2 𝑡−2. The next member of the ARCH-family models that were considered was the GARCH (m, s) models. Table 4.3.4 gives the various suggested models for the GARCH (m, s) model with their respective fit statistics. Table 4.3.4: Comparison of suggested GARCH (m, s) models with fit statistics Model AIC BIC GARCH(1,1) 5.17 5.22 GARCH(1,2) 5.16 5.22 GARCH(2,1) 5.12 5.18 GARCH(2,2) 5.17 5.24 Source: Researcher’s Calculation based on sampled data From Table 4.3.4, the GARCH (2, 1) have the smallest AIC and BIC values of 5.12 and 5.18 respectively and hence GARCH (2, 1) model is the most appropriate among the GARCH (m, s) models. Therefore the most adequate model is specified by 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (4.2a) 𝜎2𝑡 = 𝛼0 + 𝛼1𝑥 2 𝑡 −1 + 𝛼2𝑥 2 𝑡 −2+ 𝛽1𝜎 2 𝑡 −1 (4.2b) University of Ghana http://ugspace.ug.edu.gh 95 Assuming that the residuals are normally distributed, the model output obtained for GARCH (2, 1) is given by Table 4.3.5. Table 4.3.5: Model output for GARCH (2, 1) Coefficient Std. Error z-Statistic Prob. C -0.050480 Mean Equation 0.061193 -0.824919 0.4094 AR(1) 0.354733 0.206899 1.714528 0.0864 AR(2) 0.168479 0.084851 1.985587 0.0471 MA(1) -0.053134 0.218579 -0.243089 0.8079 Variance Equation 𝛼0 -0.009901 0.004244 -2.332809 0.0197 𝛼1 0.711921 0.048894 14.56061 0.0000 𝛼2 -0.503197 0.046471 -10.82820 0.0000 𝛽1 0.864969 0.008080 107.0518 0.0000 Source: Researcher’s Calculation based on sampled data From Table 4.3.5, the estimates of the coefficients of GARCH (2, 1) are all significant at the 5% level of significance. Hence from Table 4.3.5 equations (4.2a) and (4.2b) can now be written as 𝑥𝑡 = 𝜎𝑡𝜀𝑡 𝜎2𝑡 = −0.0099 + 0.7119𝑥 2 𝑡−1 – 0.5032𝑥 2 𝑡−2 + 0.8650𝜎 2 𝑡 −1. Lastly the EGARCH (m, s) models were considered. Table 4.3.6 gives the various suggested models for the EGARCH (m, s) model with their respective fit statistics. Table 4.3.6: Comparison of suggested EGARCH (m, s) models with fit statistics Model AIC BIC EGARCH(1,1) 5.16 5.23 EGARCH(1,2) 5.15 5.22 EGARCH(2,1) 5.09 5.16 EGARCH(2,2) 5.16 5.24 Source: Researcher’s Calculation based on sampled data University of Ghana http://ugspace.ug.edu.gh 96 From Table 4.3.6, the EGARCH (2, 1) have the smallest AIC and BIC values of 5.09 and 5.16 respectively and hence EGARCH (2, 1) model is the most appropriate among the EGARCH (m, s) models. Therefore the most adequate model using the alternative model form is specified by 𝑥𝑡 = 𝜎𝑡𝜀𝑡 (4.3a) ln(𝜎2𝑡) = 𝛼0 + 𝛼1|𝑥𝑡−1| 𝜎𝑡−1 + 𝛼2|𝑥𝑡−2| 𝜎𝑡−2 + 𝛾 𝑥𝑡−1 𝜎𝑡−1 +𝛽1 ln(𝜎 2 𝑡−1) (4.3b) Assuming that the residuals are normally distributed, the model output obtained for EGARCH (2, 1) is given by Table 4.3.7 Table 4.3.7: Model output for EGARCH (2, 1) Coefficient Std. Error z-Statistic Prob. C 0.106006 Mean Equation 0.126240 0.839721 0.4011 AR(1) 0.697105 0.203489 3.425760 0.0006 AR(2) 0.007368 0.085391 0.086289 0.9312 MA(1) -0.458052 0.195324 -2.345089 0.0190 Variance Equation 𝛼0 -0.168646 0.017756 -9.497944 0.0000 𝛼1 0.866121 0.069383 12.48327 0.0000 𝛼2 -0.597416 0.069788 -8.560454 0.0000 𝛾 0.124346 0.024108 5.157891 0.0000 𝛽1 0.991669 0.002699 367.4403 0.0000 Source: Researcher’s Calculation based on sampled data From Table 4.3.7, the estimates of EGARCH (2, 1) as seen by the variance equation are all significant at the 5% level of significance. From Table 4.3.5 equations (4.2a) and (4.2b) can now be written as 𝑥𝑡 = 𝜎𝑡𝜀𝑡 and ln(𝜎2𝑡) = −0.1686 + 0.8661|𝑥𝑡−1| 𝜎𝑡−1 − 0.5974|𝑥𝑡−2| 𝜎𝑡−2 + 0.1243 ( 𝑥𝑡−1 𝜎𝑡−1 ) + 0.9917 ln(𝜎2𝑡−1) University of Ghana http://ugspace.ug.edu.gh 97 Having constructed the most adequate model for each ARCH-family type model, the next task was to assess how well these models fit the data (model diagnostics or adequacy). The model diagnostic check was based on the residuals of the constructed model. 4.4 Diagnostic Checks and Adequacy for estimated Models The model diagnostic checks are performed to determine the adequacy of a chosen model. These checks were done through the analysis of the residuals from the fitted model. If the model fits the data well, the residuals are expected to be random, independent and identically distributed following the normal distribution. Plots of the residuals such as the histogram, the normal probability plot and the time plot of residuals were used. The histogram of residuals as well as the normal probability plot was used to check for normality approximately symmetric. The ACF and the PACF of the standardized residuals were used for checking the adequacy of the conditional variance model whilst the Lagrange multiplier and the Ljung Box Q-test were used to check the validity of the ARCH effects in the data. 4.4.1 Diagnostic Checks and Adequacy for the ARCH (2) Model The time plot of the residuals was used to check whether the residuals were random and the result is given by Figure 4.4.1.1. From the plot, the residuals exhibit random variation about their mean and hence it can be concluded that the residuals appear to be random. Also from the normal probability plot of the standardized residuals in University of Ghana http://ugspace.ug.edu.gh 98 Figure 4.4.1.2 the plot of the standardized residuals were almost linear. The linearity of the plot implied the distribution of the standardised residuals is normal. This was confirmed by the histogram of the standardised residuals shown by Figure 4.4.1.3. From Figure 4.4.1.3, the histogram is symmetric implying that the standardised residuals have a normal distribution. Furthermore, the results of Lagrange Multiplier test for ARCH effects as shown by Table 4.4.1.1 leads to the rejection of the null hypothesis of no ARCH effects since the test statistic was 2.5549 with a probability value of 0.0000, which is less than 5% significance level. Figure 4.4.1.1: Time plot of the residuals from ARCH (2) model -60 - 0 -20 0 20 40 -40 -20 0 20 40 100 200 300 400 500 Residual Actual Fitted University of Ghana http://ugspace.ug.edu.gh 99 5.02.50.0-2.5-5.0 99.99 99 95 80 50 20 5 1 0.01 standardized residuals P e r c e n t Mean 0.07700 StDev 0.9979 N 575 AD 12.550 P-Value <0.005 Figure 4.4.1.2: Normal Probability Plot of the Standardised Residuals from ARCH (2) Figure 4.4.1.3: A histogram of the Standardized Residuals from ARCH (2) Table 4.4.1.1: Lagrange Multiplier ARCH Test for ARCH (2) model F-statistic 2.554944 Probability 0.000004 Obs*R-squared 83.25909 Probability 0.000013 Source: Researcher’s Calculation based on sampled data 0 40 80 120 160 200 -6 -4 -2 0 2 4 6 University of Ghana http://ugspace.ug.edu.gh 100 The Ljung-Box (Q) statistic had a probability value of less than 5% level of significance for most of the lags as shown by Table 13C in the Appendix C. Hence the null hypothesis of no autocorrelation was rejected. Thus it can be concluded that the ARCH (2) model do not provide an adequate representation of the data since most of the model adequacy conditions were not satisfied. 4.4.2 Diagnostic Checks and Adequacy for the GARCH (2, 1) Model The time plot of the residuals was used to check whether the standardized residuals were random and the result is given by Figure 4.4.2.1. From the plot, the standardized residuals exhibit random variation about their mean and hence it can be concluded that the standardized residuals appear to be random. -60 -40 -20 0 20 40 -40 -20 0 20 40 100 200 300 400 500 Residual Actual Fitted University of Ghana http://ugspace.ug.edu.gh 101 Figure 4.4.2.1: A Time plot of the standardized residuals from the GARCH (2, 1) model 7.55.02.50.0-2.5-5.0 99.99 99 95 80 50 20 5 1 0.01 standardized residuals P e r c e n t Mean 0.05134 StDev 1.022 N 575 AD 7.458 P-Value <0.005 Figure 4.4.2.2: Normal Probability plot of the standardized residuals from the GARCH (2, 1) model Figure 4.4.2.3: A histogram of the standardized residuals from the GARCH (2, 1) model Table 4.4.2.1: Lagrange Multiplier ARCH Test for GARCH (2, 1) model F-statistic 0.649902 Probability 0.943659 0 40 80 120 160 200 -4 -2 0 2 4 6 8 University of Ghana http://ugspace.ug.edu.gh 102 Obs*R-squared 24.01883 Probability 0.936672 Source: Researcher’s Calculation based on sampled data Also from the normal probability plot of the standardized residuals in Figure 4.4.2.2 the plot of the standardized residuals was almost linear. The linearity of the plot implied the distribution of the standardised residuals is normal. This was confirmed by the histogram of the standardised residuals shown by Figure 4.4.2.3. From Figure 4.4.2.3, the histogram is symmetric implying that the standardised residuals have a normal distribution. Furthermore, the results of the Lagrange Multiplier test for ARCH effects as shown by Table 4.4.2.1 leads to failing to reject the null hypothesis of no ARCH effects since the test statistic was 0.6499 with a probability value of 0.9437, which is greater than 5% significance level. The Ljung-Box (Q) statistic had a probability value of less than 5% level of significance for almost all the lags as shown by Table 15C in the Appendix C. Hence the null hypothesis of no autocorrelation was rejected. Thus it can be concluded that the GARCH (2, 1) model provides an adequate representation of the data since most of the model adequacy conditions were satisfied. 4.4.3 Diagnostic Checks and Adequacy for the EGARCH (2, 1) Model The time plot of the standardized residuals was used to check whether the standardized residuals were random and the result is given by Figure 4.4.3.1. From the plot, the standardized residuals exhibit random variation about their mean and hence it can be concluded that the standardized residuals appear to be random. Also from the normal probability plot of the standardized residuals in Figure 4.4.3.2 the University of Ghana http://ugspace.ug.edu.gh 103 plot of the standardized residuals were almost linear. The linearity of the plot implied the distribution of the standardised residuals is normal. This was confirmed by the histogram of the standardised residuals shown by Figure 4.4.3.3. Figure 4.4.3.1: Time plot of the standardized residuals from the EGARCH (2, 1) model 7.55.02.50.0-2.5-5.0 99.99 99 95 80 50 20 5 1 0.01 standardized residuals P e r c e n t Mean 0.002032 StDev 1.003 N 575 AD 6.134 P-Value <0.005 Figure 4.4.3.2: Normal Probability plot of the standardized residuals from EGARCH (2, 1) -60 -40 -20 0 20 40 -40 -20 0 20 40 100 200 300 400 500 Residual Actual Fitted University of Ghana http://ugspace.ug.edu.gh 104 Figure 4.4.3.3: A histogram of the standardised residuals from EGARCH (2, 1) model Table 4.4.3.1: Lagrange Multiplier ARCH Test for the EGARCH (2, 1) F-statistic 0.509935 Probability 0.992536 Obs*R-squared 19.03354 Probability 0.990923 Source: Researcher’s Calculation based on sampled data From Figure 4.4.3.3, the histogram is symmetric implying that the standardised residuals have a normal distribution. Furthermore, the results of the Lagrange Multiplier test for ARCH effects as shown by Table 4.4.3.1 leads to the failure to the reject the null hypothesis of no ARCH effects since the test statistic was 0.5099 with a probability value of 0.9925, which is greater than 5% significance level. The Ljung-Box (Q) statistic had a probability value of less than 5% level of significance for almost all the lags as shown by Table 17C in the Appendix C. Hence the null hypothesis of no autocorrelation was rejected. Thus it can be concluded that 0 40 80 120 160 200 -4 -2 0 2 4 6 University of Ghana http://ugspace.ug.edu.gh 105 the EGARCH (2, 1) model provides an adequate representation of the data since most of the model adequacy conditions were satisfied. 4.5 Most Appropriate Model Selection Based on the diagnostics checks above, it is seen that among the three selected models (i.e. ARCH (2), GARCH (2, 1) and EGARCH (2, 1) models), the ARCH (2) model did not provide an adequate representation of the data. However, both the GARCH (2, 1) and EGARCH (2,1) provided an adequate representation of the data. Hence to choose the most appropriate model out of these two models, various selection criteria as mentioned earlier under section 3.5 were used. Table 4.5.1 gives the values of AIC and BIC of the GARCH (2, 1) and EGARCH (2, 1) models. Table 4.5.1: Selection Criteria Values for GARCH (2, 1) and EGARCH (2, 1) models Model AIC BIC GARCH(2,1) 5.12 5.18 EGARCH(2,1) 5.09 5.16 Source: Researcher’s Calculation based on sampled data From Table 4.5.1, it can be seen that the EGARCH (2, 1) had smaller values in both the AIC and BIC as compared to the GARCH (2, 1) model. Hence it can be concluded that the EGARCH (2, 1) model is the most appropriate representation of the data and therefore it would be used to forecast the future values of the monthly rates of inflation series. University of Ghana http://ugspace.ug.edu.gh 106 4.6 Forecasting Evaluation and Accuracy Criteria The models were also evaluated in terms of their forecasting ability of future monthly rates of inflation. This was necessitated as most previous research studies had found that the selected model was not necessary the model that provides best forecasting. Common measures of forecast evaluation such as the RMSE, MAE, MAPE and TIC were used. The model that exhibits the lowest values of the error measurements is considered to be the best. In Table 4.6.1, the results of the forecast performance are shown. Table 4.6.1 revealed that the EGARCH (2, 1) had the least value of all but one of the performance measurements. It had a RMSE of 5.79, MAE of 2.88, and MAPE of 15.70% and a TIC of 0.07. Thus the EGARCH (2, 1) model outperformed all the other models and was adjudged the best performing model. This results confirmed the earlier conclusion obtained based on the selection criteria such as the AIC, BIC, etc. The ARCH (2) model performed the least in forecasting the conditional volatility of the monthly rates of inflation. Table 4.6.1 Forecast Performance of Estimated Models Measure ARCH(2) GARCH(2,1) EGARCH(2,1) Root Mean Squared Error (RMSE) 5.75 5.70 5.79 Mean Absolute Error (MAE) 2.89 2.89 2.88 Mean Abs. Percent Error (MAPE) 15.88% 15.91% 15.70% Theil’s Inequality Coefficient (TIC) 0.07 0.07 0.07 Overall Model Performance rank 3 2 1 Source: Researcher’s Calculation based on sampled data University of Ghana http://ugspace.ug.edu.gh 107 4.7 Comparison of the EGARCH (2, 1) and ARIMA (2, 1, 1) Models To provide a base for comparison of the EGARCH (2,1) model to models that assume constant mean and variance, the ARIMA (2,1,1) model was estimated (details of the output are shown by Table 18C under appendix C). Table 4.7.1 shows the AIC, BIC, RMSE, MAE, MAPE and TIC values of the ARIMA (2, 1, 1) and EGARCH (2, 1) models. Table 4.7.1 Comparison of Performance Results of ARIMA (2, 1, 1) and EGARCH (2, 1) Statistic ARIMA (2,1,1) EGARCH (2,1) AIC 6.4 5.1 BIC 6.4 5.2 RMSE 5.7 5.8 MAE 2.9 2.9 MAPE 16.1% 15.7% TIC 0.07 0.07 Source: Researcher’s Calculation based on sampled data From Table 4.7.1, it was revealed that the EGARCH (2, 1) had the least AIC, BIC and MAPE whereas the ARIMA (2, 1) had the least RMSE. However, the MAE and TIC values were the same. Thus the EGARCH (2, 1) model outperformed the ARIMA (2, 1) model in three of the selection criteria used. Hence it was concluded that the EGARCH (2, 1) model was the best performing model on the basis of these results. Hence a one year out-sample forecast of the monthly rates of inflation based on the EGARCH (2, 1) model was obtained. Table 4.7.2 shows the actual monthly rates of inflation for January, 2012 to December, 2012 and the corresponding forecast values for the same period based on the EGARCH (2, 1) model as well as the 95% confidence interval of the forecasted values. Comparing the forecasted values to the actual values, it can be seen that the forecasted values are close to the actual values. Also, all the actual values fall within the confidence interval. Hence it is concluded University of Ghana http://ugspace.ug.edu.gh 108 that the EGARCH (2, 1), model is adequate to be used to forecast the monthly rates of inflation in Ghana. Table 4.7.2: One Year out – sample forecast of the monthly inflation rate from the EGARCH (2, 1) Model Month/Year Actual Forecast Forecast Error 95% Confidence Interval Lower Limit Upper Limit Jan-2012 8.70 8.70 0.00 8.0 9.3 Feb-2012 8.60 8.80 0.20 8.2 9.4 Mar-2012 8.80 8.70 -0.10 8.0 9.3 Apr-2012 9.10 8.90 0.20 8.3 9.5 May-2012 9.30 9.30 0.00 8.5 10.0 Jun-2012 9.40 9.50 0.10 8.9 10.0 Jul-2012 9.50 9.50 0.00 9.0 10.1 Aug-2012 9.50 9.60 0.10 9.1 10.1 Sep-2012 9.40 9.60 0.20 9.0 10.1 Oct-2012 9.20 9.40 0.20 8.9 10.0 Nov-2012 9.30 9.20 0.10 8.6 9.8 Dec-2012 8.80 9.40 0.60 8.9 9.9 Source: Researcher’s calculation based on sampled data University of Ghana http://ugspace.ug.edu.gh 109 4.8 Discussion of Results The implications of the results obtained are discussed under this section. However since the GARCH (2,1) and EGARCH (2,1) proved to be an adequate representation of the data whilst there was evidence that the ARCH (2) model do not provide adequate representation of the data, the discussion would neglect the results from the ARCH (2,1) model and focus on the results obtained from the GARCH (2,1) and the EGARCH (2,1) models. The results from the GARCH (2, 1) revealed that the volatility in the current month’s rate of inflation is explained by approximately 86% of the volatility in the previous month’s rate of inflation. Also there was no evidence of weakly stationarity in the volatility in the monthly rates of inflation as the sum of the ARCH parameters and the GARCH parameter exceeds one (i.e.0.7119 − 0.5032 + 0.8650 = 1.0737). This implies that there is volatility persistence in the monthly rates of inflation and that the volatility in the monthly rates of inflation is explosive and dies slowly. The persistence in the volatility in the monthly rate of inflation means that the impact of new shocks or information on the monthly rate of inflation will last for a longer period. The EGARCH (2, 1) model also provided evidence to the effect that the volatility in the current month’s rate of inflation is perfectly explained by the volatility in the previous month’s rate of inflation. Also the volatility persistence in the monthly rates of inflation observed in the GARCH (2, 1) model is also confirmed by the EGARCH (2, 1) model and that the volatility persistence was explosive as the sum of the ARCH parameters (0.8661,-0.5974) and the GARCH parameter (0.9917) is greater than one (i.e. 0.8661 − 0.5974 + 0.9917 = 1.2604). Furthermore, there was the presence of University of Ghana http://ugspace.ug.edu.gh 110 asymmetric effects on the volatility of the monthly rates of inflation. Thus positive shocks (news) and negative shocks (news) would have different impacts on the volatility of the monthly rates of inflation. However, there was no evidence of leverage effects as the asymmetric term positive (0.1243). The absence of leverage effects indicates that the impact of a positive shock on the volatility of the monthly rates of inflation exceeds that of a negative shock of equal magnitude. From the results, a positive shock would change the volatility in the monthly rates of inflation by 0.9738 while a negative shock of the same magnitude would change the volatility in the monthly rates of inflation by 0.7584. The monthly rates of inflation over the period were mostly low and centred around 20%. However, there is a high amount of variation in the monthly rates of inflation over the period and this might pose great challenges to other economic variables such as exchange rates, interest rates, stock returns, insurance premium, etc. Also the autoregressive Heteroscedastic models were superior in forecasting the monthly rates of inflation. This is consistent with Amos (2010), Awogbemi and Oluwaseyi (2011), Ezzat (2012), Igogo (2010), Su (2010), Lee and Brorsen (1996), Karanasos and Kim (2003), Alberg et al (2008); and Angelidis et al (2003). The superiority in performance of the autoregressive models is attributed to their ability to capture the stochastic nature of the monthly rates of inflation as evident in the pattern of the forecast errors. Lastly, looking at the upward trend of the out – sample forecasts, it can be predicted that Ghana would experience double digit inflation for the year 2013. This would impact on several aspects of the economy and could erode the economic gains made in the year 2012. The policy maker (Bank of Ghana) needs to put in place coherent monetary and fiscal policies that would put the anticipated increase in inflation under control. University of Ghana http://ugspace.ug.edu.gh 111 CHAPTER FIVE SUMMARY, CONCLUSIONS AND RECOMMEMNDTIONS 5.0 Introduction The chapter presents a summary of the findings from the study as well as the conclusions, recommendations and the areas for future research. Thus the main results and findings on the performance of the three selected Heteroscedastic models are presented. 5.1 Summary Most empirical researches carried out to investigate the volatilities of financial series have been carried out using the Autoregressive integrated moving averages (ARIMA) models of Box and Jenkins (1976) until the introduction of the Heteroscedastic models. In Ghana, several of such empirical researches have been carried out in the area of inflation modelling and forecasting. However, these Box and Jenkins models are based on the assumption of constant variance, which is uncharacteristic of most financial series. This study was designed to investigate the relative performance of three selected autoregressive Heteroscedastic models – Autoregressive Heteroscedastic Conditional (ARCH), Generalised ARCH (GARCH) and the Exponential GARCH (EGARH) models – in modelling the monthly rates of inflation in Ghana from January 1965 to December 2012. Also the study sought to compare the performance of the best fit University of Ghana http://ugspace.ug.edu.gh 112 model among the three selected autoregressive Heteroscedastic models with models from the Box and Jenkins family type models such as the ARIMA and finally to check for the presence of asymmetric and leverage effects in the volatility in the monthly rates of inflation. Series of tentative models were developed for each of the selected models based on the different values of the order (m, n) of the model. For the ARCH (m) models, the order 1 2, and 3 were developed whilst in the case of the GARCH (m, n) and EGARCH (m, n) models, four different combinations of the order (m, n) were developed. These order combinations were order (1, 1), (1, 2), (2, 1) and (2, 2). After considering the AIC and BIC values of the tentative models, the model with the minimum AIC and BIC was adjudged the best fit model among each of the ARCH - family type models. Based on the data analyses, the key findings are summarized as follows:  The average monthly rate of inflation for the period under study was 29.82 with a standard error of 1.28 and a standard deviation of 30.73. The high value of the standard deviation implied that there is a greater amount of variability among the monthly rates of inflation in Ghana over the period under study. The distribution of the monthly rates of inflation was leptokurtic with a long – tail to the right.  The distribution of the monthly rate of inflation was characterised by a non constant mean and an unstable variance implying a non-stationary series. Also a decreasing trend was observed for the series. The series also exhibited the presence of heteroscedasticity (ARCH effects) and autocorrelation. University of Ghana http://ugspace.ug.edu.gh 113  The ARCH (1), ARCH (2) and ARCH (3) models were developed among the ARCH (m) models. The AIC values for ARCH (1), ARCH (2) and ARCH (3) were 5.36, 5.31 and 5.31 respectively whilst the BIC values were also 5.41, 5.36 and 5.37 respectively. Hence the ARCH (2) had the minimum value for both AIC and BIC among the ARCH (m) model developed.  The GARCH (1, 1), GARCH (1, 2), GARCH (2, 1) and GARCH (2, 2) had the AIC values of 5.17, 5.16, 5.12 and 5.17 respectively. For the BIC values, GARCH (1, 1) had 5.22, GARCH (1, 2) had 5.22, and GARCH (2, 1) had 5.18 whilst GARCH (2, 2) had 5.24. The GARCH (2, 1) had the minimum value for both AIC and BIC values among the GARCH (m, s) models developed.  The EGARCH (1, 1), EGARCH (1, 2), EGARCH (2, 1) and EGARCH (2, 2) had AIC values of 5.16, 5.15, 5.09 and 5.16 respectively whilst their corresponding BIC values were 5.23, 5.22, 5.16 and 5.24. The EGARCH (2, 1), therefore, had the minimum value for both AIC and BIC values.  The EGARCH (2, 1) model was the overall most appropriate model as it had the minimum value for both AIC and BIC values among the ARCH –type models fitted.  Though there was asymmetric effects in the volatility in the monthly rates of inflation, there was however an absence of leverage effects.  The ARIMA (2, 1, 1) model was fitted for the Box and Jenkins family models. The AIC and BIC values were 6.35 and 6.38 respectively for the ARIMA (2, 1 1) model. University of Ghana http://ugspace.ug.edu.gh 114 5.2 Conclusion The ARCH (2) model was selected as the best fit model for predicting the monthly rate of inflation amongst the ARCH (m) models. In the case of the GARCH (m, n) and EGARCH (m, n) models, the order (1, 2) was the best choice amongst the four different order combinations. Thus the GARCH (1,2) and EGARCH (1,2) models were selected as the best fit models amongst the GARCH (m, n) and EGARCH (m, n) models respectively. With respect to the Box and Jenkins models, the ARIMA (2, 1, 1) model was adjudged the best fit model for modelling monthly rates of inflation in Ghana Subsequently, the three selected autoregressive Heteroscedastic best fit models – AR (2), GARCH (2, 1) and EGARCH (2, 1) - were compared based on their forecast performance. The goodness of fit models that were used included the root mean squared error, mean absolute error, mean absolute percent error and the Theil’s Inequality coefficient. The EGARCH (2, 1) was adjudged the most appropriate model amongst the three best fit models in modelling the monthly rates of inflation in Ghana as it had the minimum value for all the goodness of fit statistics. An asymmetric effect was also evident in the volatility in the monthly rates of inflation. However, there was an absence of leverage effects as positive shock changed the volatility in the monthly rate of inflation more than a negative shock of equal magnitude. Finally, when the EGACRH (2, 1) model was compared to the ARIMA (2, 1, 1) model, the EGACRH (2, 1) was found to be superior in modelling the rate of inflation in Ghana. University of Ghana http://ugspace.ug.edu.gh 115 5.3 Recommendations Based on the findings and conclusions made from the study, the following recommendations are made both in the area of policy formulation and further research. First of all, policy makers, industry players and all those interested in modelling and forecasting future rates of inflation in Ghana should consider using the Heteroscedastic models instead of the traditional Box and Jenkins models since the Heteroscedastic models are able to capture the volatilities in the monthly rates of inflation. Also by using the Heteroscedastic models, policy makers and industry players would be able to properly capture the volatility persistence in the monthly rates of inflation and leverage effects and hence forecasts and estimates would be more accurate. Secondly in the area of researchers, similar studies could be carried out using other Heteroscedastic models such as the Glosten-Jagannathan and Runkle (GJR) model, Threshold ARCH (TARCH) model, Power ARCH (PARCH) model, Integrated PARCH (IPARCH) and others. This would help identify the best model among the family of Heteroscedastic models that fits the monthly rate of inflation better for Ghana. Finally, it is recommended that multivariate time series models, where other economic variables that could influence the volatilities in the monthly rate of inflation such as exchange rates, amount of money supply, interest rates and others will be modelled along the rates of inflation. The inclusion of these other variables could help identify which of them contribute more to the variability in the monthly rates of inflation in Ghana. University of Ghana http://ugspace.ug.edu.gh 116 REFRENCES Abledu, G. K., & Agbodah, K. (2012). Stochastic Forecasting and Modelling of Volatility of Oil Prices in Ghana using ARIMA Time series model. European Journal of Business and Management, 4 (16), 122-131. Aidoo, E. (2010). Modelling and Forecasting inflation rates in Ghana: An application of SARIMA models. (Unpublished Master’s Thesis). School of Technology and Business Studies, Hogskolen , Dalarma. Akaike, H (1974). A New Look at the Statistical Model Identification. I.E.E.E. 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Seminar in Financial Data Analysis. University of Ghana http://ugspace.ug.edu.gh 125 APPENDICES APPENDIX A Table 1A: Monthly Consumer Price Index, Rates of Inflation and the first difference of the Rates of Inflation in Ghana from January 1965 to December 2012. OBS YEAR/MONTH CPI INFLATION FIRST DIFFERENCE 1 1965M01 128.10 13.30 NA 2 1965M02 135.30 18.40 5.10 3 1965M03 137.80 18.50 0.10 4 1965M04 145.20 23.90 5.40 5 1965M05 152.00 28.10 4.20 6 1965M06 155.60 27.00 -1.10 7 1965M07 162.80 33.80 6.80 8 1965M08 157.50 31.00 -2.80 9 1965M09 159.40 31.40 0.40 10 1965M10 158.50 31.20 -0.20 11 1965M11 158.50 28.80 -2.40 12 1965M12 165.00 29.80 1.00 13 1966M01 168.20 31.30 1.50 14 1966M02 172.30 27.30 -4.00 15 1966M03 169.30 22.90 -4.40 16 1966M04 173.00 19.10 -3.80 17 1966M05 176.20 15.90 -3.20 18 1966M06 184.90 18.80 2.90 19 1966M07 183.70 12.80 -6.00 20 1966M08 177.00 12.40 -0.40 21 1966M09 164.70 3.30 -9.10 22 1966M10 162.10 2.30 -1.00 23 1966M11 163.20 3.00 0.70 24 1966M12 161.90 -1.90 -4.90 25 1967M01 158.80 -5.60 -3.70 26 1967M02 156.10 -9.40 -3.80 27 1967M03 155.30 -8.30 1.10 28 1967M04 157.50 -9.00 -0.70 29 1967M05 159.40 -9.50 -0.50 30 1967M06 163.70 -11.50 -2.00 31 1967M07 161.50 -12.10 -0.60 32 1967M08 157.10 -11.20 0.90 33 1967M09 151.50 -8.00 3.20 34 1967M10 153.10 -5.60 2.40 35 1967M11 153.10 -6.20 -0.60 36 1967M12 157.20 -2.90 3.30 37 1968M01 161.60 1.80 4.70 38 1968M02 161.00 3.10 1.30 39 1968M03 160.10 3.10 0.00 40 1968M04 160.30 1.80 -1.30 41 1968M05 164.00 2.90 1.10 42 1968M06 169.20 3.40 0.50 University of Ghana http://ugspace.ug.edu.gh 126 43 1968M07 175.40 8.60 5.20 44 1968M08 174.40 11.00 2.40 45 1968M09 175.00 15.50 4.50 46 1968M10 179.30 17.10 1.60 47 1968M11 178.50 16.60 -0.50 48 1968M12 176.80 12.50 -4.10 49 1969M01 174.30 7.90 -4.60 50 1969M02 174.60 8.40 0.50 51 1969M03 174.60 9.10 0.70 52 1969M04 179.40 11.90 2.80 53 1969M05 183.80 12.10 0.20 54 1969M06 191.70 13.30 1.20 55 1969M07 195.40 11.40 -1.90 56 1969M08 185.50 6.40 -5.00 57 1969M09 178.00 1.70 -4.70 58 1969M10 180.50 0.70 -1.00 59 1969M11 181.30 1.60 0.90 60 1969M12 182.50 3.20 1.60 61 1970M01 182.50 3.20 0.00 62 1970M02 186.40 6.80 3.60 63 1970M03 186.20 6.60 -0.20 64 1970M04 186.90 4.20 -2.40 65 1970M05 193.70 5.40 1.20 66 1970M06 194.90 1.70 -3.70 67 1970M07 193.60 -0.90 -2.60 68 1970M08 195.20 5.20 6.10 69 1970M09 188.40 5.80 0.60 70 1970M10 185.60 2.80 -3.00 71 1970M11 185.00 2.00 -0.80 72 1970M12 183.80 0.70 -1.30 73 1971M01 185.90 1.90 1.20 74 1971M02 194.20 4.20 2.30 75 1971M03 194.70 4.60 0.40 76 1971M04 199.50 6.70 2.10 77 1971M05 212.80 9.90 3.20 78 1971M06 225.60 15.80 5.90 79 1971M07 222.70 15.00 -0.80 80 1971M08 211.90 8.60 -6.40 81 1971M09 208.30 10.90 2.30 82 1971M10 207.10 11.60 0.70 83 1971M11 205.30 11.00 -0.60 84 1971M12 203.80 10.90 -0.10 85 1972M01 218.30 17.40 6.50 86 1972M02 214.70 10.60 -6.80 87 1972M03 219.40 12.70 2.10 88 1972M04 229.50 15.00 2.30 89 1972M05 231.10 8.60 -6.40 90 1972M06 252.10 11.70 3.10 91 1972M07 227.50 2.20 -9.50 92 1972M08 223.40 5.40 3.20 93 1972M09 224.00 7.50 2.10 University of Ghana http://ugspace.ug.edu.gh 127 94 1972M10 224.10 8.20 0.70 95 1972M11 226.90 10.50 2.30 96 1972M12 229.60 12.70 2.20 97 1973M01 231.40 6.00 -6.70 98 1973M02 238.10 10.90 4.90 99 1973M03 243.90 11.20 0.30 100 1973M04 254.10 10.70 -0.50 101 1973M05 259.90 12.50 1.80 102 1973M06 267.10 6.00 -6.50 103 1973M07 289.60 27.30 21.30 104 1973M08 284.00 27.10 -0.20 105 1973M09 281.60 25.70 -1.40 106 1973M10 277.40 23.80 -1.90 107 1973M11 282.30 24.40 0.60 108 1973M12 287.80 25.30 0.90 109 1974M01 285.90 23.60 -1.70 110 1974M02 292.60 22.90 -0.70 111 1974M03 295.70 21.20 -1.70 112 1974M04 304.70 19.90 -1.30 113 1974M05 313.20 20.50 0.60 114 1974M06 319.50 20.50 0.00 115 1974M07 319.50 19.60 -0.90 116 1974M08 331.00 14.30 -5.30 117 1974M09 331.00 16.50 2.20 118 1974M10 324.20 15.10 -1.40 119 1974M11 322.60 16.30 1.20 120 1974M12 329.40 16.70 0.40 121 1975M01 334.10 16.10 -0.60 122 1975M02 343.80 20.30 4.20 123 1975M03 348.80 19.20 -1.10 124 1975M04 380.10 20.30 1.10 125 1975M05 390.10 24.60 4.30 126 1975M06 411.50 28.80 4.20 127 1975M07 425.50 28.50 -0.30 128 1975M08 433.40 30.90 2.40 129 1975M09 442.20 36.40 5.50 130 1975M10 447.70 38.80 2.40 131 1975M11 459.00 39.30 0.50 132 1975M12 469.00 40.40 1.10 133 1976M01 495.80 44.20 3.80 134 1976M02 519.40 48.90 4.70 135 1976M03 533.80 50.00 1.10 136 1976M04 556.40 46.40 -3.60 137 1976M05 587.60 50.60 4.20 138 1976M06 627.30 52.40 1.80 139 1976M07 690.70 62.30 9.90 140 1976M08 704.40 62.50 0.20 141 1976M09 721.90 63.30 0.80 142 1976M10 725.10 62.00 -1.30 143 1976M11 725.80 58.10 -3.90 144 1976M12 783.30 67.00 8.90 University of Ghana http://ugspace.ug.edu.gh 128 145 1977M01 814.10 64.20 -2.80 146 1977M02 868.20 67.20 3.00 147 1977M03 964.50 80.70 13.50 148 1977M04 1128.70 102.90 22.20 149 1977M05 1249.80 112.70 9.80 150 1977M06 1534.60 144.60 31.90 151 1977M07 1764.20 155.40 10.80 152 1977M08 1729.20 145.50 -9.90 153 1977M09 1741.40 141.20 -4.30 154 1977M10 1589.90 119.30 -21.90 155 1977M11 1591.60 119.30 0.00 156 1977M12 1613.60 106.00 -13.30 157 1978M01 127.20 103.90 -2.10 158 1978M02 132.70 99.40 -4.50 159 1978M03 144.20 95.10 -4.30 160 1978M04 153.20 77.10 -18.00 161 1978M05 159.90 66.90 -10.20 162 1978M06 167.70 42.60 -24.30 163 1978M07 170.10 25.80 -16.80 164 1978M08 172.70 30.30 4.50 165 1978M09 183.50 37.50 7.20 166 1978M10 199.90 64.00 26.50 167 1978M11 222.70 82.60 18.60 168 1978M12 243.30 96.70 14.10 169 1979M01 258.60 103.30 6.60 170 1979M02 269.50 103.10 -0.20 171 1979M03 310.90 94.40 -8.70 172 1979M04 284.80 69.80 -24.60 173 1979M05 310.90 94.40 24.60 174 1979M06 284.80 69.80 -24.60 175 1979M07 241.60 42.00 -27.80 176 1979M08 233.40 35.10 -6.90 177 1979M09 234.90 28.00 -7.10 178 1979M10 242.90 21.50 -6.50 179 1979M11 264.20 18.60 -2.90 180 1979M12 287.80 18.30 -0.30 181 1980M01 303.30 17.30 -1.00 182 1980M02 307.00 13.90 -3.40 183 1980M03 329.90 16.80 2.90 184 1980M04 356.60 20.10 3.30 185 1980M05 377.30 21.40 1.30 186 1980M06 400.30 40.60 19.20 187 1980M07 404.70 67.50 26.90 188 1980M08 416.20 78.30 10.80 189 1980M09 428.30 82.30 4.00 190 1980M10 449.80 85.20 2.90 191 1980M11 500.10 89.30 4.10 192 1980M12 540.60 87.80 -1.50 193 1981M01 611.80 101.70 13.90 194 1981M02 673.80 119.50 17.80 195 1981M03 733.60 122.40 2.90 University of Ghana http://ugspace.ug.edu.gh 129 196 1981M04 783.50 119.70 -2.70 197 1981M05 828.40 119.60 -0.10 198 1981M06 873.50 118.20 -1.14 199 1981M07 893.80 120.90 2.70 200 1981M08 922.00 121.50 0.60 201 1981M09 955.70 123.10 1.60 202 1981M10 1014.40 125.50 2.40 203 1981M11 1050.80 110.10 -15.40 204 1981M12 1083.30 100.40 -9.70 205 1982M01 1047.40 71.20 -29.20 206 1982M02 949.10 40.90 -30.30 207 1982M03 955.40 30.20 -10.70 208 1982M04 978.60 24.90 -5.30 209 1982M05 1016.50 22.70 -2.20 210 1982M06 1028.00 17.70 -5.00 211 1982M07 1078.30 20.60 2.90 212 1982M08 1048.60 13.70 -6.90 213 1982M09 1080.90 13.10 -0.60 214 1982M10 1131.50 11.50 -1.60 215 1982M11 1171.10 11.40 -0.10 216 1982M12 1264.10 16.70 5.30 217 1983M01 1388.00 32.50 15.80 218 1983M02 1543.60 62.60 30.10 219 1983M03 1743.80 82.50 19.90 220 1983M04 2039.60 108.40 25.90 221 1983M05 2517.80 147.70 39.30 222 1983M06 2818.20 174.10 26.40 223 1983M07 2599.30 141.10 -33.00 224 1983M08 2520.30 140.30 -0.80 225 1983M09 2570.00 137.80 -2.50 226 1983M10 2728.60 141.10 3.30 227 1983M11 2875.10 145.50 4.40 228 1983M12 3064.40 142.40 -3.10 229 1984M01 3136.80 126.00 -16.40 230 1984M02 3357.40 117.50 -8.50 231 1984M03 3459.90 98.40 -19.10 232 1984M04 3552.30 74.20 -24.20 233 1984M05 3606.80 43.30 -30.90 234 1984M06 3491.20 23.90 -19.40 235 1984M07 3352.10 29.00 5.10 236 1984M08 3132.00 24.30 -4.70 237 1984M09 3106.50 20.90 -3.40 238 1984M10 3105.10 13.80 -7.10 239 1984M11 3102.10 7.90 -5.90 240 1984M12 3247.90 6.00 -1.90 241 1985M01 3408.30 8.70 2.70 242 1985M02 3495.40 4.10 -4.60 243 1985M03 3575.90 3.40 -0.70 244 1985M04 3613.50 1.70 -1.70 245 1985M05 3647.90 1.10 -0.60 246 1985M06 3728.00 6.80 5.70 University of Ghana http://ugspace.ug.edu.gh 130 247 1985M07 3693.40 10.20 3.40 248 1985M08 3665.10 17.00 6.80 249 1985M09 3648.20 17.40 0.40 250 1985M10 3672.10 18.30 0.90 251 1985M11 3737.00 20.50 2.20 252 1985M12 3881.20 19.50 -1.00 253 1986M01 4047.20 18.70 -0.80 254 1986M02 4179.00 19.60 0.90 255 1986M03 4301.50 20.30 0.70 256 1986M04 4394.10 21.60 1.30 257 1986M05 4499.20 23.30 1.70 258 1986M06 4590.90 23.10 -0.20 259 1986M07 4575.20 23.90 0.80 260 1986M08 4542.30 23.90 0.00 261 1986M09 4517.80 23.80 -0.10 262 1986M10 4734.90 28.90 5.10 263 1986M11 4959.80 32.70 3.80 264 1986M12 5175.00 33.30 0.60 265 1987M01 5400.40 33.40 0.10 266 1987M02 5616.30 34.40 1.00 267 1987M03 5914.20 37.50 3.10 268 1987M04 6195.60 41.00 3.50 269 1987M05 6404.30 42.30 1.30 270 1987M06 6605.00 43.90 1.60 271 1987M07 6634.00 45.00 1.10 272 1987M08 6610.90 45.50 0.50 273 1987M09 6591.80 45.90 0.40 274 1987M10 6596.00 39.30 -6.60 275 1987M11 6710.50 35.30 -4.00 276 1987M12 6943.80 34.20 -1.10 277 1988M01 7232.00 33.90 -0.30 278 1988M02 7533.60 34.10 0.20 279 1988M03 7875.00 33.20 -0.90 280 1988M04 8208.00 32.50 -0.70 281 1988M05 8553.50 33.60 1.10 282 1988M06 8808.00 32.50 -1.10 283 1988M07 8722.00 31.50 -1.00 284 1988M08 8625.60 30.50 -1.00 285 1988M09 8571.50 30.00 -0.50 286 1988M10 8569.80 29.90 -0.10 287 1988M11 8639.30 28.70 -1.20 288 1988M12 8787.80 26.60 -2.10 289 1989M01 9132.70 26.30 -0.30 290 1989M02 9456.90 25.50 -0.80 291 1989M03 9829.00 24.80 -0.70 292 1989M04 10209.10 24.40 -0.40 293 1989M05 10544.20 23.30 -1.10 294 1989M06 10813.10 22.80 -0.50 295 1989M07 10775.00 23.50 0.70 296 1989M08 10677.40 23.80 0.30 297 1989M09 10663.20 24.40 0.60 University of Ghana http://ugspace.ug.edu.gh 131 298 1989M10 10764.10 25.60 1.20 299 1989M11 11062.70 28.10 2.50 300 1989M12 11464.40 30.50 2.40 301 1990M01 12150.40 33.00 2.50 302 1990M02 12856.30 35.90 2.90 303 1990M03 13374.20 36.10 0.20 304 1990M04 13885.10 36.00 -0.10 305 1990M05 14298.60 35.60 -0.40 306 1990M06 14751.00 36.40 0.80 307 1990M07 14978.00 39.00 2.60 308 1990M08 14971.00 40.20 1.20 309 1990M09 15073.40 41.40 1.20 310 1990M10 14996.80 39.30 -2.10 311 1990M11 15182.80 37.20 -2.10 312 1990M12 15580.30 35.90 -1.30 313 1991M01 15841.70 30.40 -5.50 314 1991M02 16277.90 26.60 -3.80 315 1991M03 16701.90 24.90 -1.70 316 1991M04 16980.70 22.30 -2.60 317 1991M05 17125.70 19.80 -2.50 318 1991M06 17302.80 17.30 -2.50 319 1991M07 17267.20 15.30 -2.00 320 1991M08 17157.80 14.60 -0.70 321 1991M09 17066.40 13.20 -1.40 322 1991M10 17088.50 13.90 0.70 323 1991M11 17140.00 12.90 -1.00 324 1991M12 17178.70 10.30 -2.60 325 1992M01 17217.70 8.70 -1.60 326 1992M02 17535.30 7.70 -1.00 327 1992M03 17925.40 7.30 -0.40 328 1992M04 18374.70 8.20 0.90 329 1992M05 18645.10 8.90 0.70 330 1992M06 18751.30 8.40 -0.50 331 1992M07 19034.40 10.20 1.80 332 1992M08 19164.10 11.70 1.50 333 1992M09 19025.60 11.50 -0.20 334 1992M10 19090.90 11.70 0.20 335 1992M11 19324.20 12.70 1.00 336 1992M12 19469.00 13.30 0.60 337 1993M01 20912.40 21.50 8.20 338 1993M02 21562.70 23.00 1.50 339 1993M03 22096.70 23.30 0.30 340 1993M04 22609.30 23.00 -0.30 341 1993M05 23106.70 23.90 0.90 342 1993M06 23634.90 26.00 2.10 343 1994M07 23826.70 25.20 -0.80 344 1993M08 23997.20 25.20 0.00 345 1993M09 24141.20 26.90 1.70 346 1993M10 24148.90 26.50 -0.40 347 1993M11 24466.50 26.60 0.10 348 1993M12 24853.20 27.70 1.10 University of Ghana http://ugspace.ug.edu.gh 132 349 1994M01 25682.00 22.80 -4.90 350 1994M02 26298.60 22.00 -0.80 351 1994M03 26855.00 21.50 -0.50 352 1994M04 27368.80 21.10 -0.40 353 1994M05 27958.40 21.00 -0.10 354 1994M06 28572.50 20.90 -0.10 355 1994M07 29145.40 22.30 1.40 356 1994M08 29680.90 23.70 1.40 357 1994M09 30441.70 26.10 2.40 358 1994M10 31258.90 29.40 3.30 359 1994M11 32223.20 31.70 2.30 360 1994M12 33347.70 26.10 -5.60 361 1995M01 34819.50 35.60 9.50 362 1995M02 36394.40 38.40 2.80 363 1995M03 38561.40 43.60 5.20 364 1995M04 41034.40 49.90 6.30 365 1995M05 43647.60 56.10 6.20 366 1995M06 46246.00 61.90 5.80 367 1995M07 48731.20 67.20 5.30 368 1995M08 50438.60 69.90 2.70 369 1995M09 51691.00 69.80 -0.10 370 1995M10 52871.40 69.10 -0.70 371 1995M11 54856.00 70.20 1.10 372 1995M12 56964.20 70.80 0.60 373 1996M01 58914.00 69.20 -1.60 374 1996M02 61154.40 68.00 -1.20 375 1996M03 63543.00 64.80 -3.20 376 1996M04 65763.00 60.30 -4.50 377 1996M05 67323.30 54.20 -6.10 378 1996M06 68639.20 48.40 -5.80 379 1996M07 69511.80 42.60 -5.80 380 1996M08 70218.00 39.20 -3.40 381 1996M09 70564.70 36.50 -2.70 382 1996M10 71001.40 34.30 -2.20 383 1996M11 73050.70 33.20 -1.10 384 1996M12 75569.70 32.70 -0.50 385 1997M01 77477.10 31.50 -1.20 386 1997M02 79841.00 30.60 -0.90 387 1997M03 82108.00 29.20 -1.40 388 1997M04 84894.30 29.10 -0.10 389 1997M05 87232.50 29.60 0.50 390 1997M06 88576.50 29.00 -0.60 391 1997M07 89788.80 29.20 0.20 392 1997M08 90033.10 28.20 -1.00 393 1997M09 90133.40 27.70 -0.50 394 1997M10 90467.00 27.40 -0.30 395 1997M11 90723.50 24.20 -3.20 396 1997M12 91311.80 20.80 -3.40 397 1998M01 103.00 19.80 -1.00 398 1998M02 106.00 19.60 -0.20 399 1998M03 109.60 20.30 0.70 University of Ghana http://ugspace.ug.edu.gh 133 400 1998M04 115.90 23.10 2.80 401 1998M05 119.00 22.90 -0.20 402 1998M06 119.70 21.80 -1.10 403 1998M07 118.20 18.70 -3.10 404 1998M08 118.40 18.60 -0.10 405 1998M09 117.40 17.40 -1.20 406 1998M10 115.80 17.10 -0.30 407 1998M11 115.60 16.20 -0.90 408 1998M12 116.90 15.70 -0.50 409 1999M01 118.70 15.30 -0.40 410 1999M02 121.90 15.00 -0.30 411 1999M03 124.60 13.70 -1.30 412 1999M04 127.80 10.20 -3.50 413 1999M05 130.20 9.40 -0.80 414 1999M06 132.00 10.30 0.90 415 1999M07 133.20 12.70 2.40 416 1999M08 132.60 12.00 -0.70 417 1999M09 131.20 11.80 -0.20 418 1999M10 130.40 12.60 0.80 419 1999M11 130.90 13.20 0.60 420 1999M12 133.00 13.80 0.60 421 2000M01 135.70 14.30 0.50 422 2000M02 140.10 14.90 0.60 423 2000M03 144.00 15.60 0.70 424 2000M04 150.10 17.50 1.90 425 2000M05 154.50 18.70 1.20 426 2000M06 158.20 19.80 1.10 427 2000M07 162.60 22.10 2.30 428 2000M08 167.90 26.60 4.50 429 2000M09 173.60 32.30 5.70 430 2000M10 179.20 37.40 5.10 431 2000M11 182.70 39.50 2.10 432 2000M12 187.00 40.50 1.00 433 2001M01 191.20 40.90 0.40 434 2001M02 196.40 40.10 -0.80 435 2001M03 204.40 41.90 1.80 436 2001M04 209.40 39.50 -2.40 437 2001M05 213.10 37.90 -1.60 438 2001M06 216.50 36.80 -1.10 439 2001M07 219.40 34.90 -1.90 440 2001M08 221.70 32.00 -2.90 441 2001M09 222.70 28.30 -3.70 442 2001M10 225.00 25.60 -2.70 443 2001M11 226.00 23.70 -1.90 444 2001M12 226.80 21.30 -2.40 445 2002M01 229.20 19.90 -1.40 446 2002M02 232.30 18.30 -1.60 447 2002M03 237.10 16.00 -2.30 448 2002M04 240.60 14.90 -1.10 449 2002M05 243.70 14.30 -0.60 450 2002M06 246.10 13.70 -0.60 University of Ghana http://ugspace.ug.edu.gh 134 451 2002M07 249.00 13.50 -0.20 452 2002M08 250.80 13.10 -0.40 453 2002M09 251.40 12.90 -0.20 454 2002M10 254.70 13.20 0.30 455 2002M11 257.60 14.00 0.80 456 2002M12 261.20 15.20 1.20 457 2003M01 109.40 13.50 -1.70 458 2003M02 121.30 25.50 12.00 459 2003M03 126.50 29.80 4.30 460 2003M04 127.10 29.30 -0.50 461 2003M05 130.70 31.60 2.30 462 2003M06 132.60 32.90 1.30 463 2003M07 132.60 33.00 0.10 464 2003M08 134.70 33.60 0.60 465 2003M09 132.20 29.80 -3.40 466 2003M10 135.40 33.20 0.40 467 2003M11 137.00 33.60 -2.30 468 2003M12 138.50 31.30 -2.30 469 2004M01 141.10 29.00 -10.40 470 2004M02 143.90 18.60 -3.00 471 2004M03 146.30 15.60 1.70 472 2004M04 149.00 17.30 0.30 473 2004M05 153.70 17.60 0.40 474 2004M06 156.50 18.00 0.40 475 2004M07 152.50 15.00 -3.00 476 2004M08 158.40 17.50 2.50 477 2004M09 158.20 19.60 2.10 478 2004M10 158.30 16.90 -2.70 479 2004M11 159.60 16.50 -0.40 480 2004M12 161.30 16.40 -0.10 481 2005M01 164.80 16.80 0.40 482 2005M02 168.30 17.00 0.20 483 2005M03 172.30 17.80 0.80 484 2005M04 174.60 17.10 -0.70 485 2005M05 176.00 14.50 -2.60 486 2005M06 178.50 14.00 -0.50 487 2005M07 179.00 17.30 3.30 488 2005M08 179.50 13.30 -4.00 489 2005M09 180.80 14.30 1.00 490 2005M10 181.90 14.90 0.60 491 2005M11 183.00 14.70 -0.20 492 2005M12 183.70 13.90 -0.80 493 2006M01 185.80 12.80 -1.10 494 2006M02 188.90 12.30 -0.50 495 2006M03 191.70 11.30 -1.00 496 2006M04 194.20 11.20 -0.10 497 2006M05 196.70 11.70 0.50 498 2006M06 198.80 11.40 -0.30 499 2006M07 202.10 12.90 1.50 500 2006M08 202.00 12.60 -0.30 501 2006M09 201.90 11.70 -0.90 University of Ghana http://ugspace.ug.edu.gh 135 502 2006M10 201.70 10.90 -0.80 503 2006M11 202.60 10.70 -0.20 504 2006M12 203.80 10.90 0.20 505 2007M01 206.10 10.90 0.00 506 2007M02 208.60 10.40 -0.50 507 2007M03 211.30 10.20 -0.20 508 2007M04 214.50 10.50 0.30 509 2007M05 218.40 11.00 0.50 510 2007M06 220.00 10.70 -0.30 511 2007M07 222.60 10.10 -0.60 512 2007M08 223.00 10.40 0.30 513 2007M09 222.50 10.20 -0.20 514 2007M10 222.10 10.10 -0.10 515 2007M11 225.70 11.40 1.30 516 2007M12 229.80 12.70 1.30 517 2008M01 232.50 12.80 0.10 518 2008M02 236.20 13.20 0.40 519 2008M03 240.40 13.80 0.60 520 2008M04 247.40 15.30 1.50 521 2008M05 255.30 16.90 1.60 522 2008M06 260.50 18.40 1.50 523 2008M07 263.40 18.30 -0.10 524 2008M08 263.40 18.10 -0.20 525 2008M09 262.30 17.90 -0.20 526 2008M10 260.60 17.30 -0.60 527 2008M11 265.10 17.40 0.10 528 2008M12 271.50 18.10 0.70 529 2009M01 278.60 19.90 1.80 530 2009M02 284.20 20.30 0.40 531 2009M03 289.80 20.50 0.20 532 2009M04 298.20 20.60 0.10 533 2009M05 306.50 20.10 -0.50 534 2009M06 314.60 20.70 0.60 535 2009M07 317.30 20.50 -0.20 536 2009M08 315.10 19.60 -0.90 537 2009M09 310.50 18.40 -1.20 538 2009M10 307.60 18.00 -0.40 539 2009M11 309.90 16.90 -1.10 540 2009M12 314.80 16.00 -0.90 541 2010M01 319.80 14.80 -1.20 542 2010M02 324.70 14.20 -0.60 543 2010M03 328.40 13.30 -0.90 544 2010M04 333.00 11.70 -1.60 545 2010M05 339.20 10.70 -1.00 546 2010M06 344.50 9.50 -1.20 547 2010M07 347.30 9.50 0.00 548 2010M08 344.90 9.40 -0.10 549 2010M09 339.70 9.40 0.00 550 2010M10 336.40 9.40 0.00 551 2010M11 338.00 9.10 -0.30 552 2010M12 341.80 8.60 -0.50 University of Ghana http://ugspace.ug.edu.gh 136 553 2011M01 348.90 9.10 0.50 554 2011M02 354.40 9.20 0.10 555 2011M03 358.34 9.13 -0.70 556 2011M04 363.02 9.02 -0.11 557 2011M05 369.41 8.90 -0.12 558 2011M06 374.13 8.59 -0.31 559 2011M07 376.50 8.39 -0.20 560 2011M08 373.88 8.41 0.02 561 2011M09 368.18 8.40 -0.01 562 2011M10 365.22 8.56 0.16 563 2011M11 366.90 8.55 -0.01 564 2011M12 371.16 8.58 0.03 565 2012M01 379.30 8.70 0.12 566 2012M02 385.00 8.60 -0.10 567 2012M03 389.80 8.80 0.20 568 2012M04 396.10 9.10 0.30 569 2012M05 403.90 9.30 0.20 570 2012M06 409.50 9.40 0.10 571 2012M07 412.40 9.50 0.10 572 2012M08 409.20 9.50 0.00 573 2012M09 402.90 9.40 -0.10 574 2012M10 399.00 9.20 -0.20 575 2012M11 401.10 9.30 -0.10 576 2012M12 404.00 8.80 -0.50 University of Ghana http://ugspace.ug.edu.gh 137 APPENDIX B Figure 1B: Histogram of Monthly Rates of Inflation in Ghana from January 1965 to December 2012 1501209060300-30 180 160 140 120 100 80 60 40 20 0 INFLATION F r e q u e n c y Mean 29.82 StDev 30.73 N 576 Histogram (with Normal Curve) of INFLATION Figure 2B: Seasonal Component Analysis of Monthly Rates of Inflation in Ghana from January 1965 to December 2012 5754603452301151 150 100 50 0 Index 5754603452301151 150 100 50 0 Index 5754603452301151 6 4 2 0 Index 5754603452301151 160 80 0 Index Component Analysis for INFLATION Multiplicative Model Original Data Seasonally Adjusted Data Detrended Data Seas. Adj. and Detr. Data University of Ghana http://ugspace.ug.edu.gh 138 Figure 3B: Seasonal Component Analysis of Monthly Rates of Inflation in Ghana from January 1965 to December 2012 121110987654321 1.02 1.00 0.98 121110987654321 7.5 5.0 2.5 0.0 121110987654321 6 4 2 0 121110987654321 160 80 0 Seasonal Analysis for INFLATION Multiplicative Model Seasonal Indices Percent Variation by Season Detrended Data by Season Residuals by Season Figure 4B: Residual Plots for the first difference of Monthly Rates of Inflation in Ghana from January 1965 to December 2012 40200-20-40 99.99 99 90 50 10 1 0.01 Residual P e r c e n t 0.20.10.0-0.1-0.2 40 20 0 -20 -40 Fitted Value R e s i d u a l 403020100-10-20-30 300 200 100 0 Residual F r e q u e n c y 550500450400350300250200150100501 40 20 0 -20 -40 Observation Order R e s i d u a l Normal Probability Plot Versus Fits Histogram Versus Order Residual Plots for 1ST DIFF University of Ghana http://ugspace.ug.edu.gh 139 Figure 5B: Trend Analysis for the first difference of Monthly Rates of Inflation in Ghana from January 1965 to December 2012 Year Month 205320452037202920212013 FebFebFebFebFebFeb 40 30 20 10 0 -10 -20 -30 -40 1 S T D I F F MAPE 109.956 MAD 3.316 MSD 42.629 Accuracy Measures Actual Fits Variable Trend Analysis Plot for 1ST DIFF Linear Trend Model Yt = 0.189 - 0.000682*t Figure 6B: Histogram of the first difference of the monthly rates of inflation series. 0 40 80 120 160 200 240 -25.0 -12.5 0.0 12.5 25.0 37.5 Series: Residuals Sample 2 576 Observations 575 Mean 2.31e-16 Median -0.092174 Maximum 39.30783 Minimum -32.99217 Std. Dev. 6.535746 Skewness 0.154286 Kurtosis 13.84925 Jarque-Bera 2822.325 Probability 0.000000 University of Ghana http://ugspace.ug.edu.gh 140 Table 1B: Test for Heteroscedasticity (ARCH Effects) in Monthly rates of Inflation series in its level form Lag Q-Statistic P-value 1 551.52 0.0000 2 1053.90 0.0000 3 1494.20 0.0000 4 1877.00 0.0000 5 2175.40 0.0000 6 2425.30 0.0000 7 2623.60 0.0000 8 2776.40 0.0000 9 2890.70 0.0000 10 2973.20 0.0000 11 3031.60 0.0000 12 3073.10 0.0000 13 3106.00 0.0000 14 3134.80 0.0000 15 3162.40 0.0000 16 3191.90 0.0000 17 3225.00 0.0000 18 3263.20 0.0000 19 3306.60 0.0000 20 3356.10 0.0000 21 3410.90 0.0000 22 3471.20 0.0000 23 3537.00 0.0000 24 3608.50 0.0000 25 3686.20 0.0000 26 3769.70 0.0000 27 3857.50 0.0000 University of Ghana http://ugspace.ug.edu.gh 141 28 3946.80 0.0000 29 4035.10 0.0000 30 4120.20 0.0000 31 4200.80 0.0000 32 4275.20 0.0000 33 4342.80 0.0000 34 4403.20 0.0000 35 4455.30 0.0000 36 4499.30 0.0000 University of Ghana http://ugspace.ug.edu.gh 142 APPENDIX C Table 1C: Model Output for ARCH (1) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:18 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 31 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 Coefficient Std. Error z-Statistic Prob. C -0.498281 0.073911 -6.741625 0.0000 AR(1) 0.844158 0.032263 26.16456 0.0000 AR(2) -0.026945 0.017772 -1.516176 0.1295 MA(1) -0.704173 0.024205 -29.09191 0.0000 Variance Equation C 2.847795 0.192694 14.77883 0.0000 RESID(-1)^2 2.373269 0.144997 16.36766 0.0000 R-squared 0.140449 Mean dependent var -0.017683 Adjusted R-squared 0.132705 S.D. dependent var 6.613337 S.E. of regression 6.158914 Akaike info criterion 5.364787 Sum squared resid 21052.38 Schwarz criterion 5.411094 Log likelihood -1498.823 F-statistic 18.13714 Durbin-Watson stat 1.370483 Prob(F-statistic) 0.000000 Inverted AR Roots .81 .03 Inverted MA Roots .70 University of Ghana http://ugspace.ug.edu.gh 143 Table 2C: Model Output for ARCH (2) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:30 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 22 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*RESID(-2)^2 Coefficient Std. Error z-Statistic Prob. C -0.154409 0.100296 -1.539531 0.1237 AR(1) 0.038025 0.080133 0.474520 0.6351 AR(2) 0.247293 0.038061 6.497345 0.0000 MA(1) 0.236626 0.091648 2.581898 0.0098 Variance Equation C 2.581578 0.169496 15.23090 0.0000 RESID(-1)^2 1.071298 0.093814 11.41942 0.0000 RESID(-2)^2 0.598491 0.061488 9.733461 0.0000 R-squared 0.219029 Mean dependent var -0.017683 Adjusted R-squared 0.210571 S.D. dependent var 6.613337 S.E. of regression 5.875939 Akaike info criterion 5.305401 Sum squared resid 19127.77 Schwarz criterion 5.359426 Log likelihood -1481.165 F-statistic 25.89555 Durbin-Watson stat 1.741881 Prob(F-statistic) 0.000000 Inverted AR Roots .52 -.48 Inverted MA Roots -.24 University of Ghana http://ugspace.ug.edu.gh 144 Table 3C: Model Output for ARCH (3) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:30 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 103 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*RESID(-2)^2 + C(8)*RESID( -3)^2 Coefficient Std. Error z-Statistic Prob. C -0.212855 0.113236 -1.879746 0.0601 AR(1) 0.892542 0.093720 9.523500 0.0000 AR(2) -0.110620 0.045480 -2.432313 0.0150 MA(1) -0.627033 0.080582 -7.781275 0.0000 Variance Equation C 2.533706 0.169868 14.91570 0.0000 RESID(-1)^2 1.111763 0.104060 10.68385 0.0000 RESID(-2)^2 0.644038 0.101071 6.372114 0.0000 RESID(-3)^2 -0.001810 0.015143 -0.119515 0.9049 R-squared 0.202374 Mean dependent var -0.017683 Adjusted R-squared 0.192278 S.D. dependent var 6.613337 S.E. of regression 5.943630 Akaike info criterion 5.305566 Sum squared resid 19535.68 Schwarz criterion 5.367309 Log likelihood -1480.211 F-statistic 20.04393 Durbin-Watson stat 1.651470 Prob(F-statistic) 0.000000 Inverted AR Roots .74 .15 Inverted MA Roots .63 University of Ghana http://ugspace.ug.edu.gh 145 Table 4C: Model Output for GARCH (1, 1) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:35 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 112 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*GARCH(-1) Coefficient Std. Error z-Statistic Prob. C -0.194598 0.202762 -0.959734 0.3372 AR(1) 0.442329 0.234397 1.887088 0.0591 AR(2) 0.177790 0.121285 1.465887 0.1427 MA(1) -0.054806 0.232681 -0.235542 0.8138 Variance Equation C 0.176700 0.031041 5.692498 0.0000 RESID(-1)^2 0.349895 0.026864 13.02445 0.0000 GARCH(-1) 0.736577 0.011384 64.70119 0.0000 R-squared 0.236821 Mean dependent var -0.017683 Adjusted R-squared 0.228555 S.D. dependent var 6.613337 S.E. of regression 5.808621 Akaike info criterion 5.168316 Sum squared resid 18692.01 Schwarz criterion 5.222341 Log likelihood -1442.713 F-statistic 28.65179 Durbin-Watson stat 1.955985 Prob(F-statistic) 0.000000 Inverted AR Roots .70 -.25 Inverted MA Roots .05 University of Ghana http://ugspace.ug.edu.gh 146 Table 5C: Model Output for GARCH (1, 2) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:37 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 172 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*GARCH(-1) + C(8) *GARCH(-2) Coefficient Std. Error z-Statistic Prob. C -0.183748 0.182732 -1.005559 0.3146 AR(1) 0.412817 0.274547 1.503629 0.1327 AR(2) 0.181130 0.133279 1.359031 0.1741 MA(1) -0.031922 0.277152 -0.115179 0.9083 Variance Equation C 0.166547 0.037240 4.472260 0.0000 RESID(-1)^2 0.430331 0.037721 11.40839 0.0000 GARCH(-1) 0.420231 0.110561 3.800907 0.0001 GARCH(-2) 0.261598 0.089936 2.908721 0.0036 R-squared 0.237631 Mean dependent var -0.017683 Adjusted R-squared 0.227980 S.D. dependent var 6.613337 S.E. of regression 5.810785 Akaike info criterion 5.160210 Sum squared resid 18672.17 Schwarz criterion 5.221953 Log likelihood -1439.439 F-statistic 24.62432 Durbin-Watson stat 1.945846 Prob(F-statistic) 0.000000 Inverted AR Roots .68 -.27 Inverted MA Roots .03 University of Ghana http://ugspace.ug.edu.gh 147 Table 6C: Model Output for GARCH (2, 1) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:38 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 93 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*RESID(-2)^2 + C(8) *GARCH(-1) Coefficient Std. Error z-Statistic Prob. C -0.050480 0.061193 -0.824919 0.4094 AR(1) 0.354733 0.206899 1.714528 0.0864 AR(2) 0.168479 0.084851 1.985587 0.0471 MA(1) -0.053134 0.218579 -0.243089 0.8079 Variance Equation C -0.009901 0.004244 -2.332809 0.0197 RESID(-1)^2 0.711921 0.048894 14.56061 0.0000 RESID(-2)^2 -0.503197 0.046471 -10.82820 0.0000 GARCH(-1) 0.864969 0.008080 107.0518 0.0000 R-squared 0.231523 Mean dependent var -0.017683 Adjusted R-squared 0.221795 S.D. dependent var 6.613337 S.E. of regression 5.834015 Akaike info criterion 5.121990 Sum squared resid 18821.76 Schwarz criterion 5.183732 Log likelihood -1428.718 F-statistic 23.80073 Durbin-Watson stat 1.790040 Prob(F-statistic) 0.000000 Inverted AR Roots .62 -.27 Inverted MA Roots .05 University of Ghana http://ugspace.ug.edu.gh 148 Table 7C: Model Output for GARCH (2, 2) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:39 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 71 iterations MA backcast: 3, Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*RESID(-2)^2 + C(8) *GARCH(-1) + C(9)*GARCH(-2) Coefficient Std. Error z-Statistic Prob. C -0.240779 0.197155 -1.221265 0.2220 AR(1) 0.400804 0.225635 1.776334 0.0757 AR(2) 0.193982 0.113446 1.709898 0.0873 MA(1) -0.002159 0.235977 -0.009148 0.9927 Variance Equation C 0.734895 0.098358 7.471611 0.0000 RESID(-1)^2 0.579129 0.048323 11.98445 0.0000 RESID(-2)^2 0.543006 0.046946 11.56649 0.0000 GARCH(-1) -0.349990 0.028228 -12.39858 0.0000 GARCH(-2) 0.564453 0.018221 30.97825 0.0000 R-squared 0.237200 Mean dependent var -0.017683 Adjusted R-squared 0.226145 S.D. dependent var 6.613337 S.E. of regression 5.817689 Akaike info criterion 5.169987 Sum squared resid 18682.72 Schwarz criterion 5.239447 Log likelihood -1441.181 F-statistic 21.45621 Durbin-Watson stat 1.980253 Prob(F-statistic) 0.000000 Inverted AR Roots .68 -.28 Inverted MA Roots .00 University of Ghana http://ugspace.ug.edu.gh 149 Table 8C: Model Output for EGARCH (1, 1) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:41 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 70 iterations MA backcast: 3, Variance backcast: ON LOG(GARCH) = C(5) + C(6)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(7)*RESID(-1)/@SQRT(GARCH(-1)) + C(8)*LOG(GARCH(-1)) Coefficient Std. Error z-Statistic Prob. C -0.348038 0.252564 -1.378021 0.1682 AR(1) 1.001650 0.108285 9.250135 0.0000 AR(2) -0.116967 0.077527 -1.508734 0.1314 MA(1) -0.683044 0.084169 -8.115105 0.0000 Variance Equation C(5) -0.220183 0.021570 -10.20764 0.0000 C(6) 0.388365 0.030889 12.57297 0.0000 C(7) 0.060083 0.026349 2.280243 0.0226 C(8) 0.979004 0.005184 188.8448 0.0000 University of Ghana http://ugspace.ug.edu.gh 150 R-squared 0.205014 Mean dependent var -0.017683 Adjusted R-squared 0.194950 S.D. dependent var 6.613337 S.E. of regression 5.933788 Akaike info criterion 5.164555 Sum squared resid 19471.04 Schwarz criterion 5.226298 Log likelihood -1440.658 F-statistic 20.37276 Durbin-Watson stat 1.745592 Prob(F-statistic) 0.000000 Inverted AR Roots .87 .13 Inverted MA Roots .68 University of Ghana http://ugspace.ug.edu.gh 151 Table 9C: Model Output for EGARCH (1, 2) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:42 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 57 iterations MA backcast: 3, Variance backcast: ON LOG(GARCH) = C(5) + C(6)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(7)*RESID(-1)/@SQRT(GARCH(-1)) + C(8)*LOG(GARCH(-1)) + C(9)*LOG(GARCH(-2)) Coefficient Std. Error z-Statistic Prob. C -0.185278 0.171929 -1.077641 0.2812 AR(1) 0.304909 0.283926 1.073905 0.2829 AR(2) 0.195520 0.135199 1.446161 0.1481 MA(1) 0.094811 0.291694 0.325035 0.7452 Variance Equation C(5) -0.369560 0.034676 -10.65742 0.0000 C(6) 0.639181 0.048335 13.22406 0.0000 C(7) -0.011265 0.033812 -0.333174 0.7390 C(8) 0.588073 0.087998 6.682786 0.0000 C(9) 0.384565 0.087501 4.394973 0.0000 R-squared 0.239104 Mean dependent var -0.017683 Adjusted R-squared 0.228077 S.D. dependent var 6.613337 S.E. of regression 5.810422 Akaike info criterion 5.152872 Sum squared resid 18636.07 Schwarz criterion 5.222333 Log likelihood -1436.381 F-statistic 21.68262 Durbin-Watson stat 1.985774 Prob(F-statistic) 0.000000 Inverted AR Roots .62 -.32 Inverted MA Roots -.09 University of Ghana http://ugspace.ug.edu.gh 152 Table 10C: Model Output for EGARCH (2, 1) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:43 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 36 iterations MA backcast: 3, Variance backcast: ON LOG(GARCH) = C(5) + C(6)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(7)*ABS(RESID(-2)/@SQRT(GARCH(-2))) + C(8)*RESID(-1) /@SQRT(GARCH(-1)) + C(9)*LOG(GARCH(-1)) Coefficient Std. Error z-Statistic Prob. C 0.106006 0.126240 0.839721 0.4011 AR(1) 0.697105 0.203489 3.425760 0.0006 AR(2) 0.007368 0.085391 0.086289 0.9312 MA(1) -0.458052 0.195324 -2.345089 0.0190 Variance Equation C(5) -0.168646 0.017756 -9.497944 0.0000 C(6) 0.866121 0.069383 12.48327 0.0000 C(7) -0.597416 0.069788 -8.560454 0.0000 C(8) 0.124346 0.024108 5.157891 0.0000 C(9) 0.991669 0.002699 367.4403 0.0000 R-squared 0.208413 Mean dependent var -0.017683 Adjusted R-squared 0.196941 S.D. dependent var 6.613337 S.E. of regression 5.926447 Akaike info criterion 5.089933 Sum squared resid 19387.77 Schwarz criterion 5.159394 Log likelihood -1418.726 F-statistic 18.16669 Durbin-Watson stat 1.624597 Prob(F-statistic) 0.000000 Inverted AR Roots .71 -.01 Inverted MA Roots .46 University of Ghana http://ugspace.ug.edu.gh 153 Table 11C: Model Output for EGARCH (2, 2) Dependent Variable: D(INFLATIO) Method: ML - ARCH (Marquardt) - Normal distribution Date: 07/26/13 Time: 17:44 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 75 iterations MA backcast: 3, Variance backcast: ON LOG(GARCH) = C(5) + C(6)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(7)*ABS(RESID(-2)/@SQRT(GARCH(-2))) + C(8)*RESID(-1) /@SQRT(GARCH(-1)) + C(9)*LOG(GARCH(-1)) + C(10) *LOG(GARCH(-2)) Coefficient Std. Error z-Statistic Prob. C -0.270043 0.156617 -1.724228 0.0847 AR(1) 0.237689 0.219705 1.081852 0.2793 AR(2) 0.269255 0.111432 2.416312 0.0157 MA(1) 0.170693 0.226417 0.753889 0.4509 Variance Equation C(5) -0.616292 0.045604 -13.51399 0.0000 C(6) 0.517074 0.037174 13.90945 0.0000 C(7) 0.552703 0.042327 13.05794 0.0000 C(8) -0.137644 0.032864 -4.188309 0.0000 C(9) 0.026609 0.025045 1.062449 0.2880 C(10) 0.929400 0.025143 36.96508 0.0000 R-squared 0.234738 Mean dependent var -0.017683 Adjusted R-squared 0.222239 S.D. dependent var 6.613337 S.E. of regression 5.832353 Akaike info criterion 5.159967 Sum squared resid 18743.01 Schwarz criterion 5.237145 Log likelihood -1437.371 F-statistic 18.77947 Durbin-Watson stat 2.009781 Prob(F-statistic) 0.000000 Inverted AR Roots .65 -.41 Inverted MA Roots -.17 University of Ghana http://ugspace.ug.edu.gh 154 Table 12C ARCH LM Test for ARCH (2) ARCH Test: F-statistic 2.554944 Probability 0.000004 Obs*R-squared 83.25909 Probability 0.000013 Test Equation: Dependent Variable: STD_RESID^2 Method: Least Squares Date: 07/27/13 Time: 03:42 Sample (adjusted): 40 564 Included observations: 525 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.381014 0.237746 1.602610 0.1097 STD_RESID^2(-1) -0.031904 0.045266 -0.704799 0.4813 STD_RESID^2(-2) -0.050002 0.045278 -1.104330 0.2700 STD_RESID^2(-3) -0.049072 0.045106 -1.087943 0.2772 STD_RESID^2(-4) -0.048959 0.045156 -1.084230 0.2788 STD_RESID^2(-5) -0.061629 0.045204 -1.363355 0.1734 STD_RESID^2(-6) -0.012361 0.045249 -0.273180 0.7848 STD_RESID^2(-7) 0.062110 0.045179 1.374755 0.1698 STD_RESID^2(-8) 0.004966 0.045186 0.109892 0.9125 STD_RESID^2(-9) -0.076530 0.045153 -1.694889 0.0907 STD_RESID^2(-10) 0.017390 0.045287 0.383998 0.7011 STD_RESID^2(-11) 0.000463 0.044818 0.010342 0.9918 STD_RESID^2(-12) 0.118461 0.044805 2.643923 0.0085 STD_RESID^2(-13) 0.102222 0.045118 2.265655 0.0239 STD_RESID^2(-14) 0.024005 0.045340 0.529442 0.5967 STD_RESID^2(-15) 0.031070 0.045339 0.685292 0.4935 STD_RESID^2(-16) 0.041521 0.045337 0.915820 0.3602 STD_RESID^2(-17) 0.324544 0.045368 7.153521 0.0000 STD_RESID^2(-18) 0.016219 0.047687 0.340115 0.7339 STD_RESID^2(-19) 0.007628 0.047674 0.160000 0.8729 STD_RESID^2(-20) 0.021533 0.045397 0.474330 0.6355 STD_RESID^2(-21) 0.031233 0.045372 0.688370 0.4915 STD_RESID^2(-22) 0.024573 0.045351 0.541844 0.5882 STD_RESID^2(-23) 0.024267 0.045356 0.535022 0.5929 STD_RESID^2(-24) -0.017210 0.045151 -0.381174 0.7032 STD_RESID^2(-25) -0.014523 0.044858 -0.323764 0.7463 STD_RESID^2(-26) 0.144309 0.044845 3.217915 0.0014 STD_RESID^2(-27) -0.004719 0.045316 -0.104139 0.9171 STD_RESID^2(-28) 0.027821 0.045171 0.615908 0.5382 STD_RESID^2(-29) -0.059968 0.045182 -1.327248 0.1850 STD_RESID^2(-30) 0.056355 0.045184 1.247227 0.2129 STD_RESID^2(-31) 0.041584 0.045250 0.918990 0.3586 STD_RESID^2(-32) 0.008435 0.045190 0.186655 0.8520 STD_RESID^2(-33) -0.015535 0.045130 -0.344234 0.7308 STD_RESID^2(-34) -0.100775 0.045077 -2.235594 0.0258 University of Ghana http://ugspace.ug.edu.gh 155 STD_RESID^2(-35) 0.015850 0.045240 0.350363 0.7262 STD_RESID^2(-36) 0.007196 0.045213 0.159162 0.8736 R-squared 0.158589 Mean dependent var 1.017323 Adjusted R-squared 0.096517 S.D. dependent var 3.184223 S.E. of regression 3.026658 Akaike info criterion 5.120665 Sum squared resid 4470.402 Schwarz criterion 5.421133 Log likelihood -1307.174 F-statistic 2.554944 Durbin-Watson stat 1.999704 Prob(F-statistic) 0.000004 Table 13C: Ljung Box Q-Statistics for ARCH (2) Date: 07/27/13 Time: 03:37 Sample: 4 564 Included observations: 561 Q-statistic probabilities adjusted for 3 ARMA term(s) Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 -0.002 -0.002 0.0031 .|. | .|. | 2 -0.031 -0.031 0.5622 .|. | .|. | 3 -0.029 -0.029 1.0382 .|. | .|. | 4 -0.003 -0.004 1.0420 0.307 .|. | .|. | 5 -0.006 -0.008 1.0624 0.588 .|. | .|. | 6 0.015 0.014 1.1954 0.754 .|* | .|* | 7 0.081 0.081 4.9535 0.292 .|. | .|. | 8 -0.018 -0.017 5.1300 0.400 .|. | .|. | 9 -0.043 -0.038 6.1880 0.402 .|. | .|. | 10 0.035 0.039 6.8871 0.441 .|. | .|. | 11 0.006 0.003 6.9076 0.547 .|* | .|* | 12 0.098 0.100 12.482 0.187 .|* | .|* | 13 0.128 0.131 21.931 0.015 .|. | .|. | 14 0.030 0.034 22.458 0.021 .|. | .|. | 15 0.003 0.023 22.463 0.033 .|. | .|. | 16 0.012 0.029 22.547 0.047 .|** | .|** | 17 0.276 0.285 66.648 0.000 .|. | .|. | 18 0.006 0.024 66.666 0.000 .|. | .|. | 19 0.004 0.017 66.673 0.000 .|. | .|. | 20 0.009 0.018 66.718 0.000 .|. | .|. | 21 -0.011 0.009 66.786 0.000 .|. | .|. | 22 -0.017 0.000 66.959 0.000 .|. | .|. | 23 0.028 0.020 67.428 0.000 .|. | .|. | 24 0.030 -0.025 67.943 0.000 .|. | .|. | 25 0.006 -0.016 67.963 0.000 .|* | .|* | 26 0.127 0.138 77.443 0.000 .|. | .|. | 27 0.004 -0.020 77.453 0.000 .|. | .|. | 28 0.021 0.022 77.712 0.000 .|. | *|. | 29 -0.006 -0.063 77.732 0.000 University of Ghana http://ugspace.ug.edu.gh 156 .|* | .|. | 30 0.121 0.055 86.399 0.000 .|. | .|. | 31 0.049 0.041 87.858 0.000 .|. | .|. | 32 0.011 0.008 87.924 0.000 .|. | .|. | 33 0.008 -0.019 87.959 0.000 .|. | *|. | 34 -0.023 -0.106 88.280 0.000 .|. | .|. | 35 0.002 0.010 88.283 0.000 .|. | .|. | 36 0.015 0.002 88.422 0.000 Table 14C: ARCH –LM Test for the GARCH (2,1) Model ARCH Test: F-statistic 0.649902 Probability 0.943659 Obs*R-squared 24.01883 Probability 0.936672 Test Equation: Dependent Variable: STD_RESID^2 Method: Least Squares Date: 07/27/13 Time: 06:50 Sample (adjusted): 40 564 Included observations: 525 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 1.132305 0.337992 3.350098 0.0009 STD_RESID^2(-1) -0.018270 0.045262 -0.403648 0.6866 STD_RESID^2(-2) 0.007529 0.045242 0.166416 0.8679 STD_RESID^2(-3) -0.007648 0.045243 -0.169035 0.8658 STD_RESID^2(-4) -0.028048 0.045245 -0.619908 0.5356 STD_RESID^2(-5) -0.017295 0.045252 -0.382183 0.7025 STD_RESID^2(-6) -0.006831 0.045259 -0.150941 0.8801 STD_RESID^2(-7) -0.027006 0.045233 -0.597043 0.5508 STD_RESID^2(-8) -0.004652 0.045251 -0.102805 0.9182 STD_RESID^2(-9) -0.030257 0.045245 -0.668743 0.5040 STD_RESID^2(-10) -0.032541 0.045242 -0.719256 0.4723 STD_RESID^2(-11) -0.009626 0.045231 -0.212808 0.8316 STD_RESID^2(-12) 0.098382 0.045223 2.175472 0.0301 STD_RESID^2(-13) -0.001776 0.045285 -0.039213 0.9687 STD_RESID^2(-14) -0.001904 0.044967 -0.042344 0.9662 STD_RESID^2(-15) -0.025881 0.044957 -0.575687 0.5651 STD_RESID^2(-16) -0.024633 0.044971 -0.547753 0.5841 STD_RESID^2(-17) -0.009271 0.044982 -0.206103 0.8368 STD_RESID^2(-18) -0.001222 0.044973 -0.027176 0.9783 STD_RESID^2(-19) -0.020014 0.044921 -0.445547 0.6561 STD_RESID^2(-20) -0.016203 0.044927 -0.360656 0.7185 STD_RESID^2(-21) -0.014759 0.044941 -0.328418 0.7427 STD_RESID^2(-22) -0.023622 0.044923 -0.525825 0.5992 STD_RESID^2(-23) 0.117809 0.044938 2.621620 0.0090 University of Ghana http://ugspace.ug.edu.gh 157 STD_RESID^2(-24) 0.084390 0.045252 1.864890 0.0628 STD_RESID^2(-25) -0.022234 0.045195 -0.491949 0.6230 STD_RESID^2(-26) -0.038882 0.045199 -0.860234 0.3901 STD_RESID^2(-27) 0.029993 0.045208 0.663454 0.5074 STD_RESID^2(-28) -0.018394 0.045203 -0.406921 0.6842 STD_RESID^2(-29) 0.002990 0.045216 0.066131 0.9473 STD_RESID^2(-30) 0.034765 0.045199 0.769162 0.4422 STD_RESID^2(-31) -0.010246 0.045218 -0.226590 0.8208 STD_RESID^2(-32) -0.022175 0.045212 -0.490465 0.6240 STD_RESID^2(-33) -0.006922 0.045203 -0.153130 0.8784 STD_RESID^2(-34) -0.010711 0.045206 -0.236939 0.8128 STD_RESID^2(-35) -0.034403 0.045211 -0.760955 0.4471 STD_RESID^2(-36) -0.013422 0.045246 -0.296640 0.7669 R-squared 0.045750 Mean dependent var 1.008184 Adjusted R-squared -0.024645 S.D. dependent var 3.558203 S.E. of regression 3.601783 Akaike info criterion 5.468605 Sum squared resid 6330.747 Schwarz criterion 5.769073 Log likelihood -1398.509 F-statistic 0.649902 Durbin-Watson stat 2.000645 Prob(F-statistic) 0.943659 Table 15C:Ljung Q- Statistics for the GARCH (2,1) Date: 07/27/13 Time: 03:52 Sample: 4 564 Included observations: 561 Q-statistic probabilities adjusted for 3 ARMA term(s) Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|. | .|. | 1 0.047 0.047 1.2543 .|. | .|. | 2 0.002 -0.000 1.2570 .|. | .|. | 3 0.058 0.058 3.1427 .|. | .|. | 4 -0.047 -0.053 4.3964 0.036 .|. | .|. | 5 0.057 0.063 6.2615 0.044 .|. | .|. | 6 0.007 -0.003 6.2873 0.098 .|. | .|. | 7 -0.009 -0.003 6.3350 0.175 .|* | .|. | 8 0.069 0.061 9.0885 0.106 .|. | .|. | 9 0.060 0.060 11.152 0.084 .|. | .|. | 10 0.035 0.028 11.860 0.105 .|. | .|. | 11 -0.007 -0.019 11.890 0.156 ***|. | ***|. | 12 -0.322 -0.327 71.565 0.000 .|. | .|. | 13 -0.036 -0.013 72.292 0.000 .|. | .|. | 14 -0.042 -0.050 73.331 0.000 *|. | .|. | 15 -0.080 -0.043 77.072 0.000 .|. | .|. | 16 0.000 -0.025 77.073 0.000 University of Ghana http://ugspace.ug.edu.gh 158 .|. | .|. | 17 -0.053 -0.022 78.700 0.000 .|. | .|. | 18 -0.033 -0.034 79.352 0.000 .|. | .|. | 19 -0.022 -0.024 79.626 0.000 .|. | .|. | 20 0.006 0.062 79.647 0.000 *|. | .|. | 21 -0.060 -0.014 81.755 0.000 .|. | .|. | 22 -0.017 0.024 81.918 0.000 *|. | *|. | 23 -0.101 -0.106 87.891 0.000 *|. | **|. | 24 -0.099 -0.201 93.628 0.000 .|. | .|. | 25 0.039 0.038 94.547 0.000 .|. | .|. | 26 0.032 0.029 95.151 0.000 .|. | .|. | 27 0.000 -0.028 95.151 0.000 .|. | .|. | 28 0.006 -0.009 95.173 0.000 .|. | .|. | 29 0.018 -0.001 95.359 0.000 .|* | .|. | 30 0.069 0.053 98.207 0.000 .|. | .|. | 31 0.055 0.053 99.977 0.000 .|. | .|. | 32 -0.027 0.021 100.41 0.000 .|. | .|. | 33 -0.005 -0.022 100.42 0.000 .|. | .|. | 34 0.012 0.007 100.51 0.000 .|* | .|. | 35 0.068 -0.015 103.26 0.000 .|* | .|. | 36 0.114 0.002 111.03 0.000 Table 16C: ARCH – LM Test for the EGARCH (2,1) Model ARCH Test: F-statistic 0.614495 Probability 0.963137 Obs*R-squared 22.74981 Probability 0.957970 Test Equation: Dependent Variable: STD_RESID^2 Method: Least Squares Date: 02/06/13 Time: 00:13 Sample (adjusted): 38 576 Included observations: 539 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 1.135220 0.329098 3.449491 0.0006 STD_RESID^2(-1) -0.023639 0.044676 -0.529127 0.5970 STD_RESID^2(-2) -0.000374 0.044678 -0.008375 0.9933 STD_RESID^2(-3) -0.011996 0.044674 -0.268513 0.7884 STD_RESID^2(-4) -0.028650 0.044668 -0.641402 0.5216 STD_RESID^2(-5) -0.034691 0.044676 -0.776508 0.4378 STD_RESID^2(-6) -0.005466 0.044700 -0.122271 0.9027 STD_RESID^2(-7) -0.028152 0.044699 -0.629805 0.5291 STD_RESID^2(-8) -0.019171 0.044703 -0.428858 0.6682 STD_RESID^2(-9) -0.042396 0.044709 -0.948269 0.3434 STD_RESID^2(-10) -0.032947 0.044747 -0.736280 0.4619 University of Ghana http://ugspace.ug.edu.gh 159 STD_RESID^2(-11) -0.006376 0.044768 -0.142433 0.8868 STD_RESID^2(-12) 0.162948 0.044763 3.640253 0.0003 STD_RESID^2(-13) 0.012889 0.045351 0.284199 0.7764 STD_RESID^2(-14) -0.002968 0.045345 -0.065443 0.9478 STD_RESID^2(-15) -0.012159 0.045343 -0.268152 0.7887 STD_RESID^2(-16) -0.012457 0.045348 -0.274689 0.7837 STD_RESID^2(-17) 0.022313 0.044983 0.496032 0.6201 STD_RESID^2(-18) 0.007660 0.044990 0.170254 0.8649 STD_RESID^2(-19) 0.016136 0.044990 0.358644 0.7200 STD_RESID^2(-20) -0.012533 0.044988 -0.278592 0.7807 STD_RESID^2(-21) -0.006908 0.044996 -0.153528 0.8780 STD_RESID^2(-22) -0.011661 0.045000 -0.259121 0.7956 STD_RESID^2(-23) 0.016000 0.045008 0.355500 0.7224 STD_RESID^2(-24) 0.024597 0.045013 0.546446 0.5850 STD_RESID^2(-25) -0.023031 0.044472 -0.517868 0.6048 STD_RESID^2(-26) -0.013946 0.044479 -0.313554 0.7540 STD_RESID^2(-27) -0.000198 0.044456 -0.004456 0.9964 STD_RESID^2(-28) 0.005024 0.044408 0.113133 0.9100 STD_RESID^2(-29) 0.025699 0.044368 0.579228 0.5627 STD_RESID^2(-30) 0.011546 0.044363 0.260256 0.7948 STD_RESID^2(-31) -0.012071 0.044356 -0.272145 0.7856 STD_RESID^2(-32) -0.023938 0.044324 -0.540063 0.5894 STD_RESID^2(-33) -0.021284 0.044324 -0.480204 0.6313 STD_RESID^2(-34) -0.008021 0.044322 -0.180974 0.8565 STD_RESID^2(-35) -0.022807 0.044319 -0.514610 0.6071 STD_RESID^2(-36) -0.016467 0.044312 -0.371619 0.7103 R-squared 0.042207 Mean dependent var 1.003799 Adjusted R-squared -0.026479 S.D. dependent var 3.100266 S.E. of regression 3.141043 Akaike info criterion 5.193163 Sum squared resid 4952.809 Schwarz criterion 5.487633 Log likelihood -1362.557 F-statistic 0.614495 Durbin-Watson stat 1.997652 Prob(F-statistic) 0.963137 University of Ghana http://ugspace.ug.edu.gh 160 Table 17C: Ljung Q-Statistics for the EGARCH (2,1) Date: 07/27/13 Time: 16:02 Sample: 4 564 Included observations: 561 Q-statistic probabilities adjusted for 3 ARMA term(s) Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|* | .|* | 1 0.077 0.077 3.3316 .|* | .|. | 2 0.070 0.065 6.1268 .|* | .|. | 3 0.070 0.060 8.8797 .|. | .|. | 4 -0.036 -0.050 9.6068 0.002 .|* | .|* | 5 0.074 0.073 12.757 0.002 .|. | .|. | 6 0.001 -0.009 12.757 0.005 .|. | .|. | 7 0.001 -0.002 12.758 0.013 .|* | .|. | 8 0.072 0.062 15.715 0.008 .|. | .|. | 9 0.060 0.059 17.798 0.007 .|. | .|. | 10 0.019 -0.004 17.998 0.012 .|. | .|. | 11 -0.030 -0.048 18.524 0.018 ***|. | ***|. | 12 -0.345 -0.352 86.874 0.000 *|. | .|. | 13 -0.066 -0.027 89.408 0.000 *|. | .|. | 14 -0.067 -0.023 91.962 0.000 *|. | .|. | 15 -0.090 -0.038 96.623 0.000 .|. | .|. | 16 -0.009 -0.017 96.666 0.000 *|. | .|. | 17 -0.083 -0.036 100.65 0.000 .|. | .|. | 18 -0.016 -0.010 100.79 0.000 .|. | .|. | 19 -0.044 -0.035 101.94 0.000 .|. | .|* | 20 -0.003 0.072 101.94 0.000 *|. | .|. | 21 -0.067 -0.014 104.61 0.000 .|. | .|. | 22 -0.027 0.016 105.03 0.000 *|. | *|. | 23 -0.083 -0.102 109.04 0.000 *|. | **|. | 24 -0.085 -0.206 113.29 0.000 .|. | .|. | 25 0.050 0.049 114.75 0.000 .|. | .|. | 26 0.036 0.053 115.53 0.000 .|. | .|. | 27 0.029 -0.003 116.03 0.000 .|. | .|. | 28 0.028 0.007 116.49 0.000 .|. | .|. | 29 0.042 0.006 117.54 0.000 .|. | .|. | 30 0.057 0.039 119.48 0.000 .|. | .|. | 31 0.055 0.037 121.28 0.000 .|. | .|. | 32 -0.018 0.017 121.47 0.000 .|. | .|. | 33 0.016 -0.005 121.63 0.000 .|. | .|. | 34 0.019 -0.008 121.84 0.000 .|. | .|. | 35 0.064 -0.026 124.33 0.000 .|* | .|. | 36 0.107 -0.023 131.27 0.000 University of Ghana http://ugspace.ug.edu.gh 161 Table 18C: Model Output for ARIMA (2, 1, 1) Dependent Variable: D(INFLATIO) Method: Least Squares Date: 07/27/13 Time: 12:26 Sample (adjusted): 4 564 Included observations: 561 after adjustments Convergence achieved after 15 iterations Backcast: 3 Variable Coefficient Std. Error t-Statistic Prob. C -0.021918 0.544724 -0.040236 0.9679 AR(1) 0.492862 0.309025 1.594893 0.1113 AR(2) 0.096179 0.154143 0.623961 0.5329 MA(1) -0.082511 0.310235 -0.265961 0.7904 R-squared 0.240747 Mean dependent var -0.017683 Adjusted R-squared 0.236658 S.D. dependent var 6.613337 S.E. of regression 5.778036 Akaike info criterion 6.353109 Sum squared resid 18595.83 Schwarz criterion 6.383981 Log likelihood -1778.047 F-statistic 58.87209 Durbin-Watson stat 1.999697 Prob(F-statistic) 0.000000 Inverted AR Roots .64 -.15 Inverted MA Roots .08 Figure 1C: Residual Plot of the ARIMA (2,1.1) Model -40 - 0 0 20 40 -40 -20 0 20 40 100 200 300 400 500 Residual Actual Fitted University of Ghana http://ugspace.ug.edu.gh 162 Figure 2C: Histogram of the Standardised Residuals of the ARIMA (2,1,1) Figure 3C: Static Forecast Graph and Performance of ARIMA (2, 1, 1). 0 40 80 120 160 200 -4 -2 0 2 4 6 Series: Standardized Residuals Sample 2 576 Observations 575 Mean 0.002032 Median -0.039047 Maximum 7.175665 Minimum -3.559597 Std. Dev. 1.002809 Skewness 0.962002 Kurtosis 10.11439 Jarque-Bera 1301.328 Probability 0.000000 -40 0 40 80 12 160 200 240 100 200 300 400 500 INFLATIOF Forecast: INFLATIOF Actual: INFLATIO Forecast sample: 1 582 Adjusted sample: 4 582 Included observations: 579 Root Mean Squared Error 5.667538 Mean Absolute Error 2.852212 Mean Abs. Percent Error 16.12051 Theil Inequality Coefficient 0.066034 Bias Proportion 0.000000 Variance Proportion 0.012588 Covariance Proportion 0.987412 University of Ghana http://ugspace.ug.edu.gh 163 Figure 4C: Static Forecast Graph and Performance for ARCH (2) Figure 5C: Static Forecast Graph and Performance for GARCH (2, 1) -50 0 50 100 150 200 250 100 200 300 400 500 INFLATIOF Forecast: INFLATIOF Actual: INFLATIO Forecast sample: 1 582 Adjusted sample: 4 582 Included observations: 579 Root Mean Squared Error 5.748092 Mean Absolute Error 2.892042 Mean Abs. Percent Error 15.88456 Theil Inequality Coefficient 0.067110 Bias Proportion 0.000195 Variance Proportion 0.006589 Covariance Proportion 0.993216 0 500 1000 1500 2000 2500 100 200 300 400 500 Forecast of Variance -50 0 50 100 150 200 250 100 200 300 400 500 INFLATIOF Forecast: INFLATIOF Actual: INFLATIO Forecast sample: 1 582 Adjusted sample: 4 582 Included observations: 579 Root Mean Squared Error 5.701873 Mean Absolute Error 2.866906 Mean Abs. Percent Error 15.90991 Theil Inequality Coefficient 0.066466 Bias Proportion 0.000010 Variance Proportion 0.010760 Covariance Proportion 0.989230 0 400 800 1200 1600 2000 100 200 300 400 500 Forecast of Variance University of Ghana http://ugspace.ug.edu.gh 164 Figure 6C: Static Forecast Graph and Performance for EGARCH (2, 1). -50 0 50 100 150 200 250 100 200 300 400 500 INFLATIOF Forecast: INFLATIOF Actual: INFLATIO Forecast sample: 1 582 Adjusted sample: 4 582 Included observations: 579 Root Mean Squared Error 5.786916 Mean Absolute Error 2.880496 Mean Abs. Percent Error 15.70252 Theil Inequality Coefficient 0.067412 Bias Proportion 0.000137 Variance Proportion 0.010423 Covariance Proportion 0.989440 0 200 400 600 800 1000 1200 1400 100 200 300 400 500 Forecast of Variance University of Ghana http://ugspace.ug.edu.gh